Tensor calculus

For more technical Wiki articles on tensors, see the section later in this article.

In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference. Tensors are of importance in physics and engineering.

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain the further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components. In particular, tensors behave in special ways under coordinate transformations.

This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail.

Background

The word tensor was introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus. The word was used in its current meaning by Woldemar Voigt in 1899.

The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry, and made accessible to many mathematicians by the publication of Tullio Levi-Civita's classic text The Absolute Differential Calculus in 1900 (in Italian; translations followed). The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General Relativity is formulated completely in the language of tensors, which Einstein had learned from Levi-Civita himself with great difficulty. But tensors are used also within other fields such as continuum mechanics, for example the strain tensor, (see linear elasticity).

Note that the word "tensor" is often used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold. To understand tensor fields, you need to first understand the basic idea of tensors.

The choice of approach

There are two ways of approaching the definition of tensors:

• The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.

• The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors.

Examples

Not all relationships in nature are linear, but most are differentiable and so may be locally approximated with sums of multilinear maps. Thus most quantities in the physical sciences can be usefully expressed as tensors.

As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The acceleration will in general not be in the same direction as the force, because of the particular shape of the ship's body. However, it turns out that the relationship between force and acceleration is linear. Such a relationship is described by a tensor of type (1,1) (that is to say, it transforms a vector into another vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. Just as the numbers which represent a vector will change if one changes the coordinate system, the numbers in the matrix that represents the tensor will also change when the coordinate system is changed.

In engineering, the stresses inside a rigid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i.e. causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.

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 representation. 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.

Approaches, in detail

There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.

• The classical approach

The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the indices of the array.

This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.

The tensor field theory can roughly be viewed, in this approach, as a further extension of the idea of the Jacobian.

• The modern approach

The modern (component-free) approach views tensors initially as abstract objects, 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. This treatment has largely replaced the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector. You could say that the slogan is 'tensors are elements of some tensor space'.

• The intermediate treatment of tensors article attempts to bridge the two extremes, and to show their relationships.

In the end the same computational content is expressed, both ways. See glossary of tensor theory for a listing of technical terms.

Tensor densities

It is also possible for a tensor field to have a "density". A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian to the r^th power. This is best explained, perhaps, using vector bundles: where the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.

Tensor rank

Rank	Alias	Element notation	Common transformation*	Geometrical interpretation
0	Scalar	a	S'=|a|S	·
1	Vector	ai	V'i=|a|aijVj
2	Matrix	aij	M'ij=|a|aikajlMkl
3	?	aijk	M'ijk=|a|ailajsakmMlsm

^* |a| is the determinant of the coefficient array a_mn or its corresponding in the given dimension. Note that quantities that transform according to column 4 are usually called metric tensor

• covariant
• contravariant
• one-form
• tensor product
• tensor derivative
• absolute differentiation
• curvature
• Riemannian geometry
• fibre bundle
• Application of tensor theory in engineering science

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