{"id":99,"date":"2021-12-15T09:53:30","date_gmt":"2021-12-15T09:53:30","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/anwendungen-2\/"},"modified":"2021-12-15T09:53:30","modified_gmt":"2021-12-15T09:53:30","slug":"anwendungen-2","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/anwendungen-2\/","title":{"raw":"Anwendungen","rendered":"Anwendungen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style>\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"ze8053bd4b90d\" class=\"sectionHead\"><span class=\"titlemark\">9.7 <\/span> <a id=\"x1-2850007\"><\/a>Anwendungen<\/h3> <a id=\"x1-285001r284\"><\/a> <h4 id=\"zda2e8561faf7\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.1 <\/span> <a id=\"x1-2860001\"><\/a>Fl\u00e4cheninhalte<\/h4> <p class=\"noindent\">Wir wollen hier nochmals Beispiele f\u00fcr Fl\u00e4chenberechnungen besprechen, welche unter anderem den Namen der Umkehrfunktionen der hyperbolischen Funktionen erkl\u00e4ren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z695f172f7ebf\"> <a id=\"x1-286001r64\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.64.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen den Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des Kreises mit Radius<\/span> <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dieser ist<\/span> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mn>2<\/mn><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">definiert (wieso?), und gemeinsam mit Bespiel<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-267001r21\"><span class=\"ecti-1095\">9.21<\/span><\/a> <span class=\"ecti-1095\">ergibt sich daraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>\u03c0<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb3c37542cdc0\"> <a id=\"x1-286002r65\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.65 <\/span>(Hyperbolische Umkehrfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir verwenden die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-286003r22\" \/><mstyle class=\"maketag\"><mtext>(9.22)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">um die<\/span> <span class=\"ecti-1095\">\u201e<\/span> <span class=\"ecti-1095\">positive H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lfte<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">der Hyperbel <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">zu parametrisieren. Wir stellen uns den Parameter<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">vorerst als Zeit vor. In diesem Sinne beschreibt<\/span> (<a href=\"..\/..\/chapter\/anwendungen#x1-286003r22\">9.22<\/a>) <span class=\"ecti-1095\">eine Bewegung im<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">wollen den Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des folgenden Gebietes in Rosa zwischen dem Ursprung und einem Teil<\/span> <span class=\"ecti-1095\">der Hyperbel berechnen.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=447&amp;url=Pictures\/fundsatz\/hyperbel.pdf\"><img id=\"z2431bd19a41f\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/hyperbel.svg\" width=\"447\"><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Dieser ist der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">des eingezeichneten Dreiecks minus dem Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt unterhalb der Hyperbel zwischen der<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in Blau. Letztere<\/span> <span class=\"ecti-1095\">Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che ist durch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gegeben. Um dieses Integral zu berechnen, verwenden wir die hyperbolische Substitution<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Somit ist der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des gesuchten Gebiets<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">arcosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rt die Namen<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Areasinus Hyperbolicus<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Areakosinus Hyperbolicus<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">der<\/span> <span class=\"ecti-1095\">Umkehrfunktionen der hyperbolischen Funktionen (wieso?).<\/span> <\/p> <\/div> <a id=\"x1-286004r286\"><\/a> <h4 id=\"z0e6bda5cfa78\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.2 <\/span> <a id=\"x1-2870002\"><\/a>Bogenl\u00e4nge<\/h4> <p class=\"noindent\">Im Folgenden m\u00f6chten wir einen stetige Funktion <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> auch <span class=\"ecbx-1095\">Weg <\/span>oder <span class=\"ecbx-1095\">Kurve <\/span>von <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nach <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> genannt, betrachten. Dabei fassen wir <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> als Zeitparameter und <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als die Position zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> auf. <\/p><p class=\"indent\">Falls alle Komponenten <math display=\"inline\"><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math> stetig differenzierbar sind, interpretieren wir f\u00fcr einen Zeitpunkt                                                                                                                                                                           <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> den Ausdruck <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt> <mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> als die Geschwindigkeit zum Zeitpunkt <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir m\u00f6chten nun die Bogenl\u00e4nge des Weges <math display=\"inline\"><mi>\u03b3<\/mi><\/math> als die gesamte Strecke, die zwischen den Zeiten <math display=\"inline\"><mi>a<\/mi><\/math> und <math display=\"inline\"><mi>b<\/mi><\/math> zur\u00fcckgelegt wurde, definieren. Dabei soll gelten, dass die zur\u00fcckgelegte Strecke zwischen gleichen Zeiten <math display=\"inline\"><mi>\u03b1<\/mi><\/math> und <math display=\"inline\"><mi>\u03b1<\/mi><\/math> Null ist und dass sich Strecken additiv verhalten, also dass die zwischen den Zeiten <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> zur\u00fcckgelegte Strecke plus die zwischen den Zeiten <math display=\"inline\"><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b3<\/mi><\/math> zur\u00fcckgelegte Strecke gerade die zwischen den Zeiten <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b3<\/mi><\/math> zur\u00fcckgelegte Strecke ist. Im Sinne von Definition&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-116001r29\">4.29<\/a> ist die zur\u00fcckgelegte Strecke also eine additive Intervallfunktion auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Des Weiteren m\u00f6chten wir nat\u00fcrlich verlangen, dass die in einem Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> zur\u00fcckgelegte Strecke zwischen <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mal die minimale Geschwindigkeit in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mal die maximale Geschwindigkeit in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> liegt. Nach Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> ist daher die einzig vern\u00fcnftige Definition der <span class=\"ecbx-1095\">Bogenl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nge <\/span>des Weges <math display=\"inline\"><mi>\u03b3<\/mi><\/math> der Ausdruck <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Anders formuliert ist also die L\u00e4nge des zur\u00fcckgelegten Weges das Integral \u00fcber die Geschwindigkeitsfunktion.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z0e4dd3a34012\"> <a id=\"x1-287001r66\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.66 <\/span>(Umfang des Kreises)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten den Weg<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wegen <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">sind der Start- und<\/span> <span class=\"ecti-1095\">der Endpunkt von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">gleich (wir<\/span> <span class=\"ecti-1095\">sagen auch, dass der Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecbi-1095\">geschlossen <\/span><span class=\"ecti-1095\">ist). Auch gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Geschwindigkeit zu jedem Zeitpunkt<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Der Weg (oder die Kurve) <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">durchl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft (wegen <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">den Einheitskreis also mit konstanter Geschwindigkeit Eins. Deswegen gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren besucht <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">jeden Punkt (bis auf den Endpunkt) genau einmal (siehe auch Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2120004\"><span class=\"ecti-1095\">7.6.4<\/span><\/a><span class=\"ecti-1095\">). Einen<\/span> <span class=\"ecti-1095\">solchen Weg nennen wir auch <\/span><span class=\"ecbi-1095\">einfach<\/span><span class=\"ecti-1095\">. Deswegen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">auch als den Umfang des Einheitskreises auffassen, der somit<\/span> <math display=\"inline\"><mn>2<\/mn><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <span class=\"ecti-1095\">Dies gilt analog f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Teilstrecken und definiert den Begriff <\/span><span class=\"ecbi-1095\">Winkel <\/span><span class=\"ecti-1095\">als Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge am<\/span> <span class=\"ecti-1095\">Einheitskreis.