Find all values of the line integral $\int_{(1,0)}^{(2, 2)}{\frac{-y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy}$, by a path that does not contains the point $(0,0)$
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Is the integrand of the form $f(z),dz$ for some analytic function $f$? – GEdgar Apr 18 '21 at 22:59
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I tried to use polar coordinates, and by doing so, i found that all possible values are of the form $\pi / 4 + 2k\pi$, but the answer in my book, had only $\pi / 4, \pi / 4 \pm 2\pi$ as results – Diego Campos Laboissiere Ville Apr 18 '21 at 23:03
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A potential function of the vector field $\frac{-y\hat{i}+x\hat{j}}{x^2+y^2}$ is $\arctan(y/x)$. You can use this if the path doesn't cross the $y-$axis. If the path does cross the $y-$axis, you could use the identities $$\arctan(y/x)=\pi/2 -\arctan(x/y)$$ if $y/x>0$ and $$\arctan(y/x)=-\pi/2-\arctan(x/y)$$ if $y/x<0$ to get around this complication. – Matthew H. Apr 19 '21 at 00:27
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For the path, take the line whose equation is $$y=\frac{2-0}{2-1}(x-1)=2x-2$$ So, $ dy=2dx$. The integrale becomes $$\int_1^2\frac{(2x-2x+2)dx}{4x^2-8x+4+x^2}$$
hamam_Abdallah
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