Flux of the vector field through the surface of a sphere?

Question

Find the flux of the vector field $F=(x^3, y^3, z^3)$ through the surface of a sphere $x^2+y^2+z^2=x$

Can someone please show me how to calculate this? Thank you in advance.

Answer 1

Since a sphere is closed, I would suggest applying the divergence theorem.

$\iint F(x,y,z)\cdot dS = \iiint \nabla\cdot F dV$

$\nabla\cdot F = 3x^2 + 3y^2 + 3 z^2$

Convert to spherical coordinates.

$\int_0^{2\pi}\int_0^{\pi}\int_0^R 3\rho^2(\rho^2\sin\phi) dV$

$\frac {12}{5}\pi R^5$

If you don't want to use Gauss's / divergence theorem.

Again convert to spherical.

$x = R\cos\theta\sin\phi\\ y = R\sin\theta\sin\phi\\ z = R\cos\phi$

$dS = ( \frac {\partial x}{\partial\theta}, \frac {\partial y}{ \partial \theta},\frac {\partial z}{\partial \theta})\times (\frac {\partial x}{\partial \phi}, \frac {\partial y}{\partial \phi},\frac {\partial z}{\partial \phi})$