Electromagnetism on Anisotropic Fractals

Understand

We derive basic equations of electromagnetic fields in fractal media which are specified by three indepedent fractal dimensions {\alpha}_{i} in the respective directions x_{i} (i=1,2,3) of the Cartesian space in which the fractal is embedded.

• To grasp the generally anisotropic structure of a fractal, we employ the product measure, so that the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving the {\alpha}_{i}'s.
• First, a formulation based on product measures is shown to satisfy the four basic identities of vector calculus.
• This allows a generalization of the Green-Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals.