Dirac tensor distributions for moving submanifolds of R n

Abstract

This paper considers nonclassical fields (tensor distributions) of the form τδΩ̃, where δΩ̃ is the Dirac delta function for a moving p-dimensional submanifold of Rn (0≤ p≤n). The density τ is a classical (smooth), rank-k tensor field on Rn+1. The main result of the paper is the development of formulas for the distributional derivatives of such fields. The derivatives considered are the absolute differential (Levi-Civita connection), the covariant derivative along a given vector field, the divergence operator, the exterior differential, and the exterior codifferential. The resulting derived fields are shown to reflect the underlying geometry of the submanifold Ω as well as the nature of its motion. In the special case p=n, it is seen that the jump conditions on fields at the boundary of the region Ω arise naturally from the distributional calculus.