Final answer only is required

Question

Use Stokes' Theorem to evaluate ∫∫_S curl F · dS.

F(x,y,z) = e^xycos(z) i + x²z j + xy k

S is the hemisphere

, oriented in the direction of the positive x-axis.

Answer 1

Stokes' Theorem tells us that this integral is equal to

∫_C F•dr

Where C is the boundary circle of the hemisphere, given by setting x=0.

Along C, F(0,y,z) = (cos(z),0,0). This is perpendicular to the yz-plane. But dr lives in the yz-plane!

More precisely, if we parametrize C as r(t) = (0, cos t, sin t), then

F(r(t))•r'(t) = (cos (sin t)), 0, 0)•(0, -sin t, cos t) = 0, so

∫_CF•dr = ∫F(r(t))•r'(t)dt = 0

Therefore ∫_CF•dr =0