Vector calculus on fractals ‒ an application to antenna theory and fractal electrodynamics

This article develops a simple but rigorous approach towards the definition of vector calculus on fractal curves, aimed at defining contour integrals over fractal curves, defined as limit set of a suitable Iterated Function System (IFS). The analysis developed here represents a generalization of the approach developed by Giona (1999) and subsequently addressed by Mendivil and Vrscay (2002). This approach is based on the definition of an Augmented IFS (AIFS) and permits the introduction of tangential measures νx(s), νy(s) as a fixed point of a contractive operator associated with the AIFS. This formulation is applied to a classical problem in electrodynamics and antenna theory, namely the solution of the generalized Pocklington equation for wire antennas.