Applied Divergence Theorem

Question

I'm struggling immensely with this problem, I can't seem to evaluate and use the Divergence Theorem correctly. I've approached this problem in both spherical and cylindrical coordinates and can't solve it. The problem states;

A non-zero scalar field $\psi$ is such that $||\vec{\nabla}\psi||^2$ = 3$\psi$ and $\vec{\nabla}\cdot(\psi\vec{\nabla}\psi)$ = 10$\psi$. Evaluate $$\iint_{S}\vec{\nabla}\psi \cdot \hat{n}dS$$ when S is the surface of the region in the first octant bounded by z = $\sqrt{x^2+y^2}$, z = $\sqrt{1-x^2-y^2}$, y = x and y = $\sqrt{3}x$. Also we are required to solve this problem using $\underline{both}$ spherical and cylindrical coordinates.

Any guidance and advice would be much appreciated, thanks in advance!

Answer 1

HINT:

Using the product rule for the divergence, we can write

$$\begin{align} \nabla \cdot \left(\psi \nabla \psi\right)&=\psi \nabla ^2(\phi)+\nabla (\psi)\cdot \nabla (\psi)\\\\ &=\psi \nabla ^2(\phi)+\left|\nabla (\psi)\right|^2 \tag 1 \end{align}$$

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SPOILER ALERT: Scroll over the highlighted area to reveal the solution.

Solving $(1)$ for $\nabla^2(\psi)$, we find that $$\nabla^2(\psi)=\frac{\nabla \cdot (\psi \nabla (\psi))-\left|\nabla (\psi)\right|^2}{\psi}=7$$Using the Divergence Theorem, the result is $7\times \,\,\text{Volume enclosed}$.

Now, solve for $\nabla^2 \cdot (\psi)$, apply the given relationships, and integrate a constant over the volume of interest.