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> ein stetig differenzierbarer Weg ausgehend von einem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Eine stetig differenzierbare <span class=\"ecbx-1095\">Reparametrisierung <\/span>von <math display=\"inline\"><mi>\u03b3<\/mi><\/math> ist ein Weg der Form <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi> <mo class=\"MathClass-punc\">,<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>\u00e3<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> ist und <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> eine stetig differenzierbare, monoton wachsende, bijektive Funktion ist. Wenn wir uns&nbsp;<math display=\"inline\"><mi>\u03b3<\/mi><\/math> als einen \u201e Fahrplan eines Autobusses\u201c  vorstellen, dann entspricht&nbsp;<math display=\"inline\"><mi>\u03c8<\/mi><\/math> einer \u201e Fahrplan\u00e4nderung\u201c. <\/p><p class=\"indent\">Intuitiv ausgedr\u00fcckt ist eine Reparametrisierung eines Weges also ein Weg mit denselben Endpunkten (da <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u00e3<\/mi> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> und <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math>) und der immer in dieselbe Richtung geht (wegen Monotonie). Anschaulich kann man deswegen erwarten, dass jede Reparametrisierung eines Weges dieselbe Bogenl\u00e4nge hat. Auch wollen wir zeigen, dass ein nie anhaltender Weg so reparametrisiert werden kann, dass der neue Weg Einheitsgeschwindigkeit hat. Falls <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> nie anh\u00e4lt oder                                                                                                                                                                           genauer falls <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> so nennen wir <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecbx-1095\">regul<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">r<\/span>. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z71fb7f17cb85\"> <a id=\"x1-287002r67\"><\/a> <span class=\"ecbx-1095\">Lemma 9.67 <\/span>(Reparametrisierungen eines Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ein stetig differenzierbarer Weg f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann hat jede Reparametrisierung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">dieselbe Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge. Falls <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">r ist, gibt es eine Reparametrisierung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit Einheitsgeschwindigkeit, welche auch die <\/span><span class=\"ecbi-1095\">Parametrisierung nach Bogenl<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">nge <\/span><span class=\"ecti-1095\">genannt<\/span> <span class=\"ecti-1095\">wird.<\/span> <\/p> <\/div> <p class=\"indent\">In Beispiel <a href=\"..\/..\/chapter\/anwendungen#x1-287001r66\">9.66<\/a> ist der betrachtete Weg bereits nach Bogenl\u00e4nge parametrisiert. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>\u00e3<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> und <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi> <mo class=\"MathClass-punc\">,<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> eine stetig differenzierbare, monoton wachsende, bijektive Funktion. Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr die zweite Aussage konstruieren wir nun eine geeignete Funktion <math display=\"inline\"><mi>\u03c8<\/mi><\/math> wie oben. Sei <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wegen <math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03d5<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> sowie <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist <math display=\"inline\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math> eine streng monoton wachsende, stetig differenzierbare Bijektion. Insbesondere ist <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u03d5<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> ebenfalls streng monoton wachsend und stetig differenzierbar. Zur Zeit <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math> berechnen wir nun die Geschwindigkeit von&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so gilt wegen&nbsp;<math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mover accent=\"true\"><mrow><mi>\u03d5<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit hat die Reparametrisierung <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math> die gew\u00fcnschte Eigenschaft. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf2a54e3eb7c4\"> <a id=\"x1-287003r68\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.68 <\/span>(Eindeutigkeit der Parametrisierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In Lemma <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-287002r67\"><span class=\"ecti-1095\">9.67<\/span><\/a> <span class=\"ecti-1095\">wird bereits von <\/span>der <span class=\"ecti-1095\">Parametrisierung nach Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge gesprochen. Wir<\/span> <span class=\"ecti-1095\">wollen                    dies                    hier                    begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden.                    Sei<\/span> <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ein stetig differenzierbarer, regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rer Weg. Nach Lemma <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-287002r67\"><span class=\"ecti-1095\">9.67<\/span><\/a> <span class=\"ecti-1095\">d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen wir annehmen, dass<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">Einheitsgeschwindigkeit  hat.  Zeigen  Sie,  dass  es  keine  weitere  Reparametrisierung  von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit Einheitsgeschwindigkeit gibt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf75c36fd54c3\"> <a id=\"x1-287004r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.69 <\/span>(Totale Variation des Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir noch eine weitere Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ndung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Definition der Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge eines<\/span> <span class=\"ecti-1095\">Weges<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">geben. Hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">interpretieren wir<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">als den<\/span> <span class=\"ecti-1095\">Abstand zweier Punkte<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die <\/span><span class=\"ecbi-1095\">totale Variation <\/span><span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist definiert als<\/span> <\/p><table id=\"z489e2ae582ae\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>V<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u2128<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">wobei das Supremum <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber alle Zerlegungen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">genommen wird. Nehmen Sie nun an, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist und zeigen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie den Mittelwertsatz f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Komponente von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in jedem Intervall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gemeinsam mit gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssiger Stetigkeit der Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msubsup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr einen Weg <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> und eine stetige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> kann ein Integral der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">auch physikalische Bedeutung haben. Zum Beispiel kann der Weg einen verbogenen Draht (mit