Notation for Surface Integral in $\mathbb{R}^3$

Question

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$:

Let $\Gamma \subset \mathbb{R}^3$ be a 2-dimensional surface, parametrized by $\varphi:\Omega \rightarrow \Gamma$ (where $\Omega$ is a domain in $\mathbb{R}^2$). Having a scalar function $f:\Gamma\rightarrow\mathbb{R}$, I would write

$$\int_{\Gamma} f(\gamma)dS(\gamma) = \int_{\Omega}f(\varphi(x))\left|\varphi_{x_1}(x)\times\varphi_{x_2}(x)\right|dx.$$

The LHS is what I wrote and the reviewers did not understand it... they "expect[ed] $dS(\gamma)$ to be $\gamma$" and other strange stuff. I have to mention, it is a paper at a computer vision conference.

What do you think? Which notation is in your view the most "standard" one? Do you think I can expect from the reader to understand a notation like $\int_{\Gamma} f(\gamma)dS(\gamma)$?

Answer 1

Maybe define $f_x(\gamma)$ instead of $f(x, \gamma)$ and go with $x\mapsto \int_\Gamma f_xdS$, unless the $dS$ part itself causes confusion.

Answer 2

A differential form is sometimes written with the point as subscript, so, for example, if we denote $dS$ the 2-form of the area element at a point $\gamma$ we can write it also as $dS_\gamma$. Remember that a 2-form takes a couple of tangent vectors (whose area it computes) as arguments written in the brackets: $dS_\gamma(u,v)$.