(adsbygoogle = window.adsbygoogle || []).push({}); Stoke's Theorem

1. The problem statement, all variables and given/known data

F(x,y,z,) = (3x+y)i - zj + y²k;

S = the union of the cylinder {(x,y,z) : x² + y² = 4, 2 < z < 4} and the hemisphere {(x,y,z) : x² + y² +(z-2)² = 4, z < 2}, oriented by P |--> n(P) with n(0,0,0) = -k

Use Stoke's Theorem to evaluate the double derivate of curl(f) [dot] ndS for the vector field F and the oriented surface.

2. Relevant equations

3. The attempt at a solution

I really don't know how to use stoke's theorem when considering a union, normally I would solve the second piece for when z = 2, thus x² + y² = 4 which gives you a parametrization of <cos(t),sin(t),2> but I don't know how to consider the cylinder. Would I just say <cos(t),sin(t),4> instead to account for the opening and then note that there is a negative orientation?

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# Stroke's Theorem

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