Disambiguation Note: The following is an overview of various approaches to the subject of tensors. For component-based "classical" treatment of tensors, see Classical treatment of tensors. Contravariant and covariant tensors A contravariant tensor of order 1(T^i is defined as \bar{T}^i = T^r\frac{\partial \bar{x}^i}{\partial x^r} See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two. In Mathematics, the modern Component-free approach to the theory of Tensors views tensors initially as Abstract objects expressing some definite type of Overview Tensor quantities may be categorized by considering the number of indices inherent in their description

A tensor is an object which extends the notion of scalar, vector, and matrix. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The term has slightly different meanings in mathematics and physics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function. In Mathematics, multilinear algebra extends the methods of Linear algebra. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable In physics and engineering, the same term usually means what a mathematician would call a tensor field: an association of a different (mathematical) tensor with each point of a geometric space, varying continuously with position. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity

History

The word tensor was introduced in 1846 by William Rowan Hamilton[1] to describe the norm operation in a certain type of algebraic system (eventually known as a Clifford algebra). Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, Clifford algebras are a type of Associative algebra. The word was used in its current meaning by Woldemar Voigt in 1899.

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and was made accessible to many mathematicians by the publication of Tullio Levi-Civita's 1900 classic text of the same name (in Italian; translations followed). History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually Gregorio Ricci-Curbastro ( January 12, 1853 - August 6, 1925) was an Italian Mathematician. Tullio Levi-Civita ( March 29, 1873 — December 29, 1941) (pronounced /'levi ˈʧivita/ was an Italian Mathematician, In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann,[2] or perhaps from Levi-Civita himself. Marcel Grossmann (born in Budapest on April 9, 1878 - died in Zurich on September 7, 1936) was a Mathematician Tensors are used also in other fields such as continuum mechanics. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e

Two usages of 'tensor'

Mathematical

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 of the particular space on which it is defined. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The word linear comes from the Latin word linearis, which means created by lines. In Computer science an array is a Data structure consisting of a group of elements that are accessed by indexing. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The intuition underlying the tensor concept is inherently geometrical: as an object in and of itself, a tensor is independent of any chosen frame of reference. See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to However, in the modern treatment, tensor theory is best regarded as a topic in multilinear algebra. In Mathematics, multilinear algebra extends the methods of Linear algebra. Engineering applications do not usually require the full, general theory, but theoretical physics now does. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world

For example, the Euclidean inner product (dot product)—a real-valued function of two vectors that is linear in each—is a mathematical tensor. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R Similarly, on a smooth curved surface such as a torus, the metric tensor (field) essentially defines a different inner product of tangent vectors at each point of the surface. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. Just as a linear transformation can be represented as a matrix of numbers with respect to given vector bases, so a tensor can be written as an organized collection of numbers. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In physics, the numbers may be obtained as physical quantities that depend on a basis, and the collection is determined to be a tensor if the quantities transform appropriately under change of basis.

Physical - tensor fields

Many mathematical structures informally called 'tensors' are actually tensor fields—a tensor valued function defined on a geometric or topological space. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This use of the term is analogous to vector fields such as electromagnetic fields, but with the 'tensor' defined so that it is invariant under a change of coordinates. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. The electromagnetic field is a physical field produced by electrically charged objects. Differential equations posed in terms of tensor quantities are basic to modern mathematical physics, so that tensor fields are usually defined on differentiable manifolds. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus.

Tensor rank

In mathematics, the term rank of a tensor may mean either of two things, and it is not always clear from the context which.

In the first definition, the rank of a tensor T is the number of indices required to write down the components of T. This is the sum of the number of covariant and contravariant indices. Expressed by means of the tensor product of multilinear algebra, this is the number of factors of the tensor product needed to express T. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector

In the second definition, the rank of a tensor is defined in a way that extends the definition of the rank of a matrix given in linear algebra. The column rank of a matrix A is the maximal number of Linearly independent columns of A. A tensor of rank 1 (also called a simple tensor) is a tensor that can be written as a tensor product of the form

$a\otimes b\otimes\cdots\otimes d$

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

$T_{ij\dots}^{k\ell\dots}=a_ib_j\cdots c^kd^\ell\cdots$

Every tensor can be expressed as a linear combination of rank 1 tensors. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In general, the rank of T is the minimum number of rank 1 tensors with which it is possible to express T as a linear combination.

For example, a matrix is a tensor with 2 indices, and so has rank 2 in the first definition. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally On the other hand, the rank of the tensor in the second definition is just the rank of the matrix. This latter meaning is possibly the intended one, whenever the array of components is two-dimensional.

To avoid this ambiguity, it is now preferred to use the terminology of tensor order to denote the number of indices, and tensor rank to designate the number of simple tensors necessary to decompose a tensor. Hence the definition of rank is now used in a way that is consistent with Linear Algebra.

