Find the curl of F = [3y+z, −3x, xyz] and G = [cosh y, sinh z, cosh x].
(a) Use Stokes’ Theorem to find the surface integral ∬S1
[xz, 1−yz, −6] ●ndA over the hemisphere
S1 = {[x, y, z]∶ x
2 + y
2 + z
2 = 25, z <= 0} where n points outside/downwards.
(b) Use Stokes’ Theorem to find ∮C H(\gamma ) ● d\gamma of H = [4z+e
x
, 4x+e
y
, 4y+e
z
] over the surface boundary C of S2 repr. by r(u, v) = [u
2
, 2u+2v, v2
] def. on R = {[u, v]∶ 0<=u<=2, 0<=v<=2−u}.
(c) Use Stokes’ Theorem to find the surface integral ∬S3
[cosh z, sinh x, sinh y] ●ndA over the ellipsoid S3 = {[x, y, z]∶ x
2+3y
2+2z
2 = 4} where n points outside.
You can use Matlab to calculate partial derivatives, cross products and curls; show where you use it.