Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, (geometric) vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another. An example is the stress, that takes one vector as input and produces another vector as output and so expresses a relationship between the input and output vectors. Tensors were first conceived by Bernhard Riemann and Elwin Bruno Christoffel, and later developed by Tullio Levi-Civita and Gregorio Ricci-Curbastro, in order to formulate the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
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. Thus a scalar is a zeroth-order tensor: its magnitude is its sole component, so it can be represented as a 0-dimensional array. A vector is a first-order tensor, being representable in coordinates as a 1-dimensional array of components. A matrix is a second-order tensor, being representable as a 2-dimensional array. And so on: an order-k tensor can be represented as a k-dimensional array of components. The order is the number of numerical indices necessary to specify an individual component of a tensor.
The term tensor is slightly ambiguous, and this often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is first of all an element of a tensor product of vector spaces. In physics, the same term often means what a mathematician would call a tensor field: an association of a different mathematical tensor with each point of a geometric space, varying continuously with position. This difference of emphasis conceals the agreement there is on the geometric nature of tensors, and in applications of tensors there may be different types of notation used, for what are actually the same underlying calculations.
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