Recall in class we have shown that, for any surface $M$ with first fundamental form $I = g_{ij} dv^i dv^j$, the Laplace operator $\Delta_M$ is given by
$\Delta_Mf\ =\ $$\sum_{i,j} f_{v^iv^j}g^{ij}$ + first order derivatives of $f$
$I\ =\ \{1+[ϕ'(u)]^2\}du^2+2aϕ'(u)dudv+(u^2+a^2)dv^2$
Find the leading term $\sum_{i,j} g^{ij}f_{v^iv^j}$ of $\Delta_Mf$
This question is an extra credit question for my class. No idea how to go about it though. More interested in learning how to do the problem rather than the extra credit.