A tensor of order n \((\in \{0,1,2,3,...\})\) in Ttensor of order three-dimensional space is defined as a mathematical entity consisting of \(3^n\) components. The numerical values of these components can vary depending on the coordinate system used to represent the tensor. These \(3^n\) numbers are called the scalar components of the tensor within the chosen coordinate system. Although these scalar components generally depend on the chosen coordinate system for their representation, the tensor itself—as a mathematical entity and as a representation of a physical quantity—remains independent of the coordinate system used. In other words, the tensor itself is invariant to the choice of the coordinate system used to express it. This property of invariance is ensured by a set of relationships between the \(3^n\) components of the tensor in one coordinate system and the corresponding set of \(3^n\) components in another coordinate system. These relationships ensure that despite the different scalar representations, the tensor remains consistent across all coordinate systems. Such a relationship is called the transformation rule of all tensors of order n. A tensor of order zero is a special case wherein the transformation rule is trivial. Such a tensor is described by one ( \(3^0\) ) number, which is independent of the choice of the working coordinate system.

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Tensors

  • Sawan S. Sinha

摘要

A tensor of order n \((\in \{0,1,2,3,...\})\) in Ttensor of order three-dimensional space is defined as a mathematical entity consisting of \(3^n\) components. The numerical values of these components can vary depending on the coordinate system used to represent the tensor. These \(3^n\) numbers are called the scalar components of the tensor within the chosen coordinate system. Although these scalar components generally depend on the chosen coordinate system for their representation, the tensor itself—as a mathematical entity and as a representation of a physical quantity—remains independent of the coordinate system used. In other words, the tensor itself is invariant to the choice of the coordinate system used to express it. This property of invariance is ensured by a set of relationships between the \(3^n\) components of the tensor in one coordinate system and the corresponding set of \(3^n\) components in another coordinate system. These relationships ensure that despite the different scalar representations, the tensor remains consistent across all coordinate systems. Such a relationship is called the transformation rule of all tensors of order n. A tensor of order zero is a special case wherein the transformation rule is trivial. Such a tensor is described by one ( \(3^0\) ) number, which is independent of the choice of the working coordinate system.