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.

中文翻译：

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高阶张量场上的二阶线性微分算子

摘要-

多维空间中任意秩的张量场自然地扩展到散度，梯度，卷曲，变形算子及其二阶叠加的概念。提出了两种将转子概括为外部产品的选择。详细考虑不改变其张量等级的二阶微分算子。引入了由微分运算符\（{\ text {Di}} {{{\ text {v}}} __ {{（l）}}} {\ text {Gra}} {{{\ text { d}}} _ {{（（k）}}} \），\（{\ text {Gra}} {{{\ text {d}}} _ {{（k）}}} {\ text {Di} } {{{\ text {v}}} _ {{（l）}}} \），并建立了他们的关系。为重复的操作器转子编写一个明确的表达式。在特定情况下，所有引入的广义算子在矢量和张量分析中的性质都与相应的经典算子一致。