I have been stuck with understanding an integral and would really appreciate if someone could help me.
Let $\Omega$ be a smooth domain in $\mathbb{R}^n$, $n= 2, 3$ and let $\Gamma = \partial \Omega.$ How do I solve or simplify the following integral (assuming that all involved functions have enough regularity so the quantities make sense) $$\int_\Gamma f\partial_\tau |g^2| dS$$ where $dS$ indicates it is a surface integral and $\tau$ is a vector in the tangent space?
I quickly talked to my advisor about this and all she said (in a rush) was "just integrate by parts on the tangential direction". Does anybody know how I could integrate by parts in the tangential direction?
Any help, comment or reference would be greatly appreciated.