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Tensors are geometric entities introduced into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices to higher orders. Tensors were first conceived by Tullio LeviCivita and Gregorio RicciCurbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.^{[1]}
Many physical quantities are naturally regarded not as vectors themselves, but as correspondences between one set of vectors and another. For example, the stress tensor T takes a direction v as input and produces the stress T^{(v)} on the surface normal to this vector as output and so expresses a relationship between these two vectors. Because they express a relationship between vectors, tensors themselves are independent of a particular choice of coordinate system. It is possible to represent a tensor by examining what it does to a coordinate basis or frame of reference; the resulting quantity is then an organized multidimensional array of numerical values. The coordinateindependence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one.
The order (or degree) of a tensor is the dimensionality of the array needed to represent it. A number is a 0dimensional array, so it is sufficient to represent a scalar, a 0thorder tensor. A coordinate vector, or 1dimensional array, can represent a vector, a 1storder tensor. A 2dimensional array, or square matrix, is then needed to represent a 2ndorder tensor. In general, an orderk tensor can be represented as a kdimensional array of components. The order of a tensor is the number of indices necessary to refer unambiguously to an individual component of a tensor.
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