Flux through a Sphere: Finding the Flux of a Vector Field Across a Unit Sphere

[pivoxa15]

Homework Statement

Find the flux of the vector field F(x,y,z)=(z,y,x) across the unit sphere x^2+y^2+z^2=1

The Attempt at a Solution

My Answer: 3(pi)^2/8

Book answer: 4(pi)/3

 

[Dick]

The easy way to do this is to use the divergence theorem which immediately gives you the book answer.

 

[nicksauce]

The book answer is correct. Can you show your work?

 

[Dmak]

divergence theorem: triple integral of the divergence of the vector field, in this case the divergence is just 1, so you're just essentially finding the volume of the sphere

 

[pivoxa15]

RIght, I just had problems with the surface integral but it's just d(theta)d(psi)

by letting x=theta, y=psi so the jacobian is 1.

All correct?

 

[Dmak]

yes but no need for spherical coordinates since its just the triple integral:

SSS1dV = volume of sphere over the domain d = { (x,y,z): x^2 + y^2 + z^2 = 1 }
( sorry no latex )

 

[Gib Z]

If dexter still posted regularly, I'm sure he would have made a point to say Volume *enclosed by* the sphere =] Welcome to PF Dmak! (And don't mind that comment, really just semantics).

EDIT: Wow Dick has 2^(12) posts =]

 

[pivoxa15]

yes but no need for spherical coordinates since its just the triple integral:

SSS1dV = volume of sphere over the domain d = { (x,y,z): x^2 + y^2 + z^2 = 1 }
( sorry no latex )

In the first attempt, I wasn't trying to use the divergence theorem but the surface integrable dS. It was the long way but I wanted to know that I could do it.

 

[Dmak]

If dexter still posted regularly, I'm sure he would have made a point to say Volume *enclosed by* the sphere =] Welcome to PF Dmak! (And don't mind that comment, really just semantics).

EDIT: Wow Dick has 2^(12) posts =]

haha thanks Gib_Z :p