How do I use Gauss' theorem to evaluate a surface integral?

[epyfathom]

Find and evaluate numerically

x^10 + y^10 + z^10 dSx^2 + y^2 + z^2 =4It says you're supposed to use gauss' divergence thm to convert surface integral to volume integral, then integrate volume integral by converting to spherical coordinates... I can do the second part but how do i use gauss' thm...? my prof was really bad at explaining this.Thanks.

 

[HallsofIvy]

Well, I would think that the first thing you would do is look up "Gauss' theorem" (perhaps better known as the "divergence theorem"). According to Wikipedia, Gauss' theorem says that
$$\int\int\int (\nabla\cdot \vec{F}) dV= \oint\int \vec{F}\cdot\vec{n}dS$$
where $\vec{n}$ is the normal vector to the surface at each point.

Here, you are not given a vector function but, fortunately, Wikipedia also notes that "Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:
$$\int\int\int \nabla f dV= \oint\int f dS$$"

So, since you are asked to use Gauss' theorem to evaluate a surface integral, you are intended to find $\nabla f$ and integrate that over the region- the ball of radius 2.

Then- first step- what is $\nabla (x^{10}+ y^{10}+ z^{10})$?