konstanter Dichte <math display=\"inline\"><mn>1<\/mn><mi>k<\/mi><mi>g<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>m<\/mi><\/math>) beschreiben. In diesem Fall gibt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die <math display=\"inline\"><mi>j<\/mi><\/math>-te Koordinate des Schwerpunktes des Drahtes an, wobei <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <a id=\"x1-287005r287\"><\/a> <\/p> <h4 id=\"z8158d033683a\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.3 <\/span> <a id=\"x1-2880003\"><\/a>Wegintegrale von Vektorfeldern<\/h4> <p class=\"noindent\">Wir kommen nun zu einem weiteren Typ von Wegintegralen, der sowohl f\u00fcr die Physik als auch f\u00fcr die weitere Analysis wichtig sein wird. Hierf\u00fcr betrachten wir nochmals reelle Zahlen <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und einen stetig differenzierbaren Weg <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Wir interpretierten <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ja bereits als Geschwindigkeit (in <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>) des Weges zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (in <math display=\"inline\"><mi>s<\/mi><\/math>) und wollen analog dazu die Ableitung <math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als den Geschwindigkeitsvektor zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> interpretieren (mit jeder Koordinate in <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>), der eben nicht nur die augenblickliche Geschwindigkeit als eindimensionale Gr\u00f6sse angibt, sondern auch die Richtung der Bewegung beschreibt. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> eine stetige Funktion (siehe Abschnitt <a href=\"..\/..\/chapter\/stetigkeit#x1-1500001\">5.4.1<\/a>), welche wir als ein <span class=\"ecbx-1095\">Kraftfeld <\/span>interpretieren und bei jedem Punkt <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> die Richtung und St\u00e4rke einer Krafteinwirkung zum Beispiel auf Grund von Wind angibt (mit jeder Koordinate in <math display=\"inline\"><mi>N<\/mi><\/math>). Wir nennen in diesem Zusammenhang <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> auch ein <span class=\"ecbx-1095\">Vektorfeld <\/span>und visualisieren f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> (und etwas schwieriger auch f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>3<\/mn><\/math>) dieses durch eine Ansammlung von Vektoren bei mehreren Punkten im Definitionsbereich, siehe folgendes Bild. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=456&amp;url=Pictures\/fundsatz\/vectorfield.pdf\"><img id=\"z12ebe3718090\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/vectorfield.svg\" width=\"456\"><\/div>  <\/div> <p class=\"indent\">Das innere Produkt <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/math> gibt damit die Leistung (in <math display=\"inline\"><mi>W<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>) an, die bei Bewegung mit vorgeschriebener Geschwindigkeit von der Krafteinwirkung zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> geleistet wird. Hierbei kann es vorkommen, dass Krafteinwirkung und Geschwindigkeit \u00e4hnliche Richtungen haben und das innere Produkt positiv ist. Ebenso kann es aber vorkommen, dass Krafteinwirkung und Geschwindigkeit entgegengesetzt sind und das innere Produkt negativ ist. In diesem Sinne (siehe auch Abschnitt <a href=\"..\/..\/chapter\/anwendungen#x1-1190004\">4.4.4<\/a>) berechnet das sogenannte <span class=\"ecbx-1095\">Wegintegral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u22c5<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mstyle><mi>s<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die Arbeit, die von der Krafteinwirkung insgesamt geleistet wurde. <\/p><p class=\"indent\">Wir werden im zweiten Semester derartige Integrale nochmals genauer untersuchen und dann zum Beispiel folgende Frage beantworten k\u00f6nnen: Wie kann man einem Kraftfeld <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> ansehen, ob das Wegintegral nur von Anfangspunkt <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und Endpunkt <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> abh\u00e4ngt und nicht von der Wahl des konkreten Weges von <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nach <span class=\"maendquote\"><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z60a1d04faa9a\"> <a id=\"x1-288001r70\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.70 <\/span>(Abh\u00e4ngigkeit von der Wahl des Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definiert<\/span> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>y<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">betrachten den Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definiert durch <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann ist das<\/span> <span class=\"ecti-1095\">Wegintegral von <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr><\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr><\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>2<\/mn><mi>t<\/mi><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>5<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben. Verwenden wir allerdings den Weg <\/span><math display=\"inline\"><mi>\u03b7<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definert durch <\/span><math display=\"inline\"><mi>\u03b7<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so sind zwar Anfangs- und Endpunkte unver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert, doch ist das Wegintegral durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>2<\/mn><mi>t<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><mn>3<\/mn><\/mrow> <mrow><mn>1<\/mn><mn>5<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z416ead4b3721\"> <a id=\"x1-288002r71\"><\/a> <span class=\"ecbx-1095\">Applet 9.71 <\/span>(Wegintegral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"574px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/XYHQhJS4\/width\/688\/height\/574\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen sowohl das Vektorfeld <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">einen verschiebbaren Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit animiertem Punkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und darunter den Graph der Funktion <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">dar.<\/span> <\/p> <\/div> <a id=\"x1-288003r288\"><\/a> <h4 id=\"zc6b0008e67a3\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.4 <\/span> <a id=\"x1-2890004\"><\/a>Volumen von Rotationsk\u00f6rpern*<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> stetig. Wir betrachten das Gebiet <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>G<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und den zugeh\u00f6rigen K\u00f6rper                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der sich aus Rotation von <math display=\"inline\"><mi>G<\/mi><\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse ergibt. Sind die beiden Zylinder <math display=\"inline\"><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> mit Radius <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> respektive <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse gegeben, so will man wegen den Enthaltungen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>K<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> dass das Volumen von <math display=\"inline\"><mi>K<\/mi><\/math> zwischen <math display=\"inline\"><mi>\u03c0<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>\u03c0<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> liegt. Wir halten dies in folgendem Bild fest, wo gemeinsam mit dem Rotationsk\u00f6rper eine von vielen \u201e Scheiben\u201c, die zusammen den K\u00f6rper approximieren, dargestellt werden. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=506&amp;url=Pictures\/fundsatz\/zylindersandwich2.pdf\"><img id=\"z3935a308c6e2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/zylindersandwich2.svg\" width=\"506\"><\/div>  <\/div> <p class=\"indent\">Deswegen (siehe auch \u00dcbung <a href=\"..\/..\/chapter\/anwendungen#x1-289003r73\">9.73<\/a>) definieren wir das <span class=\"ecbx-1095\">Volumen des Rotationsk<\/span><span class=\"ecbx-1095\">\u00f6<\/span><span class=\"ecbx-1095\">rpers<\/span> <math display=\"inline\"><mi>K<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-289001r23\" \/><mstyle class=\"maketag\"><mtext>(9.23)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zffb508b5841a\"> <a id=\"x1-289002r72\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.72 <\/span>(Volumen der Kugel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Kugel<\/span> <span class=\"ecti-1095\">mit Radius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Kugel <\/span><math display=\"inline\"><mi>K<\/mi><\/math> <span class=\"ecti-1095\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich auch als Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mittels der Funktion<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">auffassen. Ihr Volumen ist deswegen durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>4<\/mn><mi>\u03c0<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9d6778ac2238\"> <a id=\"x1-289003r73\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.73.