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

Use the surface integral in Stoke's Theorem to calculate the circulation of the Field F around the curve C in the indicated direction.

F = (x^2)i + (2x)j + (z^2)k

C: The ellipse 4x^2 + y^2 = 4 in the xy-plane, counterclockwise when viewed from above.

2. Relevant equations

3. The attempt at a solution

At first I attempted to parametrize the ellipse in terms of u and v, like r(u,v) = cos(v)i + 2sin(v)j + (u)k. But I don't think this is doing it right, and also I don't know what the bounds on u would be. So I tried another way I saw done in an example:

I tried to find the normal n, which my book states to be grad(f) / |grad(f)|. Taking f to be 4x^2 + y^2 = 4, I find n = (8x i + 2y j + 0k) / sqrt(64x^2 + 4y^2). Then I try to find curl F, which is grad x F, and I find this simply = 2k. Then taking dot product of curl F and n I end up with 0, since n has no z-component and curl F has no x and y-component. This is like I expected not the right answer.

Can someone point out what is wrong here or how I do these kind of problems the correct way? Any help is appreciated.

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# Homework Help: Stoke's theorem to calculate circulation

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