What is the flux of the vector field $\mathbf{f}(\mathbf{r})=(x- y^2z, ysinz, cosz)$ through the sphere $x^2+y^2+z^2=4$?
What I have done:
$\nabla\cdot \mathbf{f}= 1+ sinz -sinz= 1 $
I know that the flux is $\iiint\limits_{V}\nabla\cdot \mathbf{f} dxdydz= \iiint\limits_{V}1 dxdydz $. But I don't know what limits to use.
Best Answer
hint
$$\iint_S \vec{f}\cdot \vec{ds}=$$ $$\iiint_V \nabla\cdot \vec{f} dv=$$ $$\iiint_Vdxdydz=$$ volume of the sphere $$=\frac 43 \pi (\;radius\;)^3$$
but, the equation of your sphere is $$x^2+y^2+z^2=4=R^2$$