Vector differential operators in a fractional dimensional space, on fractals, and in fractal continua

This paper is devoted to the development of local vector calculus in fractional-dimensional spaces, on fractals, and in fractal continua. We conjecture that in the space of non-integer dimension one can define two different del-operators acting on the scalar and vector fields respectively. The basic vector differential operators and Laplacian in the fractional-dimensional space are expressed in terms of two del-operators in a conventional way. Likewise, we construct Laplacian and vector differential operators associated with F^α-derivatives on fractals. The conjugacy between F^α and ordinary derivatives allow us to map the vector differential operators on the fractal domain onto the vector differential calculus in the corresponding fractal continuum. These results provide a novel tool for modeling physical phenomena in complex systems.