NationMaster - Encyclopedia: Intermediate treatment of tensors

Note
The following is a modern component-based treatment of tensors (sometimes called the "classical treatment" of tensors). Read the article tensor for a simple description of tensors, or see the component-free treatment of tensors for a more abstract treatment. For an even more traditional approach, see classical treatment of tensors. Note that the word "tensor" is often used as a shorthand for "tensor field", a concept which defines a tensor value at every point in a manifold. To understand tensor fields, you need to first understand tensors.

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 numbers. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Note: This is a fairly abstract mathematical approach to tensors. ... The following is a component-based classical treatment of tensors. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Table of Geometry, from the 1728 Cyclopaedia. ... A physical quantity is either a quantity within physics that can be measured (e. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

This way of viewing tensors, called tensor analysis, was used by Einstein and is generally preferred by physicists. It is, very grossly, a generalization of the concept of vectors and matrices and allows the writing of equations independently of any given coordinate system. Einstein redirects here. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... For the square matrix section, see square matrix. ... An equation is a mathematical statement, in symbols, that two things are the same. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

1 Overview
2 Definition
3 Transformation rules
4 See also
5 Further reading

Overview

It should be noted that the array-of-numbers representation of a tensor is not the same thing as the tensor. An 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. Similarly, a given numerical representation of a tensor only makes sense in a particular coordinate system.

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. In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ...

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 (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. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... Speed is the rate of motion, or equivalently the rate of change of position, many times expressed as distance d moved per unit of time t. ... Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Fig. ... In physics, force is an influence that may cause a body to accelerate. ...

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. In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ...

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 type (or valence) of a tensor is the pair $(p, q)$ , where $p$ is the number of contravariant and $q$ the number of covariant indices, respectively. Note that a tensor of type $(p, q)$ 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 $mathbf{T}$ . The corresponding array of numbers for a type $(p, q)$ tensor is denoted by the symbol $T^{i_1ldots i_p}_{j_1ldots j_q},$ where the superscripts and subscripts are indices that vary from $1$ to $n$ . The number $n$ , the range 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. In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

Again, it must be emphasized that the tensor $mathbf{T}$ and the representing array $T^{i_1ldots i_q}_{j_1ldots j_p}$ 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 $T^{i_1ldots i_q}_{j_1ldots j_p}$ 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. In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... Partial plot of a function f. ...

Definition

The formal definition of a tensor quantity begins with a finite-dimensional vector space $mathcal{U}$ , which furnishes the uniform "building blocks" for tensors of all valences. In typical applications, $mathcal{U}$ is the tangent space at a point of a manifold; the elements of $mathcal{U}$ typically represent physical quantities such as velocities or forces. The space of $(p, q)$ -valent tensors, denoted here by $mathcal{U}^{p,q}$ is obtained by taking the tensor product of $p$ copies of $mathcal{U}$ and $q$ copies of the dual vector space $mathcal{U}^*$ . To wit, The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

$mathcal{U}^{p,q} = left{mathcal{U}otimesldotsotimesmathcal{U}right} otimes left{mathcal{U}^*otimesldotsotimesmathcal{U}^*right}$

In order to represent a tensor by a concrete array of numbers, we require a frame of reference, which is essentially a basis of $mathcal{U}$ , say $mathbf{e}_1,ldots,mathbf{e}_n in mathcal{U}.$ Every vector in $mathcal{U}$ can be "measured" relative to this basis, meaning that for every $mathbf{v}inmathcal{U}$ there exist unique scalars $v i$ , such that (note the use of the Einstein notation) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

$mathbf{v} = v^imathbf{e}_i$

These scalars are called the components of $mathbf{v}$ relative to the frame in question.

Let $varepsilon^1,ldots,varepsilon^ninmathcal{U}^*$ be the corresponding dual basis, i.e., In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. ...

$varepsilon^i(mathbf{e}_j) = delta^i {}_j,$

where the latter is the Kronecker delta array. For every covector $mathbf{alpha}inmathcal{U}^*$ there exists a unique array of components $α i$ such that In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

$mathbf{alpha} = alpha_i, varepsilon^i.$

More generally, every tensor $mathbf{T}inmathcal{U}^{p,q}$ has a unique representation in terms of components. That is to say, there exists a unique array of scalars $T^{i_1ldots i_p}_{j_1ldots j_q}$ such that

$mathbf{T} = T^{i_1ldots i_p}_{j_1ldots j_q}, mathbf{e}_{i_1} otimes ldotsotimes mathbf{e}_{i_q} otimes varepsilon^{j_1}otimesldotsotimes varepsilon^{j_p}.$

Transformation rules

Next, suppose that a change is made to a different frame of reference, say $hat{mathbf{e}}_1,ldots,hat{mathbf{e}}_ninmathcal{U}.$ Any two frames are uniquely related by an invertible transition matrix $A i j$ , having the property that for all values of $j$ we have the frame transformation rule

$hat{mathbf{e}}_j = A^i {}_j, mathbf{e}_i.$

Let $mathbf{v}inmathcal{U}$ be a vector, and let $v i$ and $hat{v}^i$ denote the corresponding component arrays relative to the two frames. From

$mathbf{v} = v^imathbf{e}_i = hat{v}^ihat{mathbf{e}}_i,$

and from the frame transformation rule we infer the vector transformation rule

$hat{v}^i = B^i {}_j, v^j,$

where $B i j$ is the matrix inverse of $A i j$ , i.e., In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

A i k B k j = δ i j .

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

$hat{v}e^i = B^i {}_j , varepsilon^j,$

and that

$v^i = varepsilon^i(mathbf{v}),$

while

$hat{v}^i = hat{v}e^i(mathbf{v}).$

The transformation rule for covector components is covariant. Let $mathbf{alpha}in mathcal{U}^*$ be a given covector, and let $α i$ and $hat{alpha}_i$ be the corresponding component arrays. Then

$hat{alpha}_j = A^i {}_j alpha_i.$

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

$alpha_i = mathbf{alpha}(mathbf{e}_i),$

and that

$hat{alpha}_j = mathbf{alpha}(hat{mathbf{e}}_j),$

and then use the transformation rule for the frame of reference.

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

$hat{T}^{i_1ldots i_q}_{,j_1ldots j_p} = A^{i_1} {}_{k_1}cdots A^{i_q} {}_{k_q} B^{l_1} {}_{j_1}cdots B^{l_1} {}_{j_p} T^{k_1ldots k_q}_{l_1ldots l_p}.$

Contents

Overview GA_googleFillSlot("encyclopedia_square");

Definition

Transformation rules

See also

Further reading

Overview