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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 rank of a particular tensor is the number of array indices required to describe such a quantity. For example, a mass, temperature, and other scalar quantities are tensors of rank 0; force, momentum and other vector-like quantities are tensors of rank 1; a linear transformation such as the relationship between force and acceleration vectors is a tensor of rank 2. Euclid, detail from The School of Athens by Raphael. ...
The word linear comes from the Latin word linearis, which means created by lines. ...
In computer programming, an array, also known as a vector or list (for one-dimensional arrays) or a matrix (for two-dimensional arrays), is one of the simplest data structures. ...
In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...
Frame of reference - Wikipedia, the free encyclopedia /**/ @import /skins-1. ...
Mass is a property of a physical object that quantifies the amount of matter it contains. ...
Temperature is also the name of a song by Sean Paul. ...
The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ...
In physics, the Newtonian definition of force is the rate of change of momentum. ...
In classical mechanics, the momentum (pl. ...
The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In physics, the Newtonian definition of force is the rate of change of momentum. ...
Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the slope of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ...
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. For a formal, axiomatic definition of tensor, see the entry entitled tensor (intrinsic definition). The entry entitled Classical treatment of tensors gives an older, less formal definition. The entry entitled intermediate treatment of tensors attempts to bridge the two extremes, and to show their relationships. 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. ...
The following is a component-based classical treatment of tensors. ...
Note: The following is a modern component-based treatment of tensors (sometimes called the classical treatment of tensors). ...
Importance and usage
Tensors are of importance in physics and engineering. 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. 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 material by a fourth rank elasticity tensor. A Superconductor demonstrating the Meissner Effect. ...
Engineering is the application of scientific and technical knowledge to solve human problems. ...
Diffusion tensor imaging (DTI) is a new magnetic resonance imaging (MRI)-based technique that allows us to visualize the location, the orientation, and the anisotropy of the brains white matter tracts. ...
Permeability has several meanings: In electromagnetism, permeability is the degree of magnetisation of a material in response to a magnetic field. ...
Comparative brain sizes In animals, the brain, or encephalon (Greek for in the head), is the control center of the central nervous system. ...
This article is in need of attention from an expert on the subject. ...
The strain tensor [ε] is a symmetric tensor used to quantify the strain of an object undergoing a 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij (i ≠ j) are the...
// Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ...
Specifically, a 2nd rank tensor quantifying stress in a 3-dimensional/solid object has components which can be conveniently represented as 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 infintesimal 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. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
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. ...
History The word "tensor" was first 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. William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ...
1846 was a common year starting on Thursday (see link for calendar). ...
Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ...
Woldemar Voigt (September 2, 1850 - December 13, 1919) was a German physicist. ...
1899 (MDCCCXCIX) was a common year starting on Sunday (see link for calendar). ...
The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry, and was 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. Einstein had learned about them only with great difficulty, perhaps from Levi-Civita himself, or, as related by Abraham Pais in his Subtle is the Lord, more particularly from the geometer Marcel Grossman. Tensors are used also in other fields such as continuum mechanics. 1890 (MDCCCXC) was a common year starting on Wednesday (see link for calendar) of the Gregorian calendar (or a common year starting on Friday of the Julian calendar). ...
Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
Tullio Levi-Civita (March 29, 1873 - December 29, 1941) was an Italian mathematician, most famous for his work on tensor calculus but who also made significant contributions in other areas, some related to this work and some not. ...
1900 (MCM) was an exceptional common year starting on Monday. ...
Albert Einstein, photographed by Oren J. Turner in 1947. ...
It has been suggested that Einsteins theory of gravitation be merged into this article or section. ...
1915 (MCMXV) was a common year starting on Friday (see link for calendar). ...
Abraham Pais (May 19, 1918 - August 4, 2000) was a Dutch-born physicist. ...
Marcel Grossman (born in Budapest on April 9th, 1878 - died in Zurich on September 7th, 1936) was a mathematician and a friend and classmate of Albert Einstein. ...
Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ...
Sometimes the word "tensor" is used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold. To better understand tensor fields, one should first understand the basic idea 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 180°. A sphere is not a Euclidean space. ...
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.
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. In category theory, see covariant functor. ...
Contravariant is a mathematical term with a precise definition in tensor analysis. ...
Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...
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. ...
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). ...
Examples A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range. These vectors in the domain are vectors of counting numbers, and these numbers are called indexes. For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the vectors range from <1, 1, 1> through <2, 5, 7>. Here, the tensor would have one value at <1, 1, 1>, another at <1, 1, 2>, and so on for a total of 70 values. (Likewise, vectors may be expressed as a sequence of values represented by a function with a scalar valued domain and a scalar valued range, and the numbers in the domain are counting numbers called indices, and the number of distinct indices is sometimes called the dimension of the vector.) In mathematics, the domain of a function is the set of all input values to the function. ...
In mathematics, the range of a function is the set of all output values produced by that function. ...
2-dimensional renderings (ie. ...
A tensor field associates a tensor value with every point on a manifold. 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 the Euclidean space as its domain. Not just any function is allowed here -- see tensor field for more coverage of these requirements. 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. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...
