# Math Help - Divergence Theorem

1. ## Divergence Theorem

Use the Divergence Theorem to find the flux of F across the surface sigma with outward orientation.

F(x,y,z) = 2xzi + yzj + z^2k, where sigma is the surface of the solid bounded above by z = sqrt(a^2-x^2-y^2) and below by the xy-plane.

2. Originally Posted by noles2188
Use the Divergence Theorem to find the flux of F across the surface sigma with outward orientation.

F(x,y,z) = 2xzi + yzj + z^2k, where sigma is the surface of the solid bounded above by $z = \sqrt{a^2-x^2-y^2}$ and below by the xy-plane.
$\text{div}(F) = 2z+z+2z=5z$

So you want to evaluate $5\int\int\int z\,dV$ bounded above by a sphere of radius $a$ and bounded below by the plane $z=0$.

Use spherical coordinates (where $\rho$ is the radius, $\phi$ is the azimuth angle, and $\theta$ is the angle around the circle.
$z=\rho\cos\phi$
$dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$

The bounds:
$\rho=0..a$
$\phi=0..\pi/2$
$\theta=0..2\pi$

So you want to evaluate $5\int_0^{2\pi}\int_0^{\pi/2}\int_0^a \rho^3\sin(\phi)\cos(\phi)\,d\rho\,d\phi\,d\theta = \frac{5}{2}\int_0^{2\pi}\int_0^{\pi/2}\int_0^a \rho^3\sin(2\phi)\,d\rho\,d\phi\,d\theta$

Evaluating this integral should be straightforward. I have included the final answer below.
Spoiler:
$\frac{5\pi a^4}{4}$