3. Let W be region bounded by the upper-half of the sphere with radius 1 and center (0,0,0) and the xy-plane. Let T be the surface of ∂W and let B be the bottom. Furthermore, let F (x , y , z)= be a vector field. a) Determine the value ∫∫T F⋅dS by direct computation. b) Verify your answer by determining the value of ∫∫B F⋅dS and using Gauss's Theorem (= Divergence Theorem, see 16.8 of Thomas).

Question

3. Let W be region bounded by the upper-half of the sphere with radius 1 and center (0,0,0) and
the xy-plane. Let T be the surface of ∂W and let B be the bottom. Furthermore, let
F (x , y , z)=< x2 , x2+y2 , z2−x > be a vector field.
a) Determine the value ∫∫T F⋅dS by direct computation.
b) Verify your answer by determining the value of ∫∫B F⋅dS and using Gauss's Theorem (=
Divergence Theorem, see 16.8 of Thomas).

jamesj · Answer

so what's giving you trouble here?

anonymous · Answer

what the method is for direct computation

jamesj · Answer

the bottom is easy. you're integration over the circle of radius 1 in the xy plane (z=0) and the outward pointing normal vector everywhere is -k.  I'd use plane polar coordinates.

jamesj · Answer

for the other part of T, ∂W, I'd use spherical polar coordinates where the outward pointing normal vector is $$ \hat{r} $$

anonymous · Answer

what would the bounds of the integral be?