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:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

-\int_{A}a_0\nabla_nbdA = \int_{V}(a\nabla^2b + \nabla{a}\cdot\nabla{b})dV

[/tex]

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

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# Divergence Theorem/Surface Gradient

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