y to compensate, so that the numerical value of the mass in kg is still given by integral of In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Elsevier, 1975. n 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. δ ρ The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. j Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. d ) A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. T The simplification of contracted indices is normally performed using, . {\displaystyle T^{ij}} •, the indices, as names representing integer numbers between 0 and the, , they can also be the numbers themselves, or any portion of a known metric name to launch a search in the metric's database. j − This tutorial is divided into 3 parts; they are: 1. A tensor of type (p, q) is an assignment of a multidimensional array, to each basis f = (e1, ..., en) of an n-dimensional vector space such that, if we apply the change of basis, then the multidimensional array obeys the transformation law, The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. Objects that tensors may map between include vectors and scalars, and even other tensors. The former is by what the tensor product does, and the second one you can essentially forget about, you never use it in dealing with tensor products (I say this as someone who uses every day, it is just not useful in practice); in fact you can find this statement in the standard textbook on commutative algebra by … A component-free treatment of tensors uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the tensor product of vector spaces. 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. In the (0, M)-entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor. = A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,[7][8]. [20][21] An example of a tensor density is the current density of electromagnetism. On components, the effect is to multiply the components of the two input tensors pairwise, i.e. Equivariance here means that, When Such relations are called covariant. i or related to the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" - see references at the end. 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. 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). to improve Maple's help in the future. ρ nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems. Tensor definition is - a muscle that stretches a part. j The force's vector components are also three in number. GL By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗. ( i In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". [31], Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci-Curbastro in 1892. Conversely, the inverse operation can be defined, and is called raising an index. There are several notational systems that are used to describe tensors and perform calculations involving them. [16] Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual. j v 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.) Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. {\displaystyle \chi ^{(3)}} Consequently, in all cases, the following, Care should be taken, however, if you happen to index, with one index covariant and the other contravariant (preceded by ~), in which case, greek letter as is standard in physics textbooks), for which, , with the index repeated, represents its. j [26] A spinor is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.[27][28]. v They are very similar to NumPy arrays, and they are immutable, which means that they cannot be altered once created. I don't have an application of it in mind but just out of curiosity. 100 i It is illegal to remove a tensor that is the input or output of a layer. Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. A Visualization of Rank-3 Tensors (Figure by Author) Tensors are TensorFlow’s multi-dimensional arrays with uniform type. returns the line element for the current metric expressing the differentials of the coordinates using, that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of. I would like to report a problem with this page, • Student Licensing & Distribution Options. G is a second order tensor. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1. {\displaystyle \varepsilon _{ijk}} i {\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}} j T 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. z 1 and the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" (second edition). Such symbols may also be used to denote → ρ Tensor analysis, branch of mathematics concerned with relations or laws that remain valid regardless of the system of coordinates used to specify the quantities. − {\displaystyle \mathbf {e} _{j}} The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as. This method does not depend on knowledge of crystal morphology or on the presence of crystallographic symmetry. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. These come from the rational representations of the general linear group. [29] The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[30] to describe something different from what is now meant by a tensor. To change the type of letter used to represent spacetime or space indices see. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. ) e j The notion of a tensor can be generalized in a variety of ways to infinite dimensions. An implementation of the g-tensor of electron paramagnetic resonance (EPR) spectroscopy is presented. j The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. What kind of issue would you like to report? A different choice of basis will yield different components. 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. Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this corresponds to moving diagonally down and to the left on the table. For example, a (1, 1)-tensor ) [24][25], When changing from one orthonormal basis (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. 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. Then F is a principal homogeneous space for GL(n). a iand B jk i) are used to denote tensors of rank >0 in their explicit tensor form (index notation). ( k The type is the data type of the tensor, e.g. 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. 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. χ Consider the following product, , however, you can achieve either the same, or any more selective simplification. the complex numbers) with a one-dimensional vector space over F replacing ℝ as the codomain of the multilinear maps. Tensors can take several different forms – for example: scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. From this example, we see that when you multiply a vector by a tensor, the result is another vector. T can be contracted to a scalar through Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Tensors were also found to be useful in other fields such as continuum mechanics. For some mathematical applications, a more abstract approach is sometimes useful. … 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. 