标题： 非整数维空间中的向量微积分及其在分形介质中的应用 显示英文标题

标题： Vector Calculus in Non-Integer Dimensional Space and its Applications to Fractal Media

摘要： 我们建议对非整数维空间的情况进行向量微积分的推广。 一阶和二阶运算，如梯度、散度、标量和向量拉普拉斯算子在非整数维空间中被定义。 为了简化，我们考虑与角度无关的标量和矢量场。 我们提出了对旋转协变标量和矢量函数的向量微积分的推广。 这种推广使我们能够在非整数维空间的连续介质模型框架内描述分形介质和材料。 作为所建议微积分的应用示例，我们考虑分形材料的弹性（分形空心球和内部和外部有压力的分形圆柱形管道）、分形介质中的稳态热分布、分形带电圆柱体的电场。 我们求解了非整数维空间模型的相关方程。 显示英文摘要

摘要： 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.