I have this problem:
A vector field $\vec{F} =\hat R\, \frac{\cos^2 (\phi)}{R^3} \ $ exists in the region between two spherical shells with same origin defined by $R=1$ and $R=2$. Find $\int \vec{F} \cdot d\vec{S}$ and $\int \nabla\cdot\vec{F} \,dV$ ( verify div. theorem)
Note spherical coords. that I use are $(R, \theta, \phi)$
I was able to find it using divergence theorem and got the answer $-14*\pi/3$ but I could not determine the correct regions and integral limits for $dS$ since $\vec{F}$ has only $\hat R$ as a unit vector I think I can only deal with $dS= R^2 \sin \theta \, d \theta \, d\phi$, but $R$ also changes in my case ! Help please..
Let $\vec F=\hat r\,\frac{\cos^2(\phi)}{r^3}$. Then, $\nabla \cdot \vec F=-\frac{2\cos^2(\phi)}{r^4}$.
Using the divergence theorem, we have
$$\begin{align} \int_V \nabla \cdot \vec F\,dV &=\oint_S \vec F \cdot \hat n\,dS\\\\ &=\int_0^{2\pi }\int_0^\pi \left(\frac{\cos^2(\phi)}{2^3}\,(2^2)-\frac{\cos^2(\phi)}{1^3}\,(1^2)\right)\,\sin(\theta)\,d\theta\,d\phi\\\\ &=-\frac12 \int_0^{2\pi}\int_0^\pi \cos^2(\phi)\,\sin(\theta)\,d\theta\,d\phi\\\\ &=-\pi \end{align}$$
We can verify this by evaluating the volume integral directly as
$$\int_V \nabla \cdot \vec F\,dV=-\int_0^{2\pi}\int_0^\pi \int_1^2 \frac{2\cos^2(\phi)}{r^4}\,r^2 \,\sin(\theta)\,dr\,d\theta\,d\phi=-\pi$$
as expected!