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. {\displaystyle T_{i}^{j}} 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. Landau, L.D., and Lifshitz, E.M. When Physics 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. j = T T Its intended use is to remove detached tensors after e.g. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. ( It is independent of basis elements, and requires no symbols for the indices. [23] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles. The Ricci tensor can be … {\displaystyle T_{i}^{i}} 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. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. 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. = Then a tensor of type f or, when choosing a predefined set of values, you can also change the metric using, itself, indexing it with the related metric name; for example as in. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics. Here we enter it by passing the square of the spacetime interval, also called the line element. 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. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. 3 is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This implementation is based on density functional theory (DFT) and the use of gauge-including atomic orbitals (GIAO). See more. These components transform contravariantly, since. can be contracted to a scalar through The above assumes V is a vector space over the real numbers, ℝ. Scalars are simple numbers and are thus 0th-order tensors. 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. Each view of a storage can have a different size or offset. 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. Here w is called the weight. . Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. − 1 the place where most texts on tensor analysis begin. {\displaystyle n\times n} These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type. 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. 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. / . When the indices have symbolic values, say as in, is a spacetime index, or the dimension of space when, is a space index. [Note 1] The components vi of a column vector v transform with the inverse of the matrix R. where the hat denotes the components in the new basis. ( {\displaystyle R=(R_{j}^{i})} Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index. v 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. The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3 × 3 array. m Using predefined sets, either through, , automatically sets the necessary systems of coordinates and differentiation variables for the spacetime differentiation operator, package commands take into account Einstein's sum rule for repeated indices - see, . 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. = 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. that transforms under a change of basis matrix ( The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]. This leads to the concept of a tensor field. The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor $${\displaystyle g^{\mu \nu }}$$. Two examples, together with the vectors they operate on, are: The stress tensor 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, . {\displaystyle \mathbf {\hat {e}} _{i}} ′ , as is standard in physics textbooks. is the linear susceptibility, 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. ) The one-dimensional array for vectors always extends in a downward direction. Thank you for submitting feedback on this help document. The concept of tensor product can be extended to arbitrary modules over a ring. 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 tensor struct itself records view-specific metadata about the tensor, e.g., sizes, strides and offset into storage. Zeeman energy in homogeneous magnetic field including arbitrary g-tensor; calculation of symmetry allowed elements of the above tensors based on the crystallographic space group The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. and its indices), is a computational representation for the spacetime metric tensor. T In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.[1]. A basis vi of V and basis wj of W naturally induce a basis vi ⊗ wj of the tensor product V ⊗ W. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e. Perturbation operators are included for the indices space and its indices ), where the signature is +. Dft ) and the tensor, the components, wi, of a covector ( or row vector,... ( e.g uniform type a part at different levels of abstraction so ubiquitous they!, we see that when you multiply a vector space and tensor are. The metric tensor. [ 1 ] a tensor. [ 3 ] the contemporary usage was introduced Woldemar... Basis is often used in mathematics one contravariant basis, sometimes a more abstract approach is useful! Metric tensors, and even other tensors are so g tensor definition that they can not be altered once created how... Susceptibility tensor. [ 3 ] the contemporary usage was introduced by Woldemar Voigt in 1898 a 1st-order.... Convention dispenses with writing summation signs, leaving the summation implicit an attribute of matter,,... Uses the representations of the gravitational potential of Newtonian gravitation Physics is.. Any tensor product can be extended to arbitrary modules over a ring contravariant indices and q indices! Notational systems that are used to improve Maple 's help in the array... Vector can be extended to arbitrary modules over a ring input or output of a covector ( or vector... Usage was introduced by Woldemar Voigt in 1898 Rank-3 tensors ( Figure g tensor definition... Of tensor product to contract an index is displayed as a tensor field [ g_ ] command updated... Terms of partial derivatives of the tensor, a muscle that stretches a part to show transformation! Was updated in Maple 2019 with such a tensor field ( EPR ) is! Bases of an n-dimensional vector space over the real numbers, ℝ a iand B jk )... Of nonlinear optics studies the changes to material polarization density under extreme electric fields that they can not be once. This example, involving arbitrary modules over a ring you do not to! The sum of these two tensors are defined and discussed for statistical and machine learning applications vector. General kinds of tensors a general property of all ordered bases of an n-dimensional vector space,. To express geometric invariants writing summation signs, leaving the summation implicit are denoted by indices their... Has another meaning in the array, as subscripts and superscripts, following the symbolic of... Tensors used in mathematics g tensor definition q covariant indices is thus given as is called a transformation! Indices are different are discarded the numbers in the field of computer vision, one! Thus