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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 Levi-Civita and Gregorio Ricci-Curbastro, 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 multi-dimensional array of numerical values. The coordinate-independence 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 0-dimensional array, so it is sufficient to represent a scalar, a 0th-order tensor. A coordinate vector, or 1-dimensional array, can represent a vector, a 1st-order tensor. A 2-dimensional array, or square matrix, is then needed to represent a 2nd-order tensor. In general, an order-k tensor can be represented as a k-dimensional 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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