The tangential gradient $\nabla_\tau f$ associated to a surface $S$ is defined as the projection of a suitable extension $\nabla f$ to the tangent plane to that surface. It seems reasonable to think that when changing coordinates in order to flatten the surface, the tangential gradient is somehow related to the actual gradient. Having no experience with this notion, I cannot imagine precisely what happens when making such a transformation. Suppose that the surface $S$ is parametrized by $x(s,t)$ where $s,t$ lie in a plane region $T$. We know the surface integral formula $$ \iint_S fdS = \iint_T f(x(s,t)) \left \| \frac{\partial x}{\partial s} \times \frac{\partial x}{\partial t}\right\|dsdt. $$ I am interested in what happens when we have to integrate a function depending on the tangential gradient, i.e. I would like to see a formula for $$ \iint_S |\nabla_\tau f|^2 dS = ... $$ My guess is that we'll have a gradient term and a Jacobian factor, but I don't have a precise idea on how to find the formula or prove its corectedness.
The questions are:
- What is the completion of the above formula?
- Do you have any useful references treating this kind of integrals? (if possible, not too advanced in differential geometry notions...)