# Vector Calculus Divergence Theorem Textbook Answer Confusion

here's a particular question I'm working on that the textbook doesn't have the same answer as me.

Use The Divergence Theorem for: $F = |r|r$, where $r = <x,y,z>$, and $S$ consists of the hemisphere $z=\sqrt{1-x^2-y^2}$ and the disk $x^2 + y^2 \leq 1$

Here's what I have using spherical coordinates:
$|r| = p$
$\nabla \times r = 3$
Multiplying them both together we get: $3p$

Using spherical coordinates we get:
$\int_{0}^{\frac{\pi}{2}} \int_{0}^{2\pi} \int_{0}^{1} 3p^3sin\phi$ $dp d\phi d\theta$

Integrating gives us: $\frac{3\pi}{2}$ but the answer in the textbook is $2\pi$, I integrated it correctly, so that means that I either didn't set the question up properly, or the textbook answer key is wrong.

Anyway, thanks.

• Small note, you should use \cdot for divergence as \times would be used for the curl – Triatticus Apr 17 '16 at 20:24
• Also you have to take the divergence of F not just r – Triatticus Apr 17 '16 at 20:44