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. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ...
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 in classical mechanics. 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 mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
For the square matrix section, see square matrix. ...
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. In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ...
A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ...
Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor, the inertia 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. ...
Moment of inertia (SI unit kilogram metre squared kg m2) quantifies the rotational inertia of an object, i. ...
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. // Degrees of freedom in mechanics In mechanics, for each particle belonging to a system, and for each independent direction in which movement is possible, two degrees of freedom, are defined, one describing the particles momentum in that direction, the other describing the particles position along an axis defined...
Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. ...
Mass is a property of a physical object that quantifies the amount of matter it contains. ...
Temperature is also the name of a song by Sean Paul. ...
In physics, the Newtonian definition of force is the rate of change of momentum. ...
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 (no indices at all), and vectors are rank one tensors.
Another example of a tensor is the Riemann curvature tensor from the theory of General Relativity, which is of rank 4 with dimensions <4, 4, 4, 4> (3 spatial + time = 4 dimensions). It can be treated as matrix (or vector) with 256 components (256 = 4 × 4 × 4 × 4). Only 20 of these components are actually independent of each other, greatly simplifying the matrix (or vector). 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. ...
General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ...
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 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 values in the array. This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.
However, to count as a tensor, the arrays need to transform correctly when the reference co-ordinate system is changed. This transformation is a generalisation of the relationship which holds for vector components, and is similarly an expression of the independence of the underlying entity from the reference frame in which it is expressed. The following is a component-based classical treatment of tensors. ...
In computer programming, an array, also known as a vector or list (for one-dimensional arrays) or a matrix (for two-dimensional arrays), is one of the simplest data structures. ...
For the square matrix section, see square matrix. ...
In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...
Partial plot of a function f. ...
In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ...
- 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'.
In the end the same computational content is expressed, both ways. See glossary of tensor theory for a listing of technical terms. 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. ...
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. ...
Note: The following is a modern component-based treatment of tensors (sometimes called the classical treatment of tensors). ...
This is a glossary of tensor theory. ...
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 rth 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. In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
See also This is a glossary of tensor theory. ...
Notation Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...
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. ...
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. ...
Voigt notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...
In tensor algebra, Mandel notation is a way to represent a symmetric tensor by reducing its rank. ...
Foundational Contravariant is a mathematical term with a precise definition in tensor analysis. ...
In category theory, see covariant functor. ...
In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...
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. ...
In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...
Applications The introduction of this article does not provide enough context for readers unfamiliar with the subject. ...
Tensors are frequently used in engineering to describe measured quantities. ...
Tensors are usede in Solid Mechanics ; if stress and strain are 3x3 matrixes , then Hooks Law which connects them with a constant has to be a tensor. ...
Curvature is the amount by which a geometric object deviates from being flat. ...
In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...
In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...
External links This article or section does not cite its references or sources. ...
Reference books - Tensors, Differential Forms, and Variational Principles (1989) David Lovelock, Hanno Rund
- Tensor Analysis on Manifolds (1981) Richard L Bishop, Samuel I. Goldberg
- Introduction to Tensor Calculus, Relativity and Cosmology (2003) D. F. Lawden
- Tensor Analysis (2003) L.P. Lebedev, Michael J. Cloud
Tensor software - GRTensorII is a computer algebra package for performing calculations in the general area of differential geometry. GRTensor II is not a stand alone package, the program runs with all versions of Maple V Release 3 through Maple 9.5. A limited version (GRTensorM) has been ported to Mathematica.
- Tensorial 3.0 Tensorial is a general purpose tensor calculus package for Mathematica 4.1 or better. Some of its features are: complete freedom in choosing tensor labels and indices; base indices may be any set of integers or symbols; tensor shortcuts for easy entry of tensors; flavored (colored or annotated) indices for different coordinate systems; CircleTimes notation available; easy methods for storing and substituting tensor values; routines for partial, covariant, total, absolute (Intrinsic) and Lie derivatives; There is extensive documentation, with a Help page and numerous examples for each command. In addition there are a number of tutorial and sample application notebooks.
You may wish to check the site occasionally for updates. A section in the Help Introduction now gives a history of the major additions and changes in usage. Load down TensorCalculus3.zip 369KB Package, StyleSheet and Documentation, 11 August 2005. TensorialReadMe.txt 3KB Instructions for installation, 11 October 2003. Related applications TMecanica, TContinuumMechanics, TGeneralRelativity, are also available on the site. August 11 is the 223rd day of the year (224th in leap years) in the Gregorian Calendar. ...
2005 (MMV) was a common year starting on Saturday of the Gregorian calendar. ...
October 11 is the 284th day of the year (285th in leap years). ...
2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ...
- MathTensor is a tensor analysis system written for the Mathematica system. It provides more than 250 functions and objects for elementary and advanced users.
- Tensors in Physics is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general Riemann-Cartan geometries.
- maxima is a free software computer algebra system which should be usable for making tensor algebra calculations
- Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
- Tela is a software package similar to Matlab and Octave, but designed specifically for tensors.
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