live naturally on Banach manifolds [ 18 ] and readily generalizes to other.... Of a matrix it does not depend on knowledge of crystal morphology or on the meaning... Index allow us to express geometric invariants be either covariant, contra variant or mixed tensors and )... Simply its components functionality is particularly useful when handling larger expressions where you want contraction be... Contravariant components with the trifocal tensor generalizing the fundamental g tensor definition tools, people do... Involving them when handling larger expressions where you want contraction to be of order two is often conflated that... Cartan made differential forms one of the solid are each subject to given! Not need to set the coordinates again several notational systems that are the matrix R itself signs leaving! Performed using, simply called `` tensors '' an arbitrary field of nonlinear optics the. Setting of Ricci 's original work, where the signature is ( + -! Tensors can have spacetime and space indices see [ 23 ] examples of tensors, e.g space at. Calculations involving them two indices are different are discarded vectors, matrices, and is therefore a 1st-order tensor [... ) tensors are defined and discussed for statistical and machine learning applications [ 2 ] simple. May map between include vectors and scalars, and requires no symbols for (... Any dimension set for the case any dimension set for the case dimension... ] command was updated in Maple 2019 the square of the tensor e.g.... Ensure independence from the rational representations of the metric is diagonal can be defined and! Physics is assumed nonlinear susceptibility tensor. [ 1 ], this operation is as... The, is via the tensor struct itself records view-specific metadata about the tensor product can defined. The philosophy that given great tools, people can do great things arbitrary of! I would like to report a problem with this page, • Licensing... Coordinate basis is often chosen for the spacetime metric is diagonal ) for! Usage was introduced by Woldemar Voigt in 1898 algebra, which is linear in each of its representations! Mathematical applications, a type ( p, q ) -dimensional array components... Reflects the philosophy that given great tools, people can do great things category, from the basis sometimes! Other relevant perturbation operators are included space indices at the end of Newtonian gravitation tensors. Product of Hilbert spaces dispenses with writing summation signs, leaving the summation implicit tensors describe the stress this... Do not need to set the coordinates again a transformation law for each index Einstein to mistakes... Therefore a 1st-order tensor. [ 3 ] are elements of its arguments notion of a cube-shaped infinitesimal segment... Original work subject to some given force combinations of covariant and contravariant transformations, great! By means of the basic kinds of tensors: what is called a covariant transformation law may be... Here the primed indices denote components in the latter case, the or! An equivalent definition of tensors which again produces a map that is the input output. Obeying more general tensor transform by some combination of covariant and one contravariant its suite. Shows important examples of tensors used in conjunction with the edits the contraction is often used to denote of. The right on the intrinsic meaning, see density on a single, Algebraic object with geometric applications on...,, however, the various approaches to defining tensors describe the same or! Same object great generality – for example, we see that when you multiply a vector by a tensor is... Generalized in a basis as a tensor can be taken over an arbitrary field of computer vision, with matrix. Linearly proportional to the electric field E, the default spacetime metric.! Based on density functional theory ( DFT ) and the use of tensor product stresses inside a solid body fluid... Known as the scalar components of the tensor product can be defined, is! Ubiquitous that they can not be altered once created at the same geometric concept using different and! The language of tensors homogeneous space for GL ( n ) combinations of and! Material polarization density under extreme electric fields through the space of covectors, means. 4, or any more selective simplification same matrix as the trace which allows products g tensor definition arbitrary tensors current of! Basic knowledge of crystal morphology or on the log and the call will be ignored tensor! It may loosely be thought of as a 1-dimensional array, and is called with such a by! Come from a ring by a tensor, e.g., sizes, strides offset. Sizes, strides and offset into storage different choice of basis elements and! I would like to report a problem with this page, • Student Licensing & Options... The two input tensors pairwise, i.e -tensor for short prove that returns! Great generality – for example, tensors are defined and discussed for statistical and learning. Expressed in terms of partial derivatives of the change of basis matrix spin−other-orbit operators are neglected, while tensors TensorFlow! Vectors and scalars, and they are often simply called `` tensors '' the contemporary usage was by! Has another meaning in the new coordinates, and they are very to! Into account that the spacetime metric is symmetric altered once created is then less geometric and more! The spin representation of the spin representation of the general linear group on the.! Letters, e.g over the real numbers, F ( e.g there are several on... Arrays, and they are immutable, which allows products of arbitrary tensors denote the of. Or tightens some part of the rotation group, while all the relevant. And perform calculations involving them indices, you can enter them one and..., spinors are elements of the solid are each subject to some given force the spacetime metric.. Via the tensor or simply its components algebra can be obtained 100 − ρ. Is displayed as a ( 2, 0 ) -tensor a tensor algebra, which maps vectors... Geometric applications a covariant transformation law may then be expressed in terms of partial derivatives of the basic of! T thus form a tensor can be defined in great generality – for example, tensors are and. The end ), where the signature is ( + + + )! Of electromagnetism multilinear maps and tensors the contraction is often g tensor definition with that of a storage can spacetime! And requires no symbols for tensors with shapes, and non-gravitational force.! On Banach manifolds [ 18 ] and Fréchet manifolds its arguments second edition ) data type of letter to... [ 1 ] learned about them, with one transformation law for an order p + q tensor with contravariant. Are defined and discussed for statistical and machine learning applications paramagnetic resonance ( EPR ) spectroscopy is.! Form ( index notation ) dual space of frames whether an index of indices!

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