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

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 9, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z−1)^2 = 9, z ≥ 1. For the vector field F = (zx + z^2y + 2 y, z^3yx+ 8 x, z^4x^2), compute doubleintM (∇×F) ·dS in any way you like.

doubleintM (∇×F) ·dS = ?

2. Relevant equations

I thought the line integral would make the most sense in solving this - I wanted to take the line integral with respect to the bottom circle. parameterized, it is (3costheta, 3sintheta, 1).

Line int = F(r) (rprime)

3. The attempt at a solution

I have no idea what's going on here - I tried to take the line integral by saying r=(3sint,3cost,0) Since z=0 for circle the only parts that matter in the F is the 2y and the 8x, but I don't think that's right.

Integral w/ limits 0 to 2pi

(2(3sintheta),8(3costheta),0) dot product (-3sintheta, 3costheta, 0)

Can anyone help me with this one?

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# Stokes Theorem

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