Stokes' theorem and alike for fractal surfaces

Has anyone ever read an article dealing with the extension of integral theorems to fractal surfaces ?

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That sounds like an interesting topic, but it's not really a question and I think it doesn't even fit more into physics than mathematics. I wonder if there is a suitable measure/differential/function algebra for such objects, that's what I would ask? Could you elaborate more how you get at this or what you already know about the topic? Maybe that's a thing and you want specific information. Or maybe you just want ideas. Related thread here, also the first paper if you google "fractal surface" integral. – NikolajK Jul 18 '12 at 9:10
I am thinking about all the integral theorems (Stokes, Ostrogradsky, etc..) used in EM. I am just curious about what would be the emf induced by a time varying B field through a fractal loop wire (a Koch snow flake for instance). The problem is interesting since the boundary has an infinite lenght but the enclosed aera is finite. So I can reformulate the question to: what is the emf induced by a time varying B field through a Koch snowflake ? – Shaktyai Jul 18 '12 at 9:17
You should look at discrete Laplacians. The mathematics involved in going to the fractal continuum limit is definitely physics, since the mathematicians haven't sorted it out (it's the same thing as rigorous QFT). The analog of the integral theorems is the summation result from finite difference calculus, and the limiting identities they give. This is not a complete answer, but it should get you started. – Ron Maimon Jul 18 '12 at 9:34
Ah, I see, might have an interesting answer. From the physical side though, I expect you will most likely not be able to handle the problem with classical EM. Assuming the most most uncomplicated charge density (a constant one, since in each other cases you'd probably have to introduce another lenght scale than the wire length), you'd also compute an infinite induced charge. Right? – NikolajK Jul 18 '12 at 9:37

1 Answer

This problem has been treated by Jenny Harrison, please see the article:

For the special case of the Gauss-Green theorems, please see the following article with Alec Norton

Prof. Harrison has more recent work concerning differential calculus on fractals, please see these lecture notes.

Please see also the replies on a similar question on mathoverflow

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I have indeed found a paper, math.berkeley.edu/~harrison/research/publications/flux.pdf but I am not sure it is of any use for practical computation. From a physical point of vue, my first idea would be to find two regular bounding curves to the fractal domain and to compute the induced emf for each one to get a bounding interval for the emf. However the infinite length of the fractal boundary is suspicious and may as well give a zero induced emf. – Shaktyai Jul 18 '12 at 11:49
The main idea in these works is that under some rather mild constraints on the types of functions to be integrated, the integration over fractals can be obtained by a limiting procedure of integrals over polyhedral approximations. Now, Maxwell equations are correct in their integral form: Since quantities like total charges and total currents should be finite, this would require their densities in some cases to have fractal dimensions. The EMF is an integral quantity, thus its calculation from the integral form of the Faraday's should not change. – David Bar Moshe Jul 18 '12 at 15:34