Question

Find the flux of f = xy i + yz j + zxk out of a sphere of radius 3 centered at the origin.

Answer 1

The sphere is presumably closed, so we can use the divergence theorem.


`\\abla\cdot\mathbf f=(\partial(xy))/(\partial x)+(\partial(yz))/(\partial y)+(\partial(xz))/(\partial z)`

`\\abla\cdot\mathbf f=x+y+z`

Now convert to spherical coordinates:


`x=\rho\cos\theta\sin\varphi`

`y=\rho\sin\theta\sin\varphi`

`z=\rho\cos\varphi`

The flux is then given by the volume integral,


`\displaystyle\iint_(x^2+y^2+z^2=9)\mathbf f\cdot\mathrm d\mathbf S=\iiint_(x^2+y^2+z^2\le9)(x+y+z)\,\mathrm dV`

`=\displaystyle\int_(\varphi=0)^(\varphi=\pi)\int_(\theta=0)^(\theta=2\pi)\int_(\rho=0)^(\rho=3)(\rho\cos\theta\sin\varphi+\rho\sin\theta\sin\varphi+\rho\cos\varphi)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=0`