I encountered the following problem in a practice Advanced Calc exam, and I have an issue.
Suppose $\phi:\mathbb{R}^3\to\mathbb{R}$ is a strictly positive function satisfying $|\nabla\phi|^2=4\phi$ and $\Delta(\phi^2)=20\phi$. Evaluate $\int_S\frac{\partial\phi}{\partial n}ds,$ where $S$ is the surface of the unit sphere centered at the origin, $\cfrac{\partial\phi}{\partial n}$ is the directional derivative of the unit outward notmal to $S$, and $ds$ is the surface measure of $S$.
I cannot for the life of me recall what the $\Delta$ means in this context. Can someone help me out?
Note that I am not (at present) looking for any hints or help in evaluating the integral, though I will update this post later if I'm still stymied even after having this notation issue cleared up.