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!

  • $\begingroup$ Are you sure you wrote down the integral correctly? It seems like the information given about $\psi$ is not useful at all. $\endgroup$ – Chee Han Nov 27 '16 at 22:54
  • $\begingroup$ @CheeHan Please see my "HINT" in the posted solution. $\endgroup$ – Mark Viola Nov 27 '16 at 22:54
  • $\begingroup$ @Dr.MV I wrote down exactly the same thing, but it doesn't correspond to the integral at all since if you were to apply Divergence theorem, you get the integral over $D$ of the Laplacian of $\psi$. (Ahhhhhh I see, nvm, good catch!) $\endgroup$ – Chee Han Nov 27 '16 at 22:56


Using the product rule for the divergence, we can write

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

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.

  • $\begingroup$ Sorry I asked such a stupid question, good hint (: $\endgroup$ – Chee Han Nov 27 '16 at 22:57
  • $\begingroup$ @CheeHan Thank you! Much appreciated. -Mark $\endgroup$ – Mark Viola Nov 27 '16 at 23:00
  • $\begingroup$ Thank you very much Dr. MV! However after I have the Laplacian of $\psi$ = 7 how would I proceed to implementing the Divergence Theorem and evaluating the triple integral with the integrand div($\psi$) over the volume of interest? I'm just not seeing it :( $\endgroup$ – Nelly Nov 28 '16 at 1:04
  • $\begingroup$ You're welcome. My pleasure. The divergence of the gradient is the Laplacian. And the Laplacian is $7$. Now integrate $7$ over the volume of interest. $\endgroup$ – Mark Viola Nov 28 '16 at 1:07
  • $\begingroup$ Oh my, I just realized what I was misinterpreting. Thank you very much for your detailed explanation and your patience! $\endgroup$ – Nelly Nov 28 '16 at 1:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.