We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and vector fields that are independent of angles. We formulate a generalization of vector calculus for rotationally covariant scalar and vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.
Applications of local fractional calculus to engineering in fractal time-space: Local fractional differential equations with local fractional derivative
Topological properties and a multiplicative Bloch-Floquet-Zak transform for scattering by self-similar or fractal media
ETH Zürich（苏黎世联邦理工学院）；Hong Kong Institute for Advanced Study, City University of Hong Kong（香港城市大学高等研究院）；University of California, Riverside（加州大学河滨分校）；City University of Hong Kong（香港城市大学）
