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.