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Divergence Theorem/Surface Gradient

  1. Jun 23, 2014 #1
    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:
    [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
     
  2. jcsd
  3. Jun 23, 2014 #2
    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.
     
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