In mathematics, the modern component-free approach to the theory of tensors 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. Euclid, detail from The School of Athens by Raphael. ... 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. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, multilinear algebra extends the methods of linear 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 references 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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... 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 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... 1) A physical property is an aspect of an object that can be experienced using one of the five human senses without changing its chemical composition: touch, taste, smell, sight or sound, or, in an extended sense, detected through any measuring device. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

Note: This article, which is fairly abstract, requires an understanding of the tensor product of vector spaces without chosen bases. The notion of a tensor product generalizes to vector spaces without chosen bases, and even further, to modules. If you find this article difficult, try reading the main tensor article and the classical treatment first. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... 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. ... The following is a component-based classical treatment of tensors. ...

Definition: Tensor Product of Vector Spaces

Let V and W be two vector spaces over a common field F. Their tensor product Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

is a vector space over the same field F together with a bilinear map In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

which is universal (i.e., the smallest possible without throwing away information) in the following sense:

for every vector space X over the field F and every F-bilinear map

there is a unique F-linear map

such that

It is easy to see that a vector space is unique up to isomorphism if it exists, and we write the instead of a tensor product. All its properties, except its existence, follow from the abstract definition, although some properties are more easily understood from an explicit model. An explicit construction is easy to give using a bases {v_i} and {w_j} respectively. The tensor product can be constructed as the vector space spanned by a basis In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

where in the basis, the symbol is alternatively seen as a formal symbol for forming a pair, and the value of the bilinear map on the basis vectors. The extention of to all of is done in the unique way compatible with bilinearity.

If V and W are both finite dimensional then the dimension of is the product of the dimensions of V and W. This tensor product can be repeated to apply to more than just two vector spaces. In mathematics, the dimension of a vector space V is the cardinality (i. ...

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

where V* is the dual space of V. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

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 of contravariant rank m and covariant rank n and total rank m+n. The tensors of rank zero are just the scalars (elements of the field F), those of contravariant rank 1 the vectors in V, and those of covariant rank 1 the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors).

The (1,1) tensors

are isomorphic in a natural way to the space of linear transformations (i.e., matrices) 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 For the square matrix section, see square matrix. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product or dot product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

called the associated metric and usually denoted g. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Alternate notation

Rather than writing out the full tensor product to denote the space of tensors of type (m,n), the literature often uses the abbreviation

Another, alternate notation for this space is in terms of linear maps from a vector space V to a vector space W. Let

denote the space of all linear maps from V to W. Thus, for example, the dual space (the space of 1-forms) may be written as (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

The set of (m,n)-tensors can then be written as

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

and

and

The notation

is often used to denote the space of invertible linear transformations from V to W; however there is no analogous notation for tensor spaces.

Tensor fields

See main article tensor field In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...

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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A Superconductor demonstrating the Meissner Effect. ... Engineering is the application of scientific and technical knowledge to solve human problems. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

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. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

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 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. ...

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 Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...

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. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

/Old Talk - still has some stuff that should likely be merged in
(Redirected from Tensor/Old) Tensors are quantities that describe a transformation between coordinate systems. ...