Second Order Linear Differential Operators over High Rank Tensor Fields

Abstract

Tensor fields of arbitrary rank in multidimensional space are naturally extended to the concepts of divergence, gradient, curl, deformer operators, as well as their second-order superpositions. Two options for generalizing the rotor as an external product are presented. Differential operators of the second order that do not change the rank of the tensor to which they are applied are considered in detail. Square matrices are introduced, consisting of differential operators $${\text{Di}}{{{\text{v}}}_{{(l)}}}{\text{Gra}}{{{\text{d}}}_{{(k)}}}$$ , $${\text{Gra}}{{{\text{d}}}_{{(k)}}}{\text{Di}}{{{\text{v}}}_{{(l)}}}$$ , and their relationship is established. An explicit expression is written for the repeated operator rotor. All introduced generalized operators in particular cases agree in their properties with the corresponding classical operators in vector and tensor analysis.