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 Fα-derivatives on fractals. The conjugacy between Fα 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. •In non-integer dimension space del-operator acting on vector fields differs from del-operator acting on scalar fields.•Basic vector differential operators in fractional dimensional space are expressed in terms of two del-operators.•Fα-calculus is generalized for fractals with the non-integer number of effective spatial degrees of freedom.•A novel version of vector differential calculus in the fractal continuum is developed.