Stoke's Theorem help

[tight pelican]

Question: Given the vector field F(r) = y e_x + z^2 e_z. Use Stoke's Theorem to compute the flux of ∇ x F through the hemisphere of radius R centered at the origin, with z ≥ 0. Orient the hemisphere with the normal pointing downards.

The image I've provided is a rough working. I am not sure if the flux is the sum of the hemisphere's flux and the circular base's flux.

[willow dome]

I do not believe it uses the base

Stokes theorem would show that they are equal. So it would be pointless to calculate both

@tight pelican

[tight pelican]

Yeah, thank you. I calculated the line integral with the boundary being the circle. I also calculated the surface integral using the domain of the hemisphere. I got both results being equal.

[willow dome]

As you should

The surface integral over the base may have been easier tho

But I'm just guessing

[tight pelican]

I found the surface integral easier to do tbh

[willow dome]

Sometimes it is

Line integrals suck

[tight pelican]

Yeah they do

Annoying you sometimes gotta break it into parts

[willow dome]

Vector calculus is always a lot of work

But tensor calculus will be even more work