Divergence Theorem to calculate flux

Question

Take the vector field given by: $F= (y^2+yz)i+(\sin(xz)+z^2)j+z^2k$

a) Calculate the divergence, $\operatorname{div}F$.

b) Use the divergence theorem to calculate the flux $$\int_S F\cdot dA $$

through a sphere or radius 2 centered at the origin oriented with an outward pointing unit normal.

For the divergence of $F$, I found it to be $2z$. I'm pretty sure I need to change the integral into spherical coordinates, but I'm not sure if that's right. I'm also not understanding how I would find the limits for the integral as well.

Answer 1

The divergence is

$$ \partial_x (y^2 + yz) + \partial_y (\sin(xz) + z^2) + \partial_z (z^2) \\ = 2z.$$

The divergence theorem tells you that the integral of the flux is equal to the integral of the divergence over the contained volume, i.e.

$$ \int_B \nabla \cdot F \,dx\,dy\,dz = \int_B 2z \,dx\,dy\,dz $$

where $B$ is the ball of radius $2$ (i.e. the interior of the sphere in question). If you want to set up limits of integration, use $x^2 + y^2 + z^z = 4$. (and integrating with respect to $z$ first seems fruitful)

Answer 2

A good parametrization for your surface is:

$$\mathbf r(\theta,\phi)=\langle x= 2\sin\phi\cos\theta,y=2\sin\phi\sin\theta,z=2\cos\phi\rangle.$$

Where $\phi$ is the angle between the positive $z$-axis and the vector $\boldsymbol v$ with tail on the origin and tip at a point on the sphere, and $\theta$ is the angle between the positive $x$-axis and the projection of the vector $\boldsymbol v$ onto the $xy$-plane. So your limits of integration would be:

$$0\leq\theta\leq 2\pi,\\0\leq\phi\leq\pi,\\0\leq\rho\leq2.$$

Divergence theorem tells you that:

$$\iint\limits_S \mathbf F \cdot d\mathbf S = \iiint\limits_E \text{div}\mathbf F\,dV.$$

The last triple integral by Fubini is the iterated integral with the bounds I proposed, do change of variables and don't forget the jacobian $\rho^2\sin\phi$.