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Motivieren Sie die Definition des Volumen eines Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers mit mehr Details in<\/span> <span class=\"ecti-1095\">Analogie zu Abschnitt <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-2870002\"><span class=\"ecti-1095\">9.7.2<\/span><\/a> <span class=\"ecti-1095\">unter Verwendung von Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <a id=\"x1-289004r289\"><\/a> <h4 id=\"z51615c26ae69\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.5 <\/span> <a id=\"x1-2900005\"><\/a>Oberfl\u00e4chen von Rotationsk\u00f6rpern*<\/h4> <p class=\"noindent\">Obwohl Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> oft ein guter Wegweiser f\u00fcr das Auffinden einer geeigneten Definition darstellt, m\u00fcssen oder k\u00f6nnen wir diese nicht immer als Grundlage w\u00e4hlen. Manchmal begn\u00fcgen wir uns mit geometrischer Intuition als Motivation der Definition.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Man kann die Sinnhaftigkeit einer Definition zwar hinterfragen, doch kann man eine Definition ohnehin nicht beweisen.<\/span><\/span> <\/p><p class=\"indent\">Wir betrachten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> eine stetig differenzierbare Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> und den Rotationsk\u00f6rper <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie im letzten Abschnitt. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=470&amp;url=Pictures\/fundsatz\/oberflaechenformel.pdf\"><img id=\"za59f12fe1b66\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/oberflaechenformel.svg\" width=\"470\"><\/div>  <\/div> <p class=\"indent\">In einem kleinen Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> der L\u00e4nge <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> ist die Funktion <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> der Tangente <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr ein <math display=\"inline\"><mi>\u03be<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> sehr nahe. Ausserdem wird die Oberfl\u00e4che des Anteils von <span class=\"maperiod\"><math display=\"inline\"><mi>K<\/mi><\/math><\/span><span class=\"period\">,<\/span> der dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math> entspricht, sehr gut durch die Aussenoberfl\u00e4che des Kegelstumpfs beschrieben, der entsteht, wenn man obiges Tangentenst\u00fcck zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse                                                                                                                                                                           rotiert. Die Aussenoberfl\u00e4che eines Kegelstumpfs ist n\u00e4herungsweise <span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi>\u2113<\/mi><\/math> die L\u00e4nge der Aussenkante des Kegelstumpfs und <math display=\"inline\"><mi>U<\/mi><\/math> der Umfang einer der beiden Kreise darstellt (wieso?). <\/p><p class=\"indent\">Die Oberfl\u00e4che sollte also n\u00e4herungsweise durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben sein, wobei <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> f\u00fcr jedes&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> einen Zwischenpunkt darstellt. Deswegen definieren wir nun die <span class=\"ecbx-1095\">Oberfl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">che des Rotationsk<\/span><span class=\"ecbx-1095\">\u00f6<\/span><span class=\"ecbx-1095\">rpers<\/span> <math display=\"inline\"><mi>K<\/mi><\/math> als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z33dc868e0615\"> <a id=\"x1-290001r74\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.74 <\/span>(Kugeloberfl\u00e4che)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie in Beispiel <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-289002r72\"><span class=\"ecti-1095\">9.72<\/span><\/a> <span class=\"ecti-1095\">betrachten wir zu <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math><span class=\"ecti-1095\">, deren Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper<\/span> <span class=\"ecti-1095\">gerade die Kugel von Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Damit ist die Kugeloberfl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che gleich<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/msqrt><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><mi>\u03c0<\/mi><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zf9258448ac31\"> <a id=\"x1-290002r75\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.75 <\/span>(Eine lange Nadel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie das Volumen und die Oberfl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che des<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">uneigentlichen Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">der entsteht, wenn man das Gebiet unter dem Graphen der Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">um die <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Achse<\/span> <span class=\"ecti-1095\">rotiert.<\/span> <\/p> <\/div> <a id=\"x1-290003r285\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: 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{vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"ze8053bd4b90d\" class=\"sectionHead\"><span class=\"titlemark\">9.7 <\/span> <a id=\"x1-2850007\"><\/a>Anwendungen<\/h3> <a id=\"x1-285001r284\"><\/a> <h4 id=\"zda2e8561faf7\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.1 <\/span> <a id=\"x1-2860001\"><\/a>Fl\u00e4cheninhalte<\/h4> <p class=\"noindent\">Wir wollen hier nochmals Beispiele f\u00fcr Fl\u00e4chenberechnungen besprechen, welche unter anderem den Namen der Umkehrfunktionen der hyperbolischen Funktionen erkl\u00e4ren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z695f172f7ebf\"> <a id=\"x1-286001r64\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.64.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir berechnen den Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des Kreises mit Radius<\/span> <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><span class=\"ecti-1095\">. Dieser ist<\/span> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mn>2<\/mn><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">definiert (wieso?), und gemeinsam mit Bespiel<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrationsmethoden#x1-267001r21\"><span class=\"ecti-1095\">9.21<\/span><\/a> <span class=\"ecti-1095\">ergibt sich daraus<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> arcsin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>r<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>x<\/mi><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>\u03c0<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb3c37542cdc0\"> <a id=\"x1-286002r65\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.65 <\/span>(Hyperbolische Umkehrfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir verwenden die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-286003r22\" \/><mstyle class=\"maketag\"><mtext>(9.22)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">um die<\/span> <span class=\"ecti-1095\">\u201e<\/span> <span class=\"ecti-1095\">positive H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lfte<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">der Hyperbel <\/span><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">zu parametrisieren. Wir stellen uns den Parameter<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">vorerst als Zeit vor. In diesem Sinne beschreibt<\/span> (<a href=\"..\/..\/chapter\/anwendungen#x1-286003r22\">9.22<\/a>) <span class=\"ecti-1095\">eine Bewegung im<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">wollen den Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des folgenden Gebietes in Rosa zwischen dem Ursprung und einem Teil<\/span> <span class=\"ecti-1095\">der Hyperbel berechnen.