1. Jun 23, 2014

### Parmenides

There is a paper in chemical physics by Overbeek in which he describes the electrostatic energy of a double layer as the "energy of the surface charges and bulk charges in a potential field"; the transformation that he provides appears to be a variant of the divergence theorem in which he introduces scalar fields $a$ and $b$ that vanish at infinity, having values $a_0$ and $b_0$, respectively, at a surface $A$. The transformation is presented as:
$$-\int_{A}a_0\nabla_nbdA = \int_{V}(a\nabla^2b + \nabla{a}\cdot\nabla{b})dV$$
Where he states that "$\nabla_n$ is directed from the surfaces into the volume V". I have never encountered the gradient as being taken with respect to a normal $n$ and so this transformation is a bit perplexing. The only thing I have been able to go on is the rare definition of the so-called 'surface gradient' on wikipedia where we have $\nabla_S{u} = \nabla{u} - \hat{n}(\hat{n}\cdot{\nabla{u}})$ for some scalar $u$ on a surface $S$, but this does not seem to be directly comparable. I can't seem to use the usual definition of the unit normal as $\hat{n} = \frac{\nabla{a}}{\|a\|}$ because $a$ (or $b$) is not equal to 0.

I seem to have forgotten some concepts in multivariate calculus. Could someone provide some clarity? The paper can be found at http://www.sciencedirect.com/science/article/pii/016666229080132N

2. Jun 23, 2014

### Parmenides

Solved. Sort of.

This is actually Green's first identity, with $\nabla_n{a}$ being the normal derivative of the scalar function $a$. As this is in the mathematics section and not physics, I won't bother with the extension to electric fields and potential.