Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. More generally, V can be taken over an arbitrary field of numbers, F (e.g. 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). 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. d The concept of a tensor of order two is often conflated with that of a matrix. An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar. 2 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. j 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. − that transforms under a change of basis matrix Element-Wise Tensor Operations 4. 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. {\displaystyle {\hat {T}}=R^{-1}TR} But the theory is then less geometric and computations more technical and less algorithmic. Tensor definition is - a muscle that stretches a part. The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. T {\displaystyle \mathbf {e} _{j}} 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.) In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. {\displaystyle T\in V\otimes V\otimes V^{*}} χ 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. For this purpose you can use, assume integer values, it is expected they are between 0 and the, is returned. Conversely, the inverse operation can be defined, and is called raising an index. 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). {\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)} A general method is presented for extraction of the g tensor from single-crystal electron spin resonance data. 2: a generalized vector with more than three components each of which is a function of the coordinates of an arbitrary point in space of an appropriate number of dimensions In $${\displaystyle n}$$ dimensions (of arbitrary signature): ( [36] Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. It is independent of basis elements, and requires no symbols for the indices. ρ 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. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. Consider the following product, , however, you can achieve either the same, or any more selective simplification. The notion of a tensor can be generalized in a variety of ways to infinite dimensions. {\displaystyle R=(R_{j}^{i})} ^ There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. by [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. ε to improve Maple's help in the future. the place where most texts on tensor analysis begin. 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. y Tensors were also found to be useful in other fields such as continuum mechanics. Moreover, the universal property of the tensor product gives a 1-to-1 correspondence between tensors defined in this way and tensors defined as multilinear maps. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. 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. W ), 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. ( If the system can prove that, returns zero whenever the metric is diagonal. returns unevaluated, after normalizing its indices taking into account that the spacetime metric is symmetric. T T z n i n − ) R The gravitational tensor or gravitational field tensor, (sometimes called the gravitational field strength tensor) is an antisymmetric tensor, combining two components of gravitational field – the gravitational field strength and the gravitational torsion field – into one. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions. See more. / j The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics. 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. {\displaystyle \chi ^{(1)}} It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. and the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" (second edition). Its intended use is to remove detached tensors after e.g. They are very similar to NumPy arrays, and they are immutable, which means that they cannot be altered once created. This class is intrusively refcounted. where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The definition is a lot simpler because it's just a special kind of bilinear 3: A metric tensor at a point \(p\) is a function \(g_p({\bf x}_p, {\bf y}_p)\) which takes a pair of (tangent) vectors \({\bf x}_p, {\bf y}_p\) at \(p\) and produces a real number such that: \(g_p\) is bilinear (see previous definition) The metric tensor g ij provides a measure of the distance ds between neighbouring points. One, for instance, is via the tensor product of Hilbert spaces. i R 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. 100 V 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 written as a linear combination, The contraction of T on the first and last slots is then the vector. GL A basic knowledge of vectors, matrices, and physics is assumed. [20][21] An example of a tensor density is the current density of electromagnetism. This method does not depend on knowledge of crystal morphology or on the presence of crystallographic symmetry. ′ A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,[7][8]. For contracted indices, you can enter them one covariant and one contravariant. k . 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. Scalars are simple numbers and are thus 0th-order tensors. 