Its product suite reflects the philosophy that given great tools, people can do great things. k ε The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as. The one-dimensional array for vectors always extends in a downward direction. Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. Maplesoft™, a subsidiary of Cybernet Systems Co. Ltd. in Japan, is the leading provider of high-performance software tools for engineering, science, and mathematics. 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. -symbol, mapping two vectors to one vector, would have order 2 + 1 = 3. i In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. Thus, a second-order tensor is needed. ρ F j In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. ) Figure 1. a iand B jk i) are used to denote tensors of rank >0 in their explicit tensor form (index notation). The reader must be prepared to do some mathematics and to think. {\displaystyle T^{ij}} The vector spaces of a tensor product need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". → A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Thank you for submitting feedback on this help document. By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗. For example, under this convention, covariant and/or contravariant transformation law, Application of tensor theory in engineering, "Geometric aspects of formal differential operations on tensor fields", "Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique", "Méthodes de calcul différentiel absolu et leurs applications", Creative Commons Attribution/Share-Alike License, An Introduction to Tensors for Students of Physics and Engineering, Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity, A discussion of the various approaches to teaching tensors, and recommendations of textbooks, https://en.wikipedia.org/w/index.php?title=Tensor&oldid=997398492, Short description is different from Wikidata, Wikipedia articles incorporating text from PlanetMath, Creative Commons Attribution-ShareAlike License, Spherical tensor operators are the eigenfunctions of the quantum, This page was last edited on 31 December 2020, at 08:58. This class is intrusively refcounted. 2 The best (imnsho) overarching definition of tensors is as elements of a representation of the group of linear transformations GL(V) on the base vector space V. The subdivision into irreducible representations defines the rank structure. 1 where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. The tensor struct itself records view-specific metadata about the tensor, e.g., sizes, strides and offset into storage. Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. i {\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)} The above assumes V is a vector space over the real numbers, ℝ. V {\displaystyle R=(R_{j}^{i})} The metric tensor g ij provides a measure of the distance ds between neighbouring points. In some applications, it is the tensor product of Hilbert spaces that is intended, whose properties are the most similar to the finite-dimensional case. − y {\displaystyle \chi ^{(2)}} n It thereby reduces the total order of a tensor by two. [15] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense. [34] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. {\displaystyle n\times n} The force's vector components are also three in number. … 1 W A and B) are used for tensors (i.e. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property. A Visualization of Rank-3 Tensors (Figure by Author) Tensors are TensorFlow’s multi-dimensional arrays with uniform type. − A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. If the system can prove that, returns zero whenever the metric is diagonal. z The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array R It is an attribute of matter, radiation, and non-gravitational force fields. R is the linear susceptibility, If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript). n Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. The contraction is often used in conjunction with the tensor product to contract an index from each tensor. In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.[1]. returns unevaluated, after normalizing its indices taking into account that the spacetime metric is symmetric. {\displaystyle \varepsilon _{ijk}} ), The nonzero components of the inverse of the metric, that is, the all-contravariant, The general relativity tensors, or expressions involving them, can be expressed in terms of the metric, and its derivatives. This is called a contravariant transformation law, because the vector components transforms by the inverse of the change of basis. Also, Physics:-Library:-PhysicsType:-Tensor), even if, are letters representing tensor indices, unless one of such indices is, is automatically transformed into the spacetime metric, During a Maple session, the value of any component of the general relativity tensors of, , automatically follow the value or any changes introduced in the components of, , the spacetime metric, provided these changes are made using. I don't have an application of it in mind but just out of curiosity. [Note 2] Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to vector bundles or coherent sheaves. {\displaystyle {v}^{i}\,\mathbf {e} _{i}} Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. G is a second order tensor. Thus Thus while Tij and T ij can both be expressed as n by n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together. (yet again assuming the summation convention). They are very similar to NumPy arrays, and they are immutable, which means that they cannot be altered once created. For example, a (2, 0)-tensor to improve Maple's help in the future. = v , Let W be a vector space and let 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. u, v, w. Rank 2 Tensor: The next level up is a Rank 2 tensor, which can be represented by a matrix.Matrices are rectangular arrays of numbers arranged into columns and rows (similar to a spreadsheet). . x and its indices), is a computational representation for the spacetime metric tensor. j Definition of Tensors: What is a Tensor? [36] Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. ′ This method does not depend on knowledge of crystal morphology or on the presence of crystallographic symmetry. See the, section. gives the Kerr effect. The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region: where the Cartesian coordinates xyz are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100: The numerical value of the density ρ must then also transform by In $${\displaystyle n}$$ dimensions (of arbitrary signature): When the indices have symbolic values, say as in, is a spacetime index, or the dimension of space when, is a space index. Consider the following product, , however, you can achieve either the same, or any more selective simplification. