- #1

- 15

- 0

## Homework Statement

Verify Stokes' theorem for the given surface S and boundary dS and vector fields F

S = x

^{2}+y

^{2}+z

^{2}, z≥0

dS= x

^{2}+y

^{2}=1

F = <y,z,x>

## Homework Equations

Stokes' theorem:

∫∫(∇×F)dS = ∫F⋅ds

## The Attempt at a Solution

1. Curl of F:

∇×F = <-1,-1,-1>

2. After getting the curl, I just treated this as a surface integral, ∫∫F⋅dS = ∫∫F⋅(T

_{u}×T

_{v})dudv

I parametrized the sphere thusly,

x = sinφcosθ

y=sinφsinθ

z=cosφ

T

_{φ}= <cosφcosθ, cosφsinθ, -sinφ>

T

_{θ}= <-sinφsinθ, sinφcosθ, 0>

T

_{φ}×T

_{θ}= <sin

^{2}φcosθ, sin

^{2}φsinθ, sinφcosθ>

Am I doing this right so far? I asked a friend what he would do, and this is what he had for dS:

<dydz, dxdz, dxdy> (Idk how to get this).

So then he got:

∫∫(∇×F)⋅dS = ∫∫(<-1, -1, -1>⋅<dydz, dxdz, dxdy>) = -∫∫dydz+dxdz+dxdy, but I don't know how to integrate that...