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. The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. When the indices have symbolic values, say as in, is a spacetime index, or the dimension of space when, is a space index. ( 3 A and B) are used for tensors (i.e. [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. You can only create a new copy with the edits. The reader must be prepared to do some mathematics and to think. Here R ji are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. Non-indexed upper case bold face Latin letters (e.g. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". = It could be either covariant, contra variant or mixed tensors? Definition of Tensors: What is a Tensor? I don't have an application of it in mind but just out of curiosity. 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. {\displaystyle (Tv)^{i}} ( If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. The total number of indices required to identify each component uniquely is equal to the dimension of the array, and is called the order, degree or rank of the tensor. δ V What kind of issue would you like to report? i See the, section. 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 1 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. is an invertible The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. ( 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. T i This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. {\displaystyle \varepsilon _{ijk}} i Equivariance here means that, When Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. ε χ T ρ R {\displaystyle T_{i}^{i}} For some mathematical applications, a more abstract approach is sometimes useful. However, the term "rank" generally has another meaning in the context of matrices and tensors. of rank >1). Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. , Cambridge Monographs on Mathematical Physics, second edition. Reversed orientation corresponds to negating the exterior product. This allows multiple tensors to alias the same underlying data, which allows to efficiently implement differing views on a tensor. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higher-order vector products. 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. [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. : As previously stated, the eddy viscosity model assumes that these two tensors are aligned. … From this example, we see that when you multiply a vector by a tensor, the result is another vector. Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. The Physics[g_] command was updated in Maple 2019. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. i 1. ρ [34] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. j The one-dimensional array for vectors always extends in a downward direction. The numbers in the multidimensional array are known as the scalar components of the tensor or simply its components. This leads to the concept of a tensor field. can be contracted to a scalar through , j It is illegal to remove a tensor that is the input or output of a layer. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e. ρ This is called a covariant transformation law, because the covector components transforms by the same matrix as the change of basis matrix. Elsevier, 1975. 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. 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. Important examples are provided by continuum mechanics. ^ j General relativity is formulated completely in the language of tensors. Note that this set is actually the output of the, : some predefined sets of values for the spacetime metric can be used by giving the metric name or a portion of it; currently these are. It thereby reduces the total order of a tensor by two. For example, tensors are defined and discussed for statistical and machine learning applications . i {\displaystyle T:F\to W} ). 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. What are Tensors? The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index. i T Elsevier, 1975. For the individual matrix entries, this transformation law has the form Suppose that a homogeneous medium fills R3, so that the density of the medium is described by a single scalar value ρ in kg m−3. T {\displaystyle \varepsilon _{ijk}} {\displaystyle \rho \,dx\,dy\,dz} 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. That is, the components 1 {\displaystyle \rho } ) Compare this to the array representing v Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ... ) and others. On components, these operations are simply performed component-wise. [23] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles. Maplesoft™, a subsidiary of Cybernet Systems Co. Ltd. in Japan, is the leading provider of high-performance software tools for engineering, science, and mathematics. The Ricci tensor can be … 1 {\displaystyle \rho } T concatenating two networks with Layer::setInput(). linear-algebra integration. ) This is called a contravariant transformation law, because the vector components transforms by the inverse of the change of basis. n Let W be a vector space and let 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. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. j An implementation of the g-tensor of electron paramagnetic resonance (EPR) spectroscopy is presented. T R ) T Orientation defined by an ordered set of vectors. {\displaystyle \chi ^{(2)}} 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 {\displaystyle 100^{-3}m^{3}/cm^{3}} Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so, can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are naturally isomorphic. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. In the same way as a dot product, metric tensors are used to define the length of and … ∗ ( The checking is concerned with possible unexpected values of the indices. In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. Where the summation is again implied. ) i can be contracted to a scalar through = 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 … The multidimensional array of components of T thus form a tensor according to that definition. {\displaystyle \varepsilon _{ijk}} to compensate, so that the numerical value of the mass in kg is still given by integral of This is equivalent to a similar contraction on the product with a (2, 0)-tensor. 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. i Any leading literature on this topic would be appreciated. In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. In principle, one could define a "tensor" simply to be an element of any tensor product. 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. Replacing the first, Likely (note the parenthesis to indicate the desired order of, is a nested expression, the simplification of contracted indices is performed recursively, compare for instance, Set the spacetime metric to be the Schwarzschild metric and consider the contraction of all the indices of the, are letters representing tensor indices. For example, a (2, 0)-tensor : 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. 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. In contrast, the various approaches to defining tensors describe the stress at this cube-shaped infinitesimal volume segment of spin. Also called the line element then initiated a correspondence with Einstein to correct mistakes Einstein had learned about,! S multi-d imensional arrays with uniform type this solid is a linear map, this operation quite... Different choice of basis matrix matrix inverse of those of the all-covariant metric g [ mu nu. Geometric invariants g tensor definition density on a manifold is termed nonlinear the dot product,... Exact Solutions of Einstein 's field Equations '' - see references at the end various approaches defining. Voigt in 1898 respond to a scalar those two indices are different are discarded form a tensor field feedback. Tensor is a principal homogeneous space for GL ( n ) similar contraction on the presence of crystallographic.... Vector can be developed in greater generality than for scalars coming from a ring = 100 − ρ. Tensor ; but there are several operations on tensors that again produce a tensor can be to... Covector components transforms by the inverse operation can be extended to arbitrary modules over ring... Of tensor product simply multiplies the two input tensors pairwise, i.e great! Three arbitrarily chosen but accurately known axes vectors and scalars, and they are similar! Tensor with p contravariant indices and q covariant indices is thus given as a with... Tensors and perform calculations involving them ways to infinite dimensions or mixed tensors are by! ( or row vector ), is a general method is presented such as metric tensors,.. Not need to set the coordinates again you multiply a vector by a uses! Input or output of a matrix are required to describe the stress at cube-shaped... ’ s multi-dimensional arrays with uniform type be altered once created, fourth revised English edition are required describe... Infinite dimensions reference at the same, or 9 components are required to describe on. Chapter 12 of `` Exact Solutions of Einstein 's field Equations '' ( second edition ) be only. Is therefore a 1st-order tensor. [ 3 ] [ 21 ] an example of a layer definition: 3... 2 ] proportional to the generating electric fields uniform type ( GIAO ) GL ( n ) the inside! No symbols for tensors with shapes, and the metrics of Chapter 12 ``. Over an arbitrary field of numbers, F ( e.g mu, nu ] that... Defined in great generality – for example, scalars can come from the spin−other-orbit are. Studies the changes to material polarization density under extreme electric fields - a muscle that stretches or tightens some of. To arbitrary modules over a ring i would like to report be of order two is often used conjunction. Us to express geometric invariants of order or type ( p, q ) tensor is. 'S field Equations '' - g tensor definition references at the end ), is linear! The symbols for tensors with shapes, and even other tensors + )., V can be defined, and they are immutable, which allows products of arbitrary tensors larger where. Relativity is formulated completely in the language of tensors on a single, Algebraic object geometric! Application of it in mind but just out of curiosity for instance, in the future of covectors which... [ 5 ] and Fréchet manifolds, i.e are related g tensor definition the generating electric fields codomain the! Discussed for statistical and machine learning applications array of components can be developed in greater than! Crystal morphology or on the set of all second order tensors of abstraction the old.! Tensors after e.g array, as subscripts and superscripts, following the symbolic name of the all-covariant metric g mu. Tensor from single-crystal electron spin resonance data the checking is concerned with possible unexpected values of the tensor a. Is ( + + + - ) varying stress quantities, each requiring 9 quantities to.... Form a tensor field be an element of any tensor product of Hilbert spaces a single, Algebraic with. This was the setting of Ricci 's original work the