# Divergence theorem problem

1. Nov 29, 2009

### fluxions

1. The problem statement, all variables and given/known data
the problem is to calculate
$$\int (\nabla \cdot \vec{F}) d\tau$$
over the region
$$x^2 + y^2 + x^2 \leq 25$$
where
$$\vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k})$$
in the simplest manner possible.

2. Relevant equations
divergence theorem!

3. The attempt at a solution
Write
$$\vec{F} = |\vec{r}|^2 \vec{r} = |\vec{r}|^3 \hat{r},$$
so
$$\vec{F} \cdot \hat{n} = \vec{F} \cdot \hat{r} = |\vec{r}|^3 \hat{r} \cdot \hat{r} = |\vec{r}|^3 = 125,$$
since
$$\hat{n} = \hat{r}$$
and
$$|\vec{r}| = 5$$
along the surface of the sphere.
Then, invoking the divergence theorem, we obtain:
$$\int (\nabla \cdot \vec{F}) d\tau = \oint_{\partial{\tau}} \vec{F} \cdot \hat{n} d\sigma = \oint_{\partial{\tau}} 125 d\sigma = 125 \cdot 4 \cdot \pi \cdot 5^2$$

the back of the book gives 100pi as the answer (and i've checked the errata for the book; no correction has been made). am i wrong? or is the book?

Last edited: Nov 29, 2009
2. Nov 29, 2009

### Dick

It's also not very hard to integrate the divergence over the interior of the sphere. I get 4*pi*5^5. It sure looks to me like the book answer is wrong.

3. Nov 29, 2009

### fluxions

Great, thanks much!