Vector calculus in non-integer dimensional space and its applications to fractal media

• Vector calculus for non-integer dimensional space is suggested.
• The gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are defined.
• Continuum models with non-integer dimensional spaces for fractal media are suggested.
• Equations for elasticity of fractal hollow ball and fractal cylindrical pipe with pressure inside and outside are solved.
• Steady distribution of heat in fractal media, and electric field of fractal charged cylinder are described.

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.