# Mathematics of Surface Divergence and Surface Curl

While studying electrodynamics I found two functions - Surface Divergence and Surface Curl - that seem to condense the formulas for superficial discontinuities of the electric and magnetic fields rather nicely. I found them on my professor's notes, defined this way:

$$surf \ curl \ \vec F = \hat n \times (\vec F\ ^+ - \vec F\ ^-) \\ surf \ div \ \vec F = \hat n \cdot (\vec F\ ^+ - \vec F\ ^-)$$

With $\vec F\ ^+$ and $\vec F\ ^-$ the fields above and below the surface, so that, for example, if $\sigma _b$ represents surface bound charge, and $\vec P$ the polarization vector,

$$surf \ div \ \vec P = -\sigma _b$$

The same can be applied for other fields.

Obviously, this is not a formal definition, and I would like to better understand the mathematics behind these functions. Unfortunately, when I asked my professor about this, she told me she was taught this years ago when she was studying, and she couldn't explain me the formalism behind it. I also couldn't find any information on the internet about "surface divergence" or "surface curl".

The name seems suggestive, and I can roughly understand the idea behind it, for (without rigour), we could say, using gauss's law for electrostatics:

$$\frac{d}{dA} \iint_S \vec E\ dS = \frac{d}{dA}(\vec E\ ^+ \cdot \vec A\ ^+ + \vec E\ ^- \cdot \vec A\ ^-)=\hat n \cdot (\vec E\ ^+ - \vec E\ ^-)$$

With $\hat n$ pointing in the same direction as $\vec E\ ^+$. The parallel is this:

$$div \ \vec E=\frac{d}{dV} \iint_S \vec E\ dS \\ surf \ div \ \vec E=\frac{d}{dA} \iint_S \vec E\ dS$$

This is my question then: Are there any books or websites were I can learn more about the formalism of surface curl and divergence? Maybe someone can shed a light on this?

• Your formulas are defined in N.Kemmer's "Vector Analysis" book, Ch.11 - "Boundary behavior of fields", Sec.1 - "Surface discontinuities". See books.google.com/…. There is also a very elegant way to do it with surface delta functions, but can't find the refs. and am kinda rusty on those unfortunately. Maybe somebody else knows of it? – udrv Dec 30 '15 at 3:57
• You can regulate the discontinuity across the surface by giving the surface 'finite but small' thickness $\epsilon$, and then consider the curl and divergence of the vector fields in that regularised surface. After that, you can consider the limit of zero thickness. The rigorous treatment of surface curl and divergence is more or less the same as rigorous treatment of the Dirac delta. – Dexter Kim Jan 5 '16 at 23:23
• Dexter Kim: yes, I've seen that approach in Griffiths. But that's not the question I'm asking. I know how to deal with discontinuity, but I wanted to know more about the above functions. – J. D. Simão Jan 6 '16 at 1:02