Mathematics: Use the fundamental theorem of line integrals to calculate r c ~f â·d~r exactly, if ~f = 2x 1/3~i+e y/7 ~j, and c is the quarter of the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).

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Use the fundamental theorem of line integrals to calculate

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strikeboystorm

14.03.2022

Mathematics

Assuming the vector field is $\mathbf f=2x^{1/3}\,\mathbf i+e^{y/7}\,\mathbf j$ , we want to find a scalar function

such that $\nabla f=\mathbf f$ . We require

$\dfrac{\partial f}{\partial x}=2x^{1/3}$
$\dfrac{\partial f}{\partial y}=e^{y/7}$

Integrating both sides of the first PDE with respect to

gives

$f(x,y)=\dfrac32x^{4/3}+g(y)$

Differentiating with respect to

gives

$\dfrac{\partial f}{\partial y}=e^{y/7}=\dfrac{\mathrm dg}{\mathrm dy}$
$\implies g(y)=7e^{y/7}+C$

$f(x,y)=\dfrac32x^{4/3}+7e^{y/7}+C$

Such a scalar function exists, so the fundamental theorem holds and any integral along any path is determined exactly by its endpoints. In this case,

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(0,1)-f(1,0)=7e^{1/7}-\frac{17}2$

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Use the fundamental theorem of line integrals to calculate r c ~f â·d~r exactly, if ~f = 2x 1/3~i+e y/7 ~j, and c is the quarter of the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).

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