(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given the vector field [itex]\vec{v} = (-y\hat{x} + x\hat{y})/(x^2+y^2)[/itex]

Show that [itex]\oint \vec{dl}\cdot\vec{v} = 2\pi\oint dl[/itex] for any closed path, where [itex]dl[/itex] is the line integral around the path.

2. Relevant equations

Stokes' Theorem: [itex]\oint_{\delta R} \vec{dl}\cdot\vec{v} = \int_R \vec{dA}\cdot(\vec{\nabla}\times\vec{v})[/itex]

3. The attempt at a solution

I tried finding the curl of v, and got that the x and y components were zero, and the z component is [itex]\frac{\delta}{\delta x}(x/(x^2+y^2)) + \frac{\delta}{\delta y}(y/(x^2+y^2))[/itex], which becomes [itex](y^2-x^2+x^2-y^2)/(x^2+y^2)^2[/itex], which is also zero. But this can't be right, because that would imply that [itex]\oint \vec{dl}\cdot\vec{v} = \int \vec{dA}\cdot0 = 0[/itex] by Stokes' Theorem, which would mean the thing I'm trying to show is false, since [itex]2\pi\oint dl[/itex] is not always zero.

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# Homework Help: Prove a Statement about the Line Integral of a Vector Field

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