A tensor is the mathematical idealization of a geometric or physical quantity whose analytic description, relative to a fixed frame of reference, consists of an array of numbers11http://aux.planetmath.org/files/objects/3112/tensor-pipe.jpg“Ceci n’est pas une pipe,” as Rene Magritte put it. The image and the object represented by the image are not the same thing. The mass of a stone is not a number. Rather the mass can be described by a number relative to some specified unit mass.. Some well known examples of tensors in geometry are quadratic formsMathworldPlanetmath, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor, and the polarization tensor.

Geometric and physical quantities may be categorized by considering the degrees of freedom inherent in their description. The scalar quantities are those that can be represented by a single number — speed, mass, temperature, for example. There are also vector-like quantities, such as force, that require a list of numbers for their description. Finally, quantities such as quadratic forms naturally require a multiply indexed array for their description. These latter quantities can only be conceived of as tensors.

Actually, the tensor notion is quite general, and applies to all of the above examples; scalars and vectors are special kinds of tensors. The feature that distinguishes a scalar from a vector, and distinguishes both of those from a more general tensor quantity is the number of indices in the representing array. This number is called the rank of a tensor. Thus, scalars are rank zero tensors (no indices at all), and vectors are rank one tensors.

It is also necessary to distinguish between two types of indices, depending on whether the corresponding numbers transform covariantly or contravariantly relative to a change in the frame of reference. Contravariant indices are written as superscripts, while the covariant indices are written as subscripts. The valencePlanetmathPlanetmathPlanetmath of a tensor is the pair (p,q), where p is the number contravariant and q the number of covariant indices, respectively.

It is customary to represent the actual tensor, as a stand-alone entity, by a bold-face symbol such as 𝖠. The corresponding array of numbers for a type (p,q) tensor is denoted by the symbol Aj1jqi1ip, where the superscripts and subscripts are indices that vary from 1 to n. This number n, the range of the indices, is called the dimensionPlanetmathPlanetmath of the tensor. The total degrees of freedom required for the specification of a particular tensor is the productPlanetmathPlanetmath of the tensor’s rank and its dimension.

Again, it must be emphasized that the tensor 𝖠 and the representing array Aj1jpi1iq are not the same thing. The values of the representing array are given relative to some frame of reference, and undergo a linear transformation when the frame is changed.

Finally, it must be mentioned that most physical and geometric applications are concerned with tensor fields, that is to say tensor valued functions, rather than tensors themselves. Some care is required, because it is common to see a tensor field called simply a tensor. There is a differencePlanetmathPlanetmath, however; the entries of a tensor array Aj1jpi1iq are numbers, whereas the entries of a tensor field are functions. The present entry treats the purely algebraicMathworldPlanetmath aspect of tensors. Tensor field concepts, which typically involved derivatives of some kind, are discussed elsewhere.


The formal definition of a tensor quantity begins with a finite-dimensional vector spaceMathworldPlanetmath U, which furnishes the uniform “building blocks” for tensors of all valences. In typical applications, U is the tangent space at a point of a manifold; the elements of U represent velocities and forces. The space of (p,q)-valent tensors, denoted here by 𝒯p,q(U) is obtained by taking the tensor product of p copies of U, and q copies of the dual vector space U*. To wit,

𝒯p,q(U)=UUp timesU*U*q times.

In order to represent a tensor by a concrete array of numbers, we require a frame of reference, which is essentially a basis of U, say e1,,enU. Every vector in U can be “measured” relative to this basis, meaning that for every 𝐯U there exist unique scalars vi, such that (note the use of the Einstein summation convention)


These scalars are called the componentsPlanetmathPlanetmathPlanetmath of 𝐯 relative to the frame in question.

Let ε1,,εnU* be the corresponding dual basisMathworldPlanetmath, i.e.,


where the latter is the Kronecker deltaMathworldPlanetmath array. For every covector αU* there exists a unique array of components αi such that


More generally, every tensor 𝖠𝒯p,q(U) has a unique description in terms of components. That is to say, there exists a unique array of scalars Aj1jpi1iq such that


Transformation rule.

Next, suppose that a change is made to a different frame of reference, say e^1,,e^nU. Any two frames are uniquely related by an invertiblePlanetmathPlanetmathPlanetmath transition matrix Xji, having the property that for all values of j we have

e^j=Xjiei. (1)

Let 𝐯U be a vector, and let vi and v^i denote the corresponding component arrays relative to the two frames. From


and from (1) we infer that

v^i=Yjivj, (2)

where Yji is the matrix inverse of Xji, i.e.,


Thus, the transformation rule for a vector’s components (2) is contravariant to the transformation rule for the frame of reference (1). It is for this reason that the superscript indices of a vector are called contravariant.

To establish (2), we note that the transformation rule for the dual basis takes the form


and that




The transformation rule for covector components is covariant. Let αU* be a given covector, and let αi and α^i be the corresponding component arrays. Then


The above relationMathworldPlanetmath is easily established. We need only remark that


and that


and then use (1).

In light of the above discussion, we see that the transformation rule for a general typePlanetmathPlanetmath (p,q) tensor takes the form

Title tensor
Canonical name Tensor
Date of creation 2013-03-22 12:47:46
Last modified on 2013-03-22 12:47:46
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 15
Author rmilson (146)
Entry type Definition
Classification msc 15A69
Related topic TensorProduct
Related topic TensorArray
Defines valence
Defines rank