Question

Use the divergence theorem to calculate the outward flux of f=r|r|f=r|r| where r=⟨x,y,z⟩r=⟨x,y,z⟩ across ss: the hemisphere z=16−x2−y2‾‾‾‾‾‾‾‾‾‾‾‾√z=16−x2−y2 and the disk x2+y2≤16x2+y2≤16 in the xyxy-plane.

Answer 1

With $\mathbf f=\mathbf r$ (?) you have divergence

$\nabla\cdot\mathbf f=\dfrac{\partial x}{\partial x}+\dfrac{\partial y}{\partial y}+\dfrac{\partial z}{\partial z}=3$

Then by the divergence theorem, the flux is

$\displaystyle\iint_S\mathbf f\cdot\mathrm d\mathbf S=3\iiint_R\mathrm dV$

where $R$ denotes the interior of the region with boundary $S$.

Convert to spherical coordinates to set up the volume integral as

$\displaystyle3\int_{\varphi=0}^{\varphi=\pi/2}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=4}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

$=\displaystyle6\pi\left(\int_0^4\rho^2\,\mathrm d\rho\right)\left(\int_0^{\pi/2}\sin\varphi\,\mathrm d\varphi\right)$

$=128\pi$