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=447&amp;url=Pictures\/fundsatz\/hyperbel.pdf\"><img decoding=\"async\" id=\"z2431bd19a41f\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/hyperbel.svg\" width=\"447\" \/><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">Dieser ist der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt <\/span><math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">des eingezeichneten Dreiecks minus dem Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt unterhalb der Hyperbel zwischen der<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">in Blau. Letztere<\/span> <span class=\"ecti-1095\">Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che ist durch <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">gegeben. Um dieses Integral zu berechnen, verwenden wir die hyperbolische Substitution<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><msup><mrow><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mfrac><mrow><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac> <mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><msup><mrow><mi class=\"qopname\">e<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\"> e<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><mi>t<\/mi><\/mrow><\/msup><\/mrow> <mrow><mn>8<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Somit ist der Fl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">cheninhalt des gesuchten Gebiets<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><msub><mrow><mi>t<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">arcosh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">arsinh<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Dies erkl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rt die Namen<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Areasinus Hyperbolicus<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">und<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">Areakosinus Hyperbolicus<\/span><span class=\"ecti-1095\">\u201c<\/span> <span class=\"ecti-1095\">der<\/span> <span class=\"ecti-1095\">Umkehrfunktionen der hyperbolischen Funktionen (wieso?).<\/span> <\/p> <\/div> <a id=\"x1-286004r286\"><\/a> <h4 id=\"z0e6bda5cfa78\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.2 <\/span> <a id=\"x1-2870002\"><\/a>Bogenl\u00e4nge<\/h4> <p class=\"noindent\">Im Folgenden m\u00f6chten wir einen stetige Funktion <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> auch <span class=\"ecbx-1095\">Weg <\/span>oder <span class=\"ecbx-1095\">Kurve <\/span>von <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nach <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> genannt, betrachten. Dabei fassen wir <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> als Zeitparameter und <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als die Position zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> auf. <\/p><p class=\"indent\">Falls alle Komponenten <math display=\"inline\"><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math> stetig differenzierbar sind, interpretieren wir f\u00fcr einen Zeitpunkt                                                                                                                                                                           <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> den Ausdruck <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt> <mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> als die Geschwindigkeit zum Zeitpunkt <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir m\u00f6chten nun die Bogenl\u00e4nge des Weges <math display=\"inline\"><mi>\u03b3<\/mi><\/math> als die gesamte Strecke, die zwischen den Zeiten <math display=\"inline\"><mi>a<\/mi><\/math> und <math display=\"inline\"><mi>b<\/mi><\/math> zur\u00fcckgelegt wurde, definieren. Dabei soll gelten, dass die zur\u00fcckgelegte Strecke zwischen gleichen Zeiten <math display=\"inline\"><mi>\u03b1<\/mi><\/math> und <math display=\"inline\"><mi>\u03b1<\/mi><\/math> Null ist und dass sich Strecken additiv verhalten, also dass die zwischen den Zeiten <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> zur\u00fcckgelegte Strecke plus die zwischen den Zeiten <math display=\"inline\"><mi>\u03b2<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b3<\/mi><\/math> zur\u00fcckgelegte Strecke gerade die zwischen den Zeiten <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b3<\/mi><\/math> zur\u00fcckgelegte Strecke ist. Im Sinne von Definition&nbsp;<a href=\"..\/..\/chapter\/anwendungen#x1-116001r29\">4.29<\/a> ist die zur\u00fcckgelegte Strecke also eine additive Intervallfunktion auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Des Weiteren m\u00f6chten wir nat\u00fcrlich verlangen, dass die in einem Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b2<\/mi><\/math> zur\u00fcckgelegte Strecke zwischen <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mal die minimale Geschwindigkeit in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b2<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b1<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> mal die maximale Geschwindigkeit in <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u03b2<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> liegt. Nach Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> ist daher die einzig vern\u00fcnftige Definition der <span class=\"ecbx-1095\">Bogenl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nge <\/span>des Weges <math display=\"inline\"><mi>\u03b3<\/mi><\/math> der Ausdruck <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Anders formuliert ist also die L\u00e4nge des zur\u00fcckgelegten Weges das Integral \u00fcber die Geschwindigkeitsfunktion.                                                                                                                                                                           <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0e4dd3a34012\"> <a id=\"x1-287001r66\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.66 <\/span>(Umfang des Kreises)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten den Weg<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wegen <\/span><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">sind der Start- und<\/span> <span class=\"ecti-1095\">der Endpunkt von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">gleich (wir<\/span> <span class=\"ecti-1095\">sagen auch, dass der Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecbi-1095\">geschlossen <\/span><span class=\"ecti-1095\">ist). Auch gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Geschwindigkeit zu jedem Zeitpunkt<\/span> <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Der Weg (oder die Kurve) <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">durchl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft (wegen <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">den Einheitskreis also mit konstanter Geschwindigkeit Eins. Deswegen gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><\/msubsup><mn>1<\/mn><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Des Weiteren besucht <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">jeden Punkt (bis auf den Endpunkt) genau einmal (siehe auch Abschnitt<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2120004\"><span class=\"ecti-1095\">7.6.4<\/span><\/a><span class=\"ecti-1095\">). Einen<\/span> <span class=\"ecti-1095\">solchen Weg nennen wir auch <\/span><span class=\"ecbi-1095\">einfach<\/span><span class=\"ecti-1095\">. Deswegen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich die Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">auch als den Umfang des Einheitskreises auffassen, der somit<\/span> <math display=\"inline\"><mn>2<\/mn><mi>\u03c0<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <span class=\"ecti-1095\">Dies gilt analog f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r Teilstrecken und definiert den Begriff <\/span><span class=\"ecbi-1095\">Winkel <\/span><span class=\"ecti-1095\">als Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge am<\/span> <span class=\"ecti-1095\">Einheitskreis.<\/span> <\/p> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> ein stetig differenzierbarer Weg ausgehend von einem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit Endpunkten <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Eine stetig differenzierbare <span class=\"ecbx-1095\">Reparametrisierung <\/span>von <math display=\"inline\"><mi>\u03b3<\/mi><\/math> ist ein Weg der Form <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi> <mo class=\"MathClass-punc\">,<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>\u00e3<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> ist und <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> eine stetig differenzierbare, monoton wachsende, bijektive Funktion ist. Wenn wir uns&nbsp;<math display=\"inline\"><mi>\u03b3<\/mi><\/math> als einen \u201e Fahrplan eines Autobusses\u201c  vorstellen, dann entspricht&nbsp;<math display=\"inline\"><mi>\u03c8<\/mi><\/math> einer \u201e Fahrplan\u00e4nderung\u201c. <\/p><p class=\"indent\">Intuitiv ausgedr\u00fcckt ist eine Reparametrisierung eines Weges also ein Weg mit denselben Endpunkten (da <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u00e3<\/mi> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> und <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math>) und der immer in dieselbe Richtung geht (wegen Monotonie). Anschaulich kann man deswegen erwarten, dass jede Reparametrisierung eines Weges dieselbe Bogenl\u00e4nge hat. Auch wollen wir zeigen, dass ein nie anhaltender Weg so reparametrisiert werden kann, dass der neue Weg Einheitsgeschwindigkeit hat. Falls <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> nie anh\u00e4lt oder                                                                                                                                                                           genauer falls <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> so nennen wir <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecbx-1095\">regul<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">r<\/span>. