# Homework Help: Divergence Theorem

1. May 9, 2012

### bugatti79

1. The problem statement, all variables and given/known data

Folks,

Verify the divergence theorem for

F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16
2. Relevant equations

$\int\int\int div(F)dV$

3. The attempt at a solution

My attempt

The radius of the sphere is 4 and div F= 1, therefore the integral becomes

$\int\int\int div(F)dV=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} 1dV=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} 2\rho^2 sin (\phi) d\rho d\phi d \theta$

Is this correct so far?
Thanks

2. May 9, 2012

### cjc0117

where did the 2 come from in the integrand? Other than that it's right.

EDIT: Also, since divF is a constant and the volume of a sphere can easily be found, you don't actually need to evaluate an integral.

3. May 9, 2012

### bugatti79

Hi, that should be a 1, ie

$...=\int_0^{2\pi} \int_0^{\pi} \int_0^{4} \rho^2 sin (\phi) d\rho d\phi d \theta$
But it says I need to verify the Divergence Theorem...so I guess I can continue and verify?

Thanks

4. May 9, 2012

### sharks

$\int_0^{2\pi} \int_0^{\pi} \int_0^{4} dV$ = volume of sphere with radius 4.

Just use the well-known formula for calculating volume of sphere. The answer should obviously be the same as the evaluation of your triple integral expression above.

Now, when you're asked to verify the Gauss Divergence Theorem, you should independently calculate the total flux using the surface integral formula, to confirm your answer.
$$\iint_S \vec F \hat n \,.d\sigma$$
Hint: To make it easier, split the sphere exactly in half laterally and evaluate the surface integral for one half. Then multiply by 2 to get the total flux.

Last edited: May 9, 2012
5. May 9, 2012

### bugatti79

Ok, thanks guys. Will respond hopefully at some stage.

Cheers

6. May 9, 2012

### sharks

Note: $d\sigma$ is the differential area.