# Divergence evaluation

#### Silverflow

Hi all,
I just completed a question using the divergence theorem, and I was wondering if someone could go over my working to make sure I was correct.

The question is
Use the divergence theorem to evaluate
$$\displaystyle \int \int_{S}(x^2+y+z)ds$$
where $$\displaystyle S$$ is the surface of the unit sphere.

First off, I calculated the unit normal vector as $$\displaystyle \hat{N} = \vec{r} = x\vec{i}+y\vec{j}+z\vec{k}$$, as the sphere is the unit sphere.

As the divergence theorem states $$\displaystyle \int \int_{S} \vec{F}\bullet \hat{N} dS = \int \int \int_{D} div \vec{F} dV$$, I need to find a $$\displaystyle \vec{F}$$ such that $$\displaystyle \vec{F}\bullet \hat{N} = x^2+y+z$$. I found that to be $$\displaystyle \vec{F} = x\vec{i}+\vec{j}+\vec{k}$$.

Let $$\displaystyle B$$ stand for the sphere bounded by unit sphere, and now evaluating the theorem,
$$\displaystyle \int \int_{S}(x^2+y+z)ds = \int \int_{S} \vec{F}\bullet \hat{N} dS = \int \int \int_{B} div \vec{F} dV$$
$$\displaystyle \int \int \int_{B} div \vec{F} dV= \int \int \int_{B} 1dV = \frac{4\pi}{3}$$

Does this look correct?
Thanks for you time.

#### HallsofIvy

MHF Helper
Yes, it does.

• Silverflow

Alright! Thanks!