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z71fb7f17cb85\"> <a id=\"x1-287002r67\"><\/a> <span class=\"ecbx-1095\">Lemma 9.67 <\/span>(Reparametrisierungen eines Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ein stetig differenzierbarer Weg f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann hat jede Reparametrisierung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">dieselbe Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge. Falls <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">r ist, gibt es eine Reparametrisierung von <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit Einheitsgeschwindigkeit, welche auch die <\/span><span class=\"ecbi-1095\">Parametrisierung nach Bogenl<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">nge <\/span><span class=\"ecti-1095\">genannt<\/span> <span class=\"ecti-1095\">wird.<\/span> <\/p> <\/div> <p class=\"indent\">In Beispiel <a href=\"..\/..\/chapter\/anwendungen#x1-287001r66\">9.66<\/a> ist der betrachtete Weg bereits nach Bogenl\u00e4nge parametrisiert. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>\u00e3<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/math> und <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>\u00e3<\/mi> <mo class=\"MathClass-punc\">,<\/mo> <mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/math> eine stetig differenzierbare, monoton wachsende, bijektive Funktion. Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>\u00e3<\/mi><\/mrow><mrow><mover accent=\"true\"><mrow><mi>b<\/mi><\/mrow><mo accent=\"true\">~<\/mo><\/mover><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">F\u00fcr die zweite Aussage konstruieren wir nun eine geeignete Funktion <math display=\"inline\"><mi>\u03c8<\/mi><\/math> wie oben. Sei <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>t<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msubsup><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo>\u2026<\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>s<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wegen <math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03d5<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> sowie <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>\u03d5<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist <math display=\"inline\"><mi>\u03d5<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math> eine streng monoton wachsende, stetig differenzierbare Bijektion. Insbesondere ist <math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u03d5<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> ebenfalls streng monoton wachsend und stetig differenzierbar. Zur Zeit <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math> berechnen wir nun die Geschwindigkeit von&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> so gilt wegen&nbsp;<math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">)<\/mo><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mover accent=\"true\"><mrow><mi>\u03d5<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msqrt><mrow><msub><mrow><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-bin\">+<\/mo><msub><mrow> <mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msub> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Somit hat die Reparametrisierung <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>\u03c8<\/mi><\/math> die gew\u00fcnschte Eigenschaft. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf2a54e3eb7c4\"> <a id=\"x1-287003r68\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.68 <\/span>(Eindeutigkeit der Parametrisierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In Lemma <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-287002r67\"><span class=\"ecti-1095\">9.67<\/span><\/a> <span class=\"ecti-1095\">wird bereits von <\/span>der <span class=\"ecti-1095\">Parametrisierung nach Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge gesprochen. Wir<\/span> <span class=\"ecti-1095\">wollen                    dies                    hier                    begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nden.                    Sei<\/span> <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ein stetig differenzierbarer, regul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">rer Weg. Nach Lemma <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-287002r67\"><span class=\"ecti-1095\">9.67<\/span><\/a> <span class=\"ecti-1095\">d<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rfen wir annehmen, dass<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">Einheitsgeschwindigkeit  hat.  Zeigen  Sie,  dass  es  keine  weitere  Reparametrisierung  von<\/span> <math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit Einheitsgeschwindigkeit gibt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf75c36fd54c3\"> <a id=\"x1-287004r69\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.69 <\/span>(Totale Variation des Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung wollen wir noch eine weitere Begr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ndung f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Definition der Bogenl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nge eines<\/span> <span class=\"ecti-1095\">Weges<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">geben. Hierf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">interpretieren wir<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>d<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">als den<\/span> <span class=\"ecti-1095\">Abstand zweier Punkte<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die <\/span><span class=\"ecbi-1095\">totale Variation <\/span><span class=\"ecti-1095\">von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist definiert als<\/span> <\/p><table id=\"z489e2ae582ae\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>V<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><\/mrow><mrow><mi>\u2128<\/mi><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">\u2225<\/mo><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-punc\">,<\/mo> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">wobei das Supremum <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber alle Zerlegungen<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mo class=\"MathClass-rel\">\u22ef<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">genommen wird. Nehmen Sie nun an, dass<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">stetig differenzierbar ist und zeigen Sie<\/span><span class=\"ecti-1095\">&nbsp;<\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie den Mittelwertsatz f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Komponente von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">in jedem Intervall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/math> <span class=\"ecti-1095\">gemeinsam mit gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssiger Stetigkeit der Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msubsup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msubsup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/details>  <\/div> <p class=\"indent\">F\u00fcr einen Weg <math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> und eine stetige Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> kann ein Integral der Form <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">auch physikalische Bedeutung haben. Zum Beispiel kann der Weg einen verbogenen Draht (mit konstanter Dichte <math display=\"inline\"><mn>1<\/mn><mi>k<\/mi><mi>g<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>m<\/mi><\/math>) beschreiben. In diesem Fall gibt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>L<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>\u03b3<\/mi><\/mrow><mrow> <mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die <math display=\"inline\"><mi>j<\/mi><\/math>-te Koordinate