Divergence Theorem to evaluate integral where S is a sphere

$\begingroup$

Use the Divergence Theorem to evaluate

$$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$$

where $S$ is the surface of the sphere $x^2+y^2+z^2=1$.

So the divergence of F is $2y^2x+2z^2y+2x^2z$. Then I tried to integrate using spherical coordinates, but the integral seemed to be very exhausting. Is there a better way to do it?

$\endgroup$ 3

1 Answer

$\begingroup$

The normal vector on the surface of the sphere is $(x,y,z)$. Thus you can view this as the integral of $(x,y,z)\cdot(xy^2,yz^2,zx^2)$. The divergence of $(xy^2,yz^2,zx^2)$ is $y^2+z^2+x^2=r^2$. The integral of $r^2$ over the unit sphere is

$$ \int_0^1\mathrm drr^4\int_{-1}^1\mathrm d\cos\theta\int_0^{2\pi}\mathrm d\phi=\frac45\pi\;. $$

$\endgroup$ 4

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy