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

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 $\int_{\Gamma} f(\gamma)dS(\gamma)$?

• Using $\Gamma$ to denote a 2-dimensional surface may be part of what's throwing them off (I did a double take myself), because it's usually used to denote 1-dimensional curves. – David H Jul 21 '14 at 13:34
• The $f(\gamma) dS(\gamma)$ confuses me as well. Rewriting it as $$\int\limits_{\Gamma} f \, dS = \int\limits_{\Omega} f(\varphi(x)) | \varphi_{x_1}(x) \times \varphi_{x_2} (x) | \,dx$$ may clear it up. – Mark Fantini Jul 21 '14 at 13:34
• I'm confused, haha. I got $\gamma \in \Gamma$ but what does that mapping do? – Mark Fantini Jul 21 '14 at 13:40
• The area element yes, but why $f(x,\gamma)?$ And I agree, the usual abbreviatory notation does not lead to confusion. Even though your area element depends on the point considered, since you are integrating it should not matter, because regardless you are summing over the entire domain. – Mark Fantini Jul 21 '14 at 13:52
• Your mapping takes a point in the domain to the integral of the function $f: \Gamma \to \Bbb R$ (which does not depend on the domain, so far) over the surface. What does it mean? – Mark Fantini Jul 21 '14 at 13:53

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: $dS_\gamma(u,v)$.
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.