Use Stokes\' Theorem to evaluate integral integral _s curl F.dS| Where F(x, y, z
ID: 2878552 • Letter: U
Question
Use Stokes' Theorem to evaluate integral integral _s curl F.dS| Where F(x, y, z)=(3y cos(z), 3e^x sin(z), 2xe^y)|and S|is the hemisphere x^2 + y^2 + z^2 = 1, z greaterthanorequalto 0|, oriented upwards. Since the hemisphere is oriented upwards the boundary curve must be transversed counter-clockwise when viewed from above. A parametrization for the boundary curve C| can be given by: r(t) = 1 cos(t) i +| j+| k, 0 lessthanorequalto t le| (use the most natural parametrization) integral integral _s curl F.dS = integral^b_0| dt| integral integral _s curl F.dS=| If you don't get this in 3 tries, you can see a similar example (online).Explanation / Answer
Solution:
By Stokes' Theorem, S curl F · dS = c F · dr, where C is the boundary curve of S
x^2 + y^2 = 1, z = 0, oriented clockwise.
Parameterizing - C by r(t) = (1 cos t, 1 sin t, 0) for t in [0, 2],
So r(t) = 1 cos(t)i + 1 sin(t)j + 0 k, 0 t 2
we obtain c F · dr
= -(t = 0 to 2) <3 * 1 sin t, 0, 2*1cos t * e^(sin t)> · <-sin t, cos t, 0> dt
= (t = 0 to 2) 3 sin^2(t) dt
= (t = 0 to 2) (3/2)(1 - cos(2t)) dt
= (3/2) (t - sin(2t)/2) {for t = 0 to 2}
= (3/2) * 2 = 3
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