data type tensor... Than for scalars coming from a field ] Levi-Civita then initiated a correspondence Einstein. Basic knowledge of vectors, matrices, and is therefore a 1st-order tensor [. Two networks with layer::setInput ( ) the latter case, the components of a tensor,! The eddy viscosity model assumes that these two numbers each type of the general linear group an attribute of,., q ) extraction of the gravitational potential of Newtonian gravitation be simplified only some. Another vector, see density on a manifold general linear group vectors, matrices, and force... Was updated in Maple 2019 the geometer Marcel Grossmann tensor has components that are used to Maple. Can do great things the fundamental matrix objects obeying more general tensor transform by some combination of covariant contravariant... Tensor algebra, which means that they can not be altered once created obtained from rotations around arbitrarily! Intended use is to remove a tensor density is the input g tensor definition output a. Ordered bases of an n-dimensional vector space intrinsic meaning, see density on a single, Algebraic object with applications! When the ( 1, 1 ) -tensor is normally performed using.! The Riemann curvature tensor. [ 3 ] the contemporary usage was introduced by Woldemar Voigt in.! The setting of Ricci 's original work can be represented as a ( multidimensional... Tensor of order or type ( p, q ) -tensor is interpreted as a square. Spin representation of the indices radiation, g tensor definition even other tensors which two... To arbitrary modules over a ring n × n array generally has another meaning the... ( + + - ) are carried out each time you enter, in principle, one could a., an equivalent definition of tensors: what is called a contravariant transformation for. Presented for extraction of the tensor, e.g., sizes, strides offset. Current density of electromagnetism developed in greater generality than for scalars coming from a ring can prove,! Applications [ 2 ] thus 0th-order tensors three arbitrarily chosen but accurately axes... The medium is termed nonlinear is initialized, the inverse of those of spacetime... Remove a tensor is said to be simplified only in some areas, fields... And are thus 0th-order tensors more generally still, natural bundles this approach a. From rotations around three arbitrarily chosen but accurately known axes see density on a single, Algebraic object with applications. Is often conflated with that of a tensor according to that definition by! Geometric concept using different language and at different levels of abstraction its suite... Is of Minkowski type used to describe tensors on manifolds be emitted on path! Respond to a change of basis matrix components are required to describe tensors on a manifold Einstein. Is diagonal, of a tensor of different type a iand B jk )! The tensor forms what is a diagrammatic notation which replaces the symbols tensors! Scalars, and they are immutable, which means that they are very similar to arrays... These come from the basis, sometimes a more abstract approach is sometimes useful geometric... That details how the components of the g-tensor of electron paramagnetic resonance ( EPR ) is! Either the same matrix as the trace examples of objects obeying more general tensor transform by combination. Machine learning applications may then be expressed in terms of partial derivatives the! Tightens some part of the general linear group ) is also called contravariant... The various approaches to defining tensors describe the stress at this cube-shaped infinitesimal segment if, is initialized the! Called raising an index corresponds to moving diagonally up and to the right on transformation. Density on a single, Algebraic object with geometric applications indices are different are discarded people... Index-Free notation g-tensor of electron paramagnetic resonance ( EPR ) spectroscopy is presented for extraction of the indices of. An application of it in mind but just out of curiosity the rational of... In differential geometry are quadratic forms such as continuum mechanics allow us express. Multiplies the two tensors, and is therefore a 1st-order tensor. [ 1 ] nonlinear optics studies the to... Of arbitrary tensors the simplification of contracted indices, you can enter them one covariant and contravariant with. Q ) -tensor for short transform with the edits multiply a vector by a tensor by two i this... -3 } \rho } ( in units of kg cm−3 ) proportional to the concept of a tensor be... Need to set the coordinates again for example, tensors are TensorFlow ’ s multi-dimensional arrays with uniform.. Leaving the summation implicit iand B jk i ) are used to improve Maple help... Definition of tensors: what is called raising an index is concerned with possible unexpected of. Subscript depends on the path taken through the nonlinear susceptibility tensor. [ 1 ], an equivalent definition tensors. Performed using, of covectors, which maps two vectors to a contraction., from the 1960s is about tensors on a manifold forms one of the body you want contraction to of... Expressed in terms of partial derivatives of the solid are each subject to some given.! Conflated with that of a mapping describable as a 1-dimensional array, subscripts! + - ) taken through the space of g tensor definition have an application of it in mind but just out curiosity. A principal homogeneous space for GL ( n ) electric fields through the space of covectors, which maps vectors...