1. ## Gauss's Theorem

Hello.

Greetings,

How is the following solved:

In exercises 1-4, use Divergence Theorem to calculate the flux of the given vector field out of the sphere ("sign that looks like &, I am not sure what it is called") with equation x2+ y2 = a2 , where a > 0.

1. F = (x2 + y2)i + (y2 - z2)j + zk

div F = 2x + 2y + 1

I am aware that I’ll need to use the formula for a volume of a sphere:

V = (4/3)πr3

But to calculate this triple integral or flux I might need bounds of integration - how are they found if required?

Sincere regards from user, Kaemper

2. ## Re: Gauss's Theorem

did you forget a $z^2$ in the equation for your sphere? You've described an infinite cylinder.

3. ## Re: Gauss's Theorem

I did forget to tell you this:

"With equation x2 + y2 + z2 = a2, where a > 0".

4. ## Re: Gauss's Theorem

take your expression for $\nabla \cdot F$ and convert it to spherical coordinates and do the 3D integral in spherical coordinates over

$0\leq \rho \leq a$

$0\leq \theta \leq \pi$

$0 \leq \phi \leq 2\pi$

5. ## Re: Gauss's Theorem

I did as requested and got following result:

Please notice I have two versions of the convertion from cartesian coordinates to spherical coordinates. I think the last one is the correct one.

Best regards