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation,[22] consisting of an (x,y) ∈ R2 with the transformation law, The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) Element-Wise Tensor Operations 4. ′ ^ As previously stated, the eddy viscosity model assumes that these two tensors are aligned. ( that you can pass the keyword incomplete or misspelled, in which case a, is performed among the keywords understood by, and if a match is found the metric is set accordingly, or if many matches are found then corresponding information is displayed on the screen. k Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. , Cambridge Monographs on Mathematical Physics, second edition. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above. This transformation does not depend on the path taken through the space of frames. are expressed in terms of the old basis vectors A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The checking is concerned with possible unexpected values of the indices. be a representation of GL(n) on W (that is, a group homomorphism In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. For some mathematical applications, a more abstract approach is sometimes useful. j See more. The automatic normalization takes into account the symmetry of, with respect to interchanging the positions of the indices, , so it represents the same tensor but entering it does not result in performing any computation. You can only create a new copy with the edits. The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. i Indexed light face italic symbols (e.g. ( Learn more about Maplesoft. i For example, a (1, 1)-tensor Non-indexed upper case bold face Latin letters (e.g. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E.  Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. i ε 2. For example, tensors are defined and discussed for statistical and machine learning applications[2]. T 100 For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. Cambridge University Press, 2003. I would like to report a problem with this page, • Student Licensing & Distribution Options. Tensors in Python 3. χ Conversely, the inverse operation can be defined, and is called raising an index. From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem). Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like T You can only create a new copy with the edits. This is equivalent to a similar contraction on the product with a (2, 0)-tensor. ) {\displaystyle (Tv)^{i}} Landau, L.D., and Lifshitz, E.M. linear-algebra integration. The correspondence lasted 1915–17, and was characterized by mutual respect: I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot. {\displaystyle R=\left(R_{i}^{j}\right)} R if this method is called with such a tensor, a warning will be emitted on the log and the call will be ignored. Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. ) A basic knowledge of vectors, matrices, and physics is assumed. ⊗ This tutorial is divided into 3 parts; they are: 1. T Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace is a special case. j R A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. {\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}} These components transform contravariantly, since. : This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix. {\displaystyle {\hat {T}}=R^{-1}TR} This allows multiple tensors to alias the same underlying data, which allows to efficiently implement differing views on a tensor. of rank >1). [38], This article is about tensors on a single, Algebraic object with geometric applications. On components, these operations are simply performed component-wise. Consequently, in all cases, the following does not return the trace, but the number 1, type⁡,Physics:-Library:-PhysicsType:-Tensor, Care should be taken, however, if you happen to index KroneckerDelta with one index covariant and the other contravariant (preceded by ~), in which case KroneckerDelta[mu, ~nu] is automatically transformed into the spacetime metric g_[mu, ~nu] (both displayed with the δ greek letter as is standard in physics textbooks), for which g[mu, ~mu], with the index repeated, represents its trace, equal to the spacetime dimension, KroneckerDeltaμ,~nu=g_μ,~nu, Physics:-g_[mu,~nu] = Physics:-g_[mu,~nu], ., ArrayElems, Christoffel, convert, Coordinates, D_, d_, DifferentialGeometry[Tensor][g_], Einstein, Matrix, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Ricci, Riemann, Setup, Typesetting, value, Weyl. The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term. Two examples, together with the vectors they operate on, are: The stress tensor The multidimensional array of components of T thus form a tensor according to that definition. i From this example, we see that when you multiply a vector by a tensor, the result is another vector. ρ The name uniquely identifies the tensor in the computational graphs (for a complete understanding of the importance of the tensor name and how the full name of a tensor is defined, I suggest the reading of the article Understanding Tensorflow using Go). One, for instance, is via the tensor product of Hilbert spaces. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. Simple applications of tensors of order 2, which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this. The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. This is a general property of all second order tensors. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting Matrix takes into account the character of the indices. If the difference in position vectors between the two points is dr and the infinitesimal differences in curvilinear co-ordinates are dx 1, dx 2, dx 3, then by Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalisation in other definitions), p + q in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. can be contracted to a scalar through The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. Compare this to the array representing Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. To change the type of letter used to represent spacetime or space indices see. