- Intermediate treatment of tensors
In
mathematics andphysics , a tensor is an idealized geometric orphysical quantity whose numerical description, relative to a particularframe of reference , consists of a multiple indexedarray of numbers. A vector, for example, is a tensor with a single index; thus, tensors can be regarded as a multi-index generalization of the vector concept. Just as it is with vectors,a change of reference frame induces a transformation of the components.This way of viewing tensors, called tensor analysis, was used by Einstein and is generally preferred by physicistsFact|date=April 2008. It is, very grossly, a generalization of the concept of vectors, matrices, and
linear transformation s, and allows the writing ofequation s independently of any givencoordinate system .Overview
Tensor quantities may be categorized by considering the number of indices inherent in their description. The scalar quantities are those that can be represented by a single number (indices are not needed) —
speed ,mass ,temperature , for example. There are also vector-like quantities such asforce that require a list of numbers for their description (one index is required so that direction can be accounted for). Finally, quantities such as quadratic forms naturally require a multiply-indexed array for their representation. These latter quantities can only be conceived of as tensors. Some well known examples of tensors in geometry arequadratic form s, and thecurvature tensor . Examples of physical tensors are theenergy-momentum tensor and thepolarization tensor .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 (or the order) of a tensor. Thus, scalars are rank zero tensors (with no indices at all) and vectors are rank one tensors.
It should be noted that the array-of-numbers representation of a tensor "is not the same thing as" the tensor. An image and theobject 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 anumber relative to some specified unit mass. Similarly, a given numerical representation of a tensor only makes sense in a particular coordinate system.
It is also necessary to distinguish between two types of indices, depending on whether the corresponding numbers transform
covariant ly orcontravariant ly relative to a change in the frame of reference. Contravariant indices are written as superscripts, while the covariant indices are written as subscripts. The type (or valence) of a tensor is the pair , where is the number of contravariant and the number of covariant indices, respectively. Note that a tensor of type has a rank of "p + q". It is customary to represent the actual tensor, as a standalone entity, by a bold-face symbol such as . The corresponding array of numbers for a type tensor is denoted by the symbol where the superscripts andsubscripts are indices that vary from to . The number , therange of the indices, is called the dimension of the tensor; the total number of degrees of freedom required for the specification of a particular tensor is the dimension of the tensor raised to the power of the tensor's rank.Again, it must be emphasized that the tensor and the representing array 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 field s, 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 difference, however; the entries of a tensor array are numbers, whereas the entries of a tensor field are functions. The present entry treats the purely algebraic aspect of tensors. Tensor field concepts, which typically involve derivatives of some kind, are discussed elsewhere.Definition
The formal definition of a tensor quantity begins with a finite-dimensional vector space , which furnishes the uniform "building blocks" for tensors of all valences. In typical applications, is the
tangent space at a point of amanifold ; the elements of typically represent physical quantities such as velocities or forces. The space of -valent tensors, denoted here by is obtained bytaking thetensor product of copies of and copies of thedual vector space . To wit,:In order to represent a tensor by a concrete array of numbers, werequire a frame of reference, which is essentially a basis of ,sayEvery vector in can be"measured" relative to this basis, meaning that for every there exist unique scalars , suchthat (note the use of the
Einstein notation ):These scalars are called the components of relative to the frame in question.
Let be the corresponding
dual basis , i.e.,:where the latter is theKronecker delta array. For every covector there exists a unique array of components suchthat:More generally, every tensor has a unique representation in terms of components. That is to say, there exists aunique array of scalars such that:
Transformation rules
Next, suppose that a change is made to a different frame of reference, sayAny two frames are uniquely related by an invertible transition matrix , having the property that for all values of we have the "frame transformation rule":
Let be a vector, and let and denote the corresponding component arrays relative tothe two frames. From:and from the frame transformation rule we infer the "vector transformation rule":
where is the
matrix inverse of , i.e.,:Thus, the transformation rule for a vector's components is contravariant to the transformation rule for the frame of reference. It is for this reason that the superscript indices of a vector are called contravariant.To establish the transformation rule for covectors, we note that the transformation rule for the dual basis takes the form:and that:while:
The transformation rule for covector components is covariant. Let be a given covector, and let and be the corresponding component arrays. Then:The above relation is easily established. We need only remark that:and that:and then use the transformation rule for the frame of reference.
In light of the above discussion, we see that the transformation rulefor a general type tensor takes the form:
ee also
*
tensor product
*tensor derivative
*absolute differentiation
*curvature
*Riemannian geometry Further reading
* Bernard Schutz, "Geometrical methods of mathematical physics", Cambridge University Press, 1980.
* "Schaum's Outline of Tensor Calculus"
* Synge and Schild, "Tensor Calculus", Toronto Press: Toronto, 1949
* James Munkres, "Analysis on Manifolds," Westview Press, 1991. Chapter six gives a "from scratch" introduction to covariant tensors.----
"An earlier version of this article was adapted from the GFDL article on tensors at http://planetmath.org/encyclopedia/Tensor.html from
PlanetMath , written by Robert Milson and others"
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