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Kurvenintegral
va-03-01-gpt
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randRangeExclude(-4,4,[0]) randRangeExclude(-4,4,[M1,-M1,0]) M1*M1 + N1*N1 randRangeExclude(-4,4,[M1,-M1,0]) randRangeExclude(-4,4,[N1,-N1,M2,,-M2,0]) M2*M2 + N2*N2 M1*M1-N1*N1 2*M1*N1 M2*M2-N2*N2 2*M2*N2
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Gegeben sei die Funktion f: \mathbb R^2 \to \mathbb R mit f(x,y) = A x + B y und die Kurve \gamma. Berechnen Sie das Kurvenintegral \displaystyle \int_\gamma f ds.



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a \displaystyle \int_\gamma f ds= ((A * X1 + B * Y1) * W1+(A * X2 + B * Y2) * W2)/2

Für die Berechnung zerlegen wir das Kurvenintegral \displaystyle \int_\gamma f ds in die Summe zweier Integrale:

\displaystyle \int_\gamma f ds = {\color{red} \int_{\gamma_1} f ds} + {\color{blue}\int_{\gamma_2} f ds} .

line( [0, 0], [X1, Y1], { stroke: RED} ); line( [X2, Y2], [0, 0], { stroke: BLUE} ); label( [ X1/1.5, Y1/1.5 ], "\\mathbf \\gamma_1", "above" ); label( [ X2/1.5, Y2/1.5 ], "\\gamma_2", "above" );

Mögliche Parametrisierungen sind:

{\color{red}\gamma_1}(t): [0,1] \to \mathbb R^2, {\color{orange}\gamma_1}(t) = \begin{pmatrix} X1 \\ Y1 \end{pmatrix} + t \cdot \begin{pmatrix} -X1 \\ -Y1 \end{pmatrix}.

{\color{blue}\gamma_2}(t): [0,1] \to \mathbb R^2, {\color{blue}\gamma_2}(t) = t \cdot \begin{pmatrix} X2 \\ Y2 \end{pmatrix}.

Berechnung des Kurvenintegrals

Es ist \displaystyle \int_{\gamma_i} f ds = \int_0^{1} f(\gamma_i(t)) \cdot |{\gamma_i}'(t)| \ dt.

Das Kurvenintegral entlang \gamma_1 ergibt:

\displaystyle {\color{red} \int_{\gamma_1} f ds} = \int_0^{1} (A \cdot (X1 + -X1 t) + B \cdot \left(Y1 + -Y1 t) \right) \cdot \sqrt{negParens(-X1)^2 + negParens(-Y1)^2} \ dt = \int_0^{1} \left(A*X1+B*Y1 + -A*X1-B*Y1 \cdot t\right) \cdot W1 \ dt = {\color{red} fractionReduce((A * X1 + B * Y1) * W1,2)}.

Das Kurvenintegral entlang \gamma_2 ergibt:

\displaystyle {\color{blue} \int_{\gamma_2} f ds} = \int_0^{1} (A \cdot (X2 t) + B \cdot (Y2 t)) \cdot \sqrt{negParens(X2)^2 + negParens(Y2)^2} \ dt = \int_0^{1} (A*X2+B*Y2 \cdot t) \cdot W2 \ dt = {\color{blue}fractionReduce((A * X2 + B * Y2) * W2,2)}.

Gesamtlösung:

\displaystyle \int_\gamma f ds = fractionReduce((A * X1 + B * Y1) * W1,2) + fractionReduce((A * X2 + B * Y2) * W2,2) = fractionReduce((A * X1 + B * Y1) * W1+(A * X2 + B * Y2) * W2,2).