tensor
Overview
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 forms, 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 valence
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
Ai1…ipj1…jq, where the superscripts and
subscripts are indices that vary from 1 to n. This number n, the
range of the indices, is called the dimension of the tensor. The
total degrees of freedom required for the specification of a
particular tensor is the product
of the tensor’s rank and its dimension.
Again, it must be emphasized that the tensor 𝖠 and the representing array Ai1…iqj1…jp 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 difference, however; the entries of a tensor array
Ai1…iqj1…jp 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
involved derivatives of some kind, are discussed elsewhere.
Definition.
The formal definition of a tensor quantity begins with a
finite-dimensional vector space 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)=p times⏞U⊗…⊗U⊗q times⏞U*⊗…⊗U*. |
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,…,en∈U. Every vector in U can be “measured” relative to this basis, meaning that for every 𝐯∈U there exist unique scalars vi, such that (note the use of the Einstein summation convention)
𝐯=viei. |
These scalars are called the components of 𝐯 relative to the frame
in question.
Let ε1,…,εn∈U* be the corresponding dual
basis, i.e.,
εi(ej)=δij, |
where the latter is the Kronecker delta array. For every covector
α∈U* there exists a unique array of components αi such
that
α=αiεi. |
More generally, every tensor 𝖠∈𝒯p,q(U) has a unique description in terms of components. That is to say, there exists a unique array of scalars Ai1…iqj1…jp such that
𝖠=Ai1…iqj1…jpei1⊗…⊗eiq⊗εj1⊗…⊗εjp. |
Transformation rule.
Next, suppose that a change is made to a different frame of
reference, say
ˆe1,…,ˆen∈U.
Any two frames are uniquely related by
an invertible transition matrix Xij, having the property that for
all values of j we have
ˆej=Xijei. | (1) |
Let 𝐯∈U be a vector, and let vi and ˆvi denote the corresponding component arrays relative to the two frames. From
𝐯=viei=ˆviˆei, |
and from (1) we infer that
ˆvi=Yijvj, | (2) |
where Yij is the matrix inverse of Xij, i.e.,
XikYkj=δij. |
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
ˆεi=Yijεj, |
and that
vi=εi(𝐯), |
while
ˆvi=ˆεi(𝐯). |
The transformation rule for covector components is covariant. Let α∈U* be a given covector, and let αi and ˆαi be the corresponding component arrays. Then
ˆαj=Xijαi. |
The above relation is easily established. We need only remark that
αi=α(ei), |
and that
ˆαj=α(ˆej), |
and then use (1).
In light of the above discussion, we see that the transformation rule
for a general type (p,q) tensor takes the form
ˆAi1…iqj1…jp=Xi1k1⋯XiqkqYl1j1⋯Yl1jpAk1…kql1…lp. |
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 |