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The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner ...
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry , algebraic geometry , general relativity , in the analysis of stress and strain in material object, and ...
[a] [1] [2] [3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. [4]
The operation of tensor contraction generalizes the trace to arbitrary tensors. Gomme and Klein (2011) define a matrix trace operator trm {\displaystyle \operatorname {trm} } that operates on block matrices and use it to compute second-order perturbation solutions to dynamic economic models without the need for tensor notation .
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. [1] There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig [2] of old ideas of Lord Kelvin ...
The earliest foundation of tensor theory – tensor index notation. [1] Order of a tensor The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank. Rank of a tensor The rank of a tensor is the minimum ...
In mathematics, a tensor is a certain kind of geometrical entity and array concept. It generalizes the concepts of scalar, vector and linear operator, in a way that is independent of any chosen frame of reference. For example, doing rotations over axis does not affect at all the properties of tensors, if a transformation law is followed.
If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order k {\displaystyle k} may be referred to as a differential k {\displaystyle k} -form , and a completely antisymmetric contravariant tensor field may be referred ...