I have a number of basic questions.
1) Consider a curve $\gamma:I \to \mathbb{R}^n$. I understand that if the arclength is $s$, the change of variables requires $ds = |\gamma_x|dx$. But what does $\partial_s = \frac{\partial_x}{|\gamma_x|}$ mean? I am told that this is the arclength derivative. What is its use?
2) Also, if $\nabla_s f$ of a vector $f$ is defined as $\nabla_s f = f_s - (f_s\cdot T)T$ where $T$ is the tangent, then this is the normal component of $f_s.$ Presumably this means taking the element-wise derivative. Does this have some meaning other than being the normal component of $f_s$?
3) Finally, I know that for normal fields $f$ and $g$ $\partial_s (f \cdot g) = \nabla_s f \cdot g + f \cdot \nabla_s g.$
How can I get an integration by parts formula for this? Is it something like $\int v\nabla_s u\;ds = -\int u \nabla_s v\;ds$ for all (not necessarily normal $u$ and $v$)?