tensor

For a gentle introduction to the nature and significance of tensors, see the tensor article. This article is at a more advanced technical level.

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.

In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally.

Note: This article requires an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article, and other treatments of tensors include the classical and intermediate level.

Definition via tensor products of vector spaces

Given a finite set { V₁, ... , V_n } of vector spaces over a common field F. One may form their tensor product V₁ ⊗ ... ⊗ V_n. An element of this tensor product is referred to as a tensor (but this is not the notion of tensor discussed in this article).

A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:

where V* is the dual space of V.

If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and covariant of order m and contravariant order n and total order m+n. The tensors of order zero are just the scalars (elements of the field F), those of covariant order 1 are the vectors in V, and those of contravariant order 1 are the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m,n) is denoted

The (1,1) tensors

are isomorphic in a natural way to the space of linear transformations from V to V. An inner product of a real vector space V; V × V → R corresponds in a natural way to a (0,2) tensor in

called the associated metric and usually denoted g.

Tensor rank

The term rank of a tensor is often used interchangeably with the order (or degree) of a tensor. However, it is also used in a different unrelated sense that extends the notion of the rank of a matrix given in linear algebra.

A tensor of rank 1 (also called a simple tensor) is a tensor that can be written as a tensor product of the form

where a, b,...,d are in V or V*. That is, if the tensor is completely factorizable. In indices, a tensor of rank 1 is a tensor of the form

Every tensor can be expressed as a linear combination of rank 1 tensors. The rank of a general tensor T is defined to be the minimum number of rank 1 tensors with which it is possible to express T as a linear combination (Bourbaki 1988, II, §7, no. 8).

The rank of an order 1 tensor is always 1 (or 0, in the case of the zero tensor). The rank of an order 2 tensor is easy to determine, e.g., using Gaussian elimination, and it agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51). The rank of an order 3 or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987).

Universal property

The space can be characterized by a universal property in terms of multilinear mappings. A scalar-valued function on a Cartesian product (or direct sum) of vector spaces

is multilinear if it is linear in each argument. The space of all multlinear mappings from the product V₁×V₂×...×V_N into W is denoted L^N(V₁,V₂,...,V_N; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V;W).

The universal characterization of the tensor product implies that, for each multilinear function

there exists a unique linear function

such that

for all v_i ∈ V and α_i ∈ V^∗.

Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism

In the formula above,the roles of V and V^* are reversed. In particular, one has

and

and

Tensor fields

See main article tensor field

Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is in fact sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point.

Basis

For any given coordinate system we have a basis {e_i} for the tangent space V (this may vary from point-to-point if the manifold is not linear), and a corresponding dual basis {eⁱ} for the cotangent space V* (see dual space). The difference between the raised and lowered indices is there to remind us of the way the components transform.

For example purposes, then, take a tensor A in the space

The components relative to our coordinate system can be written

.

Here we used the Einstein notation, a convention useful when dealing with coordinate equations: when an index variable appears both raised and lowered on the same side of an equation, we are summing over all its possible values. In physics we often use the expression

to represent the tensor, just as vectors are usually treated in terms of their components. This can be visualized as an n × n × n array of numbers. In a different coordinate system, say given to us as a basis {e_i'}, the components will be different. If (x^i'_i) is our transformation matrix (note it is not a tensor, since it represents a change of basis rather than a geometrical entity) and if (yⁱ_i') is its inverse, then our components vary per

In older texts this transformation rule often serves as the definition of a tensor. Formally, this means that tensors were introduced as specific representations of the group of all changes of coordinate systems.

References

• Abraham, Ralph; Marsden, Jerrold E. (1985), Foundations of Mechanics (2 ed.), Reading, Mass.: Addison-Wesley, ISBN 0-201-40840-6 .
• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 .
• de Groote, H. F. (1987), Lectures on the Complexity of Bilinear Problems, Lecture Notes in Computer Science, 245, Springer, ISBN 3-540-17205-X .
• Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0387900934 .