des Schwerpunktes des Drahtes an, wobei <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> <a id=\"x1-287005r287\"><\/a> <\/p> <h4 id=\"z8158d033683a\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.3 <\/span> <a id=\"x1-2880003\"><\/a>Wegintegrale von Vektorfeldern<\/h4> <p class=\"noindent\">Wir kommen nun zu einem weiteren Typ von Wegintegralen, der sowohl f\u00fcr die Physik als auch f\u00fcr die weitere Analysis wichtig sein wird. Hierf\u00fcr betrachten wir nochmals reelle Zahlen <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und einen stetig differenzierbaren Weg <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Wir interpretierten <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> ja bereits als Geschwindigkeit (in <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>) des Weges zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> (in <math display=\"inline\"><mi>s<\/mi><\/math>) und wollen analog dazu die Ableitung <math display=\"inline\"><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als den Geschwindigkeitsvektor zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> interpretieren (mit jeder Koordinate in <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>), der eben nicht nur die augenblickliche Geschwindigkeit als eindimensionale Gr\u00f6sse angibt, sondern auch die Richtung der Bewegung beschreibt. <\/p><p class=\"indent\">Sei <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> eine stetige Funktion (siehe Abschnitt <a href=\"..\/..\/chapter\/stetigkeit#x1-1500001\">5.4.1<\/a>), welche wir als ein <span class=\"ecbx-1095\">Kraftfeld <\/span>interpretieren und bei jedem Punkt <math display=\"inline\"><mstyle><mi>v<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> die Richtung und St\u00e4rke einer Krafteinwirkung zum Beispiel auf Grund von Wind angibt (mit jeder Koordinate in <math display=\"inline\"><mi>N<\/mi><\/math>). Wir nennen in diesem Zusammenhang <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> auch ein <span class=\"ecbx-1095\">Vektorfeld <\/span>und visualisieren f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> (und etwas schwieriger auch f\u00fcr <math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>3<\/mn><\/math>) dieses durch eine Ansammlung von Vektoren bei mehreren Punkten im Definitionsbereich, siehe folgendes Bild. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=456&amp;url=Pictures\/fundsatz\/vectorfield.pdf\"><img decoding=\"async\" id=\"z12ebe3718090\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/vectorfield.svg\" width=\"456\" \/><\/div>  <\/div> <p class=\"indent\">Das innere Produkt <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/math> gibt damit die Leistung (in <math display=\"inline\"><mi>W<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>N<\/mi><mi>m<\/mi><mo class=\"MathClass-bin\">\u2215<\/mo><mi>s<\/mi><\/math>) an, die bei Bewegung mit vorgeschriebener Geschwindigkeit von der Krafteinwirkung zum Zeitpunkt <math display=\"inline\"><mi>t<\/mi><\/math> geleistet wird. Hierbei kann es vorkommen, dass Krafteinwirkung und Geschwindigkeit \u00e4hnliche Richtungen haben und das innere Produkt positiv ist. Ebenso kann es aber vorkommen, dass Krafteinwirkung und Geschwindigkeit entgegengesetzt sind und das innere Produkt negativ ist. In diesem Sinne (siehe auch Abschnitt <a href=\"..\/..\/chapter\/anwendungen#x1-1190004\">4.4.4<\/a>) berechnet das sogenannte <span class=\"ecbx-1095\">Wegintegral<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>\u03b3<\/mi><\/mrow><\/msub><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-bin\">\u22c5<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mstyle><mi>s<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mover accent=\"true\"><mrow><mi>\u03b3<\/mi><\/mrow><mo accent=\"true\">\u02d9<\/mo><\/mover><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">die Arbeit, die von der Krafteinwirkung insgesamt geleistet wurde. <\/p><p class=\"indent\">Wir werden im zweiten Semester derartige Integrale nochmals genauer untersuchen und dann zum Beispiel folgende Frage beantworten k\u00f6nnen: Wie kann man einem Kraftfeld <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> ansehen, ob das Wegintegral nur von Anfangspunkt <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und Endpunkt <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> abh\u00e4ngt und nicht von der Wahl des konkreten Weges von <math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nach <span class=\"maendquote\"><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z60a1d04faa9a\"> <a id=\"x1-288001r70\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.70 <\/span>(Abh\u00e4ngigkeit von der Wahl des Weges)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definiert<\/span> <span class=\"ecti-1095\">durch <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>y<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math><span class=\"ecti-1095\">. Wir<\/span> <span class=\"ecti-1095\">betrachten den Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definiert durch <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><span class=\"ecti-1095\">. Dann ist das<\/span> <span class=\"ecti-1095\">Wegintegral von <\/span><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><\/mtr><\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">nach <\/span><math display=\"inline\"><mi>\u03b3<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>1<\/mn><\/mtd><\/mtr><\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>2<\/mn><mi>t<\/mi><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>4<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>5<\/mn><\/mrow> <mrow><mn>6<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben. Verwenden wir allerdings den Weg <\/span><math display=\"inline\"><mi>\u03b7<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">definert durch <\/span><math display=\"inline\"><mi>\u03b7<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mstyle class=\"text\"><mtext \/><mstyle class=\"math\"><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mstyle><mtext>&nbsp;<\/mtext><\/mstyle> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so sind zwar Anfangs- und Endpunkte unver<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert, doch ist das Wegintegral durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mi>t<\/mi> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>2<\/mn><mi>t<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>1<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>2<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>5<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><mn>3<\/mn><\/mrow> <mrow><mn>1<\/mn><mn>5<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z416ead4b3721\"> <a id=\"x1-288002r71\"><\/a> <span class=\"ecbx-1095\">Applet 9.71 <\/span>(Wegintegral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"574px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/XYHQhJS4\/width\/688\/height\/574\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir stellen sowohl das Vektorfeld <\/span><span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">einen verschiebbaren Weg <\/span><math display=\"inline\"><mi>\u03b3<\/mi><\/math> <span class=\"ecti-1095\">mit animiertem Punkt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und darunter den Graph der Funktion <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> \u27e8<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>\u03b3<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>\u03b3<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">\u27e9<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">dar.<\/span> <\/p> <\/div> <a id=\"x1-288003r288\"><\/a> <h4 id=\"zc6b0008e67a3\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.4 <\/span> <a id=\"x1-2890004\"><\/a>Volumen von Rotationsk\u00f6rpern*<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> ein kompaktes Intervall mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> stetig. Wir betrachten das Gebiet <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>G<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und den zugeh\u00f6rigen K\u00f6rper                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">der sich aus Rotation von <math display=\"inline\"><mi>G<\/mi><\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse ergibt. Sind die beiden Zylinder <math display=\"inline\"><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> mit Radius <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> respektive <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse gegeben, so will man wegen den Enthaltungen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>K<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>Z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> dass das Volumen von <math display=\"inline\"><mi>K<\/mi><\/math> zwischen <math display=\"inline\"><mi>\u03c0<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>\u03c0<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> liegt. Wir halten dies in folgendem Bild fest, wo gemeinsam mit dem Rotationsk\u00f6rper eine von vielen \u201e Scheiben\u201c, die zusammen den K\u00f6rper approximieren, dargestellt werden. