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In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference. Tensors are of importance in physics and engineering.

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain the further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components. In particular, tensors behave in special ways under coordinate transformations.

This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail.

1 Background

The word tensor was introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus. The word was used in its current meaning by Woldemar Voigt in 1899.

The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry, and made accessible to many mathematicians by the publication of Tullio Levi-CivitaTullio Levi-Civit ( March 29, 1873 December 29, 1941) was an Italian mathematician, most famous for his work on tensor calculus but who also made significant contributions in other areas, some related to this work and some not. He was a pupil of Gregorio's classic text The Absolute Differential Calculus in 1900 (in Italian; translations followed). The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativityGeneral relativity (GR or general relativity theory (GRT is the theory of gravitation published by Albert Einstein in 1915. The conceptual core of general relativity, from which its other consequences largely follow, is the Principle of Equivalence which, around 1915. General Relativity is formulated completely in the language of tensors, which Einstein had learned from Levi-Civita himself with great difficulty. But tensors are used also within other fields such as continuum mechanicsContinuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. liquids and gases). The fact that matter is made of atoms and that it commonly has some sort of heterogeneous micro, for example the strain tensorThe strain tensor [ε] is a symmetric tensor used to quantify the strain of an object undergoing a 3-dimensional deformation: the diagonal coefficients ε are the relative change in length in the direction of the i direction (along the x axi, (see linear elasticityContinuum mechanics Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming "small" deformations. Basic equations Linear elastodynamics is based on three tensor equations: dynamic equation constitu).

Note that the word "tensor" is often used as a shorthand for tensor fieldDifferential geometry Differential topology In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in genera, which is a tensor value defined at every point in a manifold. To understand tensor fields, you need to first understand the basic idea of tensors.

2 The choice of approach

There are two ways of approaching the definition of tensors:

• The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.

• The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors.

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TensorFor more technical Wiki articles on tensors, see the section later in this article. In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independen Tensor productAbstract algebra Algebra In mathematics, the tensor product denoted by , may be applied in different contexts to vectors, matrices, tensors and vector spaces. In each case the significance of the symbol is the same: the most general bilinear operation. Tensor (intrinsic definition)Note: This is a fairly abstract mathematical approach to tensors. If you are baffled by this article, try reading the main tensor article and the classical treatment first. The modern component-free approach to the theory of tensors views tensors initiall Tensor fieldDifferential geometry Differential topology In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in genera Tensor algebraIn mathematics, the tensor algebra is an abstract algebra construction of a unital associative algebra T(V) from a vector space V. In a sense, T(V) is the "most general" algebra containing V. If we take basis vectors for V, those become non-commuting vari Tensor product of fieldsField theory In mathematics, the theory of fields in abstract algebra lacks a direct product: the direct product of two fields, considered as ring (mathematics) is never itself a field. On the other hand it is often required to 'join' two fields K and L, Tensor product of R-algebras