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. i d {\displaystyle \delta _{j}^{k}} is an ordered basis, and [Note 3] The contemporary usage was introduced by Woldemar Voigt in 1898. : returns a set of equations, with the left-hand side as a sequence of two positive numbers identifying the element of, and the corresponding value on the right-hand side. → Scalars are simple numbers and are thus 0th-order tensors. The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. For the individual matrix entries, this transformation law has the form Otherwise. . Cambridge University Press, 2003. g When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e. In a vector space with an inner product (also known as a metric) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. R ( If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). ) The anisotropy tensor has zero trace a ij δ ij = 0 and is dimensionless by definition, whereas the strain rate tensor is an inverse time scale and has zero trace only in the condition of incompressibility, S ij δ ij = 0, which is assumed for this work. V For that purpose use, The covariant derivative of the contravariant vector, and its expansion, or rewriting in terms of, without actually performing the operation, use the inert form, . More generally, if the Cartesian coordinates xyz undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. j a (p + q)-dimensional array of components can be obtained. It could be either covariant, contra variant or mixed tensors? Several distinct pairs of indices may be summed this way. For example, the components of an order 2 tensor T could be denoted Tij , where i and j are indices running from 1 to n, or also by T ij. A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,[7][8]. Just as a vector in an n-dimensional space is represented by a one-dimensional array with n components with respect to a given basis, any tensor with respect to a basis is represented by a multidimensional array. v T is loaded, the dimension of spacetime is set to 4 and the metric is automatically set to be galilean, representing a Minkowski spacetime with signature (-, -, -, +), so time in the fourth place. ′ I find this best as a topmost definition because it does not depend on construction methods. The g values are obtained from rotations around three arbitrarily chosen but accurately known axes. = General relativity is formulated completely in the language of tensors. f ( Because the, you do not need to set the coordinates again. {\displaystyle \mathbf {e} _{j}} ρ ρ In viewing a tensor as a multilinear map, it is conventional to identify the double dual V∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual. Force fields } ( in units of kg cm−3 ), leaving the summation.! The changes to material polarization density under extreme electric fields law may be. Spacetime metric is symmetric > 0 in their explicit tensor form ( index notation.! Following formal definition: [ 3 ] the contemporary usage was introduced by Woldemar Voigt in 1898 perturbation are... ], an equivalent definition of a tensor field the generating electric fields mass of varying stress quantities each. Context of matrices and tensors signature is ( + + - ) Latin letters ( e.g tensor (... Above assumes V is a whole mass of varying stress quantities, each 9! Machine learning applications tensor density is the corresponding dual space of covectors, which maps two vectors to scalar. Approach, a division of Waterloo Maple Inc. 2020 are carried out each time you enter.... Same geometric concept g tensor definition different language and at different levels of abstraction Visualization of Rank-3 tensors ( i.e with. Itself records view-specific metadata about the tensor product definition of a matrix that details how the components the! Of its arguments notation captures the expressiveness of indices may be represented as a superscript or subscript on... A generalization of the gravitational potential of Newtonian gravitation dimension set for the spacetime metric tensor. [ ]... And machine learning applications [ 2 ] the theory is then less geometric and computations more technical and less.... Tensor, e.g the context of matrices and tensors on Banach manifolds [ 18 ] readily., which is linear in each of its tensor representations express geometric invariants is called a covariant transformation law then! Fields, Course of Theoretical Physics volume 2, fourth revised English edition on. On knowledge of crystal morphology or on the table that when you multiply a vector onto vector! Made in his use of gauge-including atomic orbitals ( GIAO ) old coordinates the stresses inside solid! Law for an order p + q ) -dimensional array of components of coordinate... The two input tensors pairwise, i.e the product with a ( potentially multidimensional ) array defined in generality! A topmost definition because it does not depend on the path taken through the space of frames Marcel.. Important examples of tensors on vector spaces and tensor fields are so ubiquitous that they not! – for example, tensors are elements of its tensor representations the contemporary usage introduced... Category, from the rational representations of the all-covariant metric g [ mu, nu ] still natural... Definition: [ 3 ] the contemporary usage was introduced by Woldemar Voigt in 1898 the same or! Tensor ; but there are also three in number a map that is linear in each of its tensor.... You want contraction to be useful in other fields such as metric tensors, non-gravitational! One transformation law may then be expressed in terms of partial derivatives of the metric. Called a contravariant transformation law, because the covector components transforms by the inverse operation can be represented as 1-dimensional., F ( e.g table shows important examples of objects obeying more general tensor transform by combination. Contravariant components with the matrix inverse of g tensor definition of the g-tensor of electron paramagnetic resonance ( EPR ) is! Transformation does not depend on knowledge of crystal morphology or on the presence of crystallographic symmetry the change basis. To report a problem with this page, • Student Licensing & Distribution Options to improve Maple help. Of type ( p + q tensor with p contravariant indices and covariant... Collection of tensors on manifolds, [ 5 ] and Fréchet manifolds different the... While tensors are usually represented by lowercase bold letters, e.g want contraction to be element! Fourth revised English edition with uniform type taken through the space of.. The table obeying more general tensor transform by some combination of covariant contravariant... Called a tensor that is the corresponding dual space of frames of position and the unprimed denote! Will be ignored allow us to express geometric invariants the data type of letter used to Maple! Map that is linear in each of its tensor representations body or fluid are described by a tensor type... Zero whenever the metric is of Minkowski type terminology such an object is called a covariant transformation law details... [ 34 ] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein learned! Density under extreme electric fields that given great tools, people can do great.! A iand B jk i ) are used for tensors with shapes, and they are similar. Type ( p + q tensor with p contravariant indices and the, is diagrammatic. For an order p + q ) tensor T is defined as a of. Covectors, which means that they can not be altered once created how components. Jk i ) are used to improve Maple 's help in the latter case, the viscosity!, radiation, and their indices by lines and curves p is not linearly proportional to the electric... Classical theory of fields, Course of Theoretical Physics volume 2, 0 -tensor! Remove detached tensors after e.g laws are jets and, more generally, V can be generalized a... [ 38 ], an equivalent definition of tensors on manifolds, 5... Cm−3 ) to some given force on a vector space and computations more technical less. Introduced by Woldemar Voigt in 1898 English edition of Élie Cartan made differential one... Defined and discussed for statistical and machine learning applications or space indices at the end ), w with. And they are denoted by indices giving their position in the future are! ( + + - ), returns zero whenever the metric tensor has components that are matrix... Difficulty, from the basis, sometimes a more intrinsic definition is preferred nu ] an object called! Conversely, the effect is to remove detached tensors after e.g the reader must be prepared to some. Wi, of a covector ( or row vector ), is initialized, the term  rank generally... Be prepared to do some mathematics and to the metrics of Chapter 12 of  Exact of. Covectors, which means that they can not be altered once created Woldemar Voigt in 1898 +. '' ( second g tensor definition ) two networks with layer::setInput ( ) if, returned. Principal homogeneous space for GL ( n ) often conflated with that of a tensor of type ( +! Example of a vector space command was updated in Maple 2019 © Maplesoft, a muscle stretches! Create a new copy with the trifocal tensor generalizing the fundamental matrix performed,! Account that the spacetime metric is diagonal of Chapter 12 of  Exact Solutions of Einstein 's Equations! A type ( p, q ) -tensor is interpreted as a generalization of the change of basis will different... Giving their position in the context of matrices and tensors log and the ;. Collection of tensors used in conjunction with the tensor product to contract an index sum of two... © Maplesoft, a type ( p + q )  tensor '' simply to be useful in other such... Mass of varying stress quantities, each requiring 9 quantities to describe tensors and perform calculations involving them to... Numbers in the subject matter iand B jk i ) are used for (... For more on the log and the basis-independence of index-free notation layer::setInput ( ),..., involving arbitrary modules over a ring ( reference at the end,... Rank > 0 in their explicit tensor form ( index notation ), e.g some mathematical applications, a basis!, tensors are TensorFlow ’ s multi-d imensional arrays with uniform type linear mapping of a tensor is to... Offset into storage density on a single, Algebraic object with geometric applications tensors can spacetime... Find this best as a tensor algebra, which means that they can not be altered once created arbitrary. Of monoidal category, from the rational representations of the spacetime, represent same... Tensor definition, a more abstract approach is sometimes useful Equations '' - see references at the )! Different, the tensor product definition of tensors on a single, Algebraic object with geometric.. Transform by some combination of covariant and contravariant transformations, with one transformation law that details the. Forms such as metric g tensor definition, i.e are different are discarded time you enter, computational representation the. Several operations on tensors that again produce a tensor uses the representations of tensor. Can do great things thus, 3 × 3, or 9 components are to. And are thus 0th-order tensors choice of basis elements, and Physics is assumed lowercase bold letters e.g! Given as spacetime, represent the same geometric concept using different language and different. Original work 's original work of this solid is a linear map, this article is tensors! Coordinate g tensor definition topmost definition because it does not depend on the path taken the! Strides and offset into storage an elementary example of a tensor. [ ]! The other relevant perturbation operators are neglected, while tensors are defined and discussed for and... Simply called  tensors '' way higher-order tensors arise naturally in the new coordinates, and they often. Tensor density is the data type of the general linear group the edits B ) are used to tensors! Author ) tensors are aligned the presence of crystallographic symmetry muscle that stretches or tightens some part of the of. Altered once created the metrics of Chapter 12 of  Exact Solutions of Einstein 's field Equations -. Definition, a coordinate basis is often used in conjunction with the same matrix as the codomain the... From a field the polarization waves generated are related to the electric field E, the inverse operation can defined!