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=506&amp;url=Pictures\/fundsatz\/zylindersandwich2.pdf\"><img decoding=\"async\" id=\"z3935a308c6e2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/zylindersandwich2.svg\" width=\"506\" \/><\/div>  <\/div> <p class=\"indent\">Deswegen (siehe auch \u00dcbung <a href=\"..\/..\/chapter\/anwendungen#x1-289003r73\">9.73<\/a>) definieren wir das <span class=\"ecbx-1095\">Volumen des Rotationsk<\/span><span class=\"ecbx-1095\">\u00f6<\/span><span class=\"ecbx-1095\">rpers<\/span> <math display=\"inline\"><mi>K<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-289001r23\" \/><mstyle class=\"maketag\"><mtext>(9.23)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zffb508b5841a\"> <a id=\"x1-289002r72\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.72 <\/span>(Volumen der Kugel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Kugel<\/span> <span class=\"ecti-1095\">mit Radius <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>r<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Kugel <\/span><math display=\"inline\"><mi>K<\/mi><\/math> <span class=\"ecti-1095\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich auch als Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper mittels der Funktion<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>r<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math> <span class=\"ecti-1095\">auffassen. Ihr Volumen ist deswegen durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi><msubsup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c0<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mfrac><mrow> <mn>4<\/mn><mi>\u03c0<\/mi><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac> <msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9d6778ac2238\"> <a id=\"x1-289003r73\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.73.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Motivieren Sie die Definition des Volumen eines Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers mit mehr Details in<\/span> <span class=\"ecti-1095\">Analogie zu Abschnitt <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-2870002\"><span class=\"ecti-1095\">9.7.2<\/span><\/a> <span class=\"ecti-1095\">unter Verwendung von Proposition <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\"><span class=\"ecti-1095\">4.30<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <a id=\"x1-289004r289\"><\/a> <h4 id=\"z51615c26ae69\" class=\"subsectionHead\"><span class=\"titlemark\">9.7.5 <\/span> <a id=\"x1-2900005\"><\/a>Oberfl\u00e4chen von Rotationsk\u00f6rpern*<\/h4> <p class=\"noindent\">Obwohl Proposition <a href=\"..\/..\/chapter\/anwendungen#x1-116007r30\">4.30<\/a> oft ein guter Wegweiser f\u00fcr das Auffinden einer geeigneten Definition darstellt, m\u00fcssen oder k\u00f6nnen wir diese nicht immer als Grundlage w\u00e4hlen. Manchmal begn\u00fcgen wir uns mit geometrischer Intuition als Motivation der Definition.<button class=\"hover-trigger\" style=\"vertical-align: super;font: smaller\">\u2020<\/button><span class=\"hover-text\"><span class=\"marginpar\">\u2020 Man kann die Sinnhaftigkeit einer Definition zwar hinterfragen, doch kann man eine Definition ohnehin nicht beweisen.<\/span><\/span> <\/p><p class=\"indent\">Wir betrachten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> eine stetig differenzierbare Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> und den Rotationsk\u00f6rper <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>K<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo><msqrt><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie im letzten Abschnitt. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=470&amp;url=Pictures\/fundsatz\/oberflaechenformel.pdf\"><img decoding=\"async\" id=\"za59f12fe1b66\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/fundsatz\/oberflaechenformel.svg\" width=\"470\" \/><\/div>  <\/div> <p class=\"indent\">In einem kleinen Teilintervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> der L\u00e4nge <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> ist die Funktion <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> der Tangente <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03be<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr ein <math display=\"inline\"><mi>\u03be<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> sehr nahe. Ausserdem wird die Oberfl\u00e4che des Anteils von <span class=\"maperiod\"><math display=\"inline\"><mi>K<\/mi><\/math><\/span><span class=\"period\">,<\/span> der dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">]<\/mo><\/math> entspricht, sehr gut durch die Aussenoberfl\u00e4che des Kegelstumpfs beschrieben, der entsteht, wenn man obiges Tangentenst\u00fcck zwischen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> um die <math display=\"inline\"><mi>x<\/mi><\/math>-Achse                                                                                                                                                                           rotiert. Die Aussenoberfl\u00e4che eines Kegelstumpfs ist n\u00e4herungsweise <span class=\"maperiod\"><math display=\"inline\"><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi>\u2113<\/mi><\/math> die L\u00e4nge der Aussenkante des Kegelstumpfs und <math display=\"inline\"><mi>U<\/mi><\/math> der Umfang einer der beiden Kreise darstellt (wieso?). <\/p><p class=\"indent\">Die Oberfl\u00e4che sollte also n\u00e4herungsweise durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u25b3<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">gegeben sein, wobei <math display=\"inline\"><msub><mrow><mi>\u03be<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> f\u00fcr jedes&nbsp;<math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> einen Zwischenpunkt darstellt. Deswegen definieren wir nun die <span class=\"ecbx-1095\">Oberfl<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">che des Rotationsk<\/span><span class=\"ecbx-1095\">\u00f6<\/span><span class=\"ecbx-1095\">rpers<\/span> <math display=\"inline\"><mi>K<\/mi><\/math> als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z33dc868e0615\"> <a id=\"x1-290001r74\"><\/a> <span class=\"ecbx-1095\">Beispiel 9.74 <\/span>(Kugeloberfl\u00e4che)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wie in Beispiel <\/span><a href=\"..\/..\/chapter\/anwendungen#x1-289002r72\"><span class=\"ecti-1095\">9.72<\/span><\/a> <span class=\"ecti-1095\">betrachten wir zu <\/span><math display=\"inline\"><mi>r<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/math><span class=\"ecti-1095\">, deren Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper<\/span> <span class=\"ecti-1095\">gerade die Kugel von Radius <\/span><math display=\"inline\"><mi>r<\/mi><\/math> <span class=\"ecti-1095\">ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>r<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Damit ist die Kugeloberfl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che gleich<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/mrow> <mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/msqrt><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup><msqrt><mrow><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/msqrt><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mi>\u03c0<\/mi><mi>r<\/mi><msubsup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>r<\/mi><\/mrow><mrow><mi>r<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-rel\">=<\/mo> <mn>4<\/mn><mi>\u03c0<\/mi><msup><mrow><mi>r<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf9258448ac31\"> <a id=\"x1-290002r75\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 9.75 <\/span>(Eine lange Nadel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie das Volumen und die Oberfl<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">che des<\/span> <span class=\"ecti-1095\">\u201e<\/span><span class=\"ecti-1095\">uneigentlichen Rotationsk<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rpers<\/span><span class=\"ecti-1095\">\u201c<\/span><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">der entsteht, wenn man das Gebiet unter dem Graphen der Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">um die <\/span><math display=\"inline\"><mi>x<\/mi><\/math><span class=\"ecti-1095\">-Achse<\/span> <span class=\"ecti-1095\">rotiert.<\/span> <\/p> <\/div> <a id=\"x1-290003r285\"><\/a> 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