Tensor valence

In physical applications, array indices are distinguished by being contravariant (superscripts) or covariant (subscripts), depending upon the type of transformation properties. The valence of a particular tensor is the number and type of array indices; tensors with the same rank but different valence are not, in general, identical. However, any given covariant index can be transformed into a contravariant one, and vice versa, by applying the metric tensor. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space This operation is generally known as raising or lowering indices. In mathematics and mathematical physics given a Tensor on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or

Importance and applications

Tensors are important in physics and engineering. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain; in this technique tensors are in effect made visible. Diffusion MRI is a Magnetic resonance imaging (MRI method that produces In vivo images of biological tissues weighted with the local microstructural The brain is the center of the Nervous system in animals All Vertebrates and the majority of Invertebrates have a brain Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear elastic material by a fourth rank elasticity tensor. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions

Specifically, a 2nd rank tensor quantifying stress in a 3-dimensional/solid object has components which can be conveniently represented as a 3x3 array. The three Cartesian faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number (being in three-space). Thus, 3x3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment (which may now be treated as a point). Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, the need for a 2nd order tensor is produced.

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond specifying that it requires a number of indexed components. In particular, tensors behave in specific ways under coordinate transformations. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra. Linear algebra is the branch of Mathematics concerned with In Mathematics, multilinear algebra extends the methods of Linear algebra.

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Contravariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the covariant vectors. In Linear algebra, a one-form on a Vector space is the same as a Linear functional on the space In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals

Physicists and engineers are among the first to recognise that vectors and tensors have a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated. Similarly, mathematicians find there are some tensor relations which are more conveniently derived in a co-ordinate notation.

Examples

Physical examples

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 relationship between force and acceleration is linear in classical mechanics. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Such a relationship is described by a rank two tensor of type (1,1) (that is to say, here it transforms a plane vector into another such vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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 solid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i. A solid body electric instrument is a String instrument such as a guitar, bass or Violin built without its normal Sound box and relying FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code 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. Surfel is an abbreviation of " surf ace el ement" In 3D computer graphics, the use of surfels is an alternative to Polygonal modeling 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), in linear elasticity, or more precisely by a tensor field of type (2,0) since the stresses may change from point to point. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions

Mathematical examples

Some well-known examples of tensors in differential geometry are quadratic forms, such as metric tensors, and the curvature tensor. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

Formally speaking, a tensor has a particular type according to the construction with tensor products that give rise to it. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector For computational purposes, it may be expressed as the sequence of values represented by a function with a tuple-valued domain and a scalar valued range. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, the range of a function is the set of all "output" values produced by that function Domain values are tuples of counting numbers, and these numbers are called indices. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the indices range from «1, 1, 1» through «2, 5, 7»; thus the tensor would have one value at «1, 1, 1», another at «1, 1, 2», and so on for a total of 70 values. As a special case, (finite-dimensional) vectors may be expressed as a sequence of values represented by a function with a scalar valued domain and a scalar valued range; the number of distinct indices is the dimension of the vector. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Using this approach, the rank 3 tensor of dimension (2,5,7) can be represented as a 3-dimensional array of size 2 × 5 × 7. In this usage, the number of "dimensions" comprising the array is equivalent to the "rank" of the tensor, and the dimensions of the tensor are equivalent to the "size" of each array dimension.

A tensor field associates a tensor value with every point on a manifold. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Thus, instead of simply having 70 values as indicated in the above example, for a rank 3 tensor field with dimensions «2, 5, 7»; every point in the space would have 70 values associated with it. In other words, a tensor field means there's some tensor-valued function which has, for example, Euclidean space as its domain.

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 defines a tensor to a collection of multidimensional arrays, such that one array is associated to each possible coordinate system of any fixed vector space. Contravariant and covariant tensors A contravariant tensor of order 1(T^i is defined as \bar{T}^i = T^r\frac{\partial \bar{x}^i}{\partial x^r} In Computer science an array is a Data structure consisting of a group of elements that are accessed by indexing. This notion generalizes scalars, vectors, matrices, linear functionals, bilinear forms, etc. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally To represent a vector x as a tensor one can simply let the array associated to any basis B be the vector of coordinates of x with respect to B.
However, to count as a tensor, the arrays need to obey a relation that precisely corresponds to how vectors, matrices, linear functionals, etc transform when one passes from one coordinate system to another.
The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. In Mathematics, the modern Component-free approach to the theory of Tensors views tensors initially as Abstract objects expressing some definite type of 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. Linear algebra is the branch of Mathematics concerned with In Mathematics, multilinear algebra extends the methods of Linear algebra. This treatment has attempted to replace 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. One could say that the slogan is 'tensors are elements of some tensor space'. Nevertheless, a component-free approach has not become fully popular, owing to the difficulties involved with giving a geometrical interpretation to higher-rank tensors.
• The intermediate treatment of tensors attempts to bridge the two extremes, and to show their relationships. Overview Tensor quantities may be categorized by considering the number of indices inherent in their description

In the end the same computational content is expressed. See glossary of tensor theory for a listing of technical terms. This is a glossary of tensor theory. For expositions of tensor theory from different points of view see Tensor Classical treatment of tensors

Tensor densities

Main article: Tensor density

It is also possible for a tensor field to have a "density". A tensor density transforms as a Tensor, except that it is additionally multiplied or weighted by a power of the Jacobian determinant In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity 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 rth power. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range. In the language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Mathematics, a line bundle expresses the concept of a line that varies from point to point of a space