Other classical groups have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well. Scalars are simple numbers and are thus 0th-order tensors. 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. For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense,[14] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. Each view of a storage can have a different size or offset. 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. 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. The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. [17] Tensors thus live naturally on Banach manifolds[18] and Fréchet manifolds. 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. ′ 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). This is a general property of all second order tensors. When working in curvilinear coordinates so that the components of the, symbols are not zero it is sometimes convenient to rewrite tensorial expressions in terms of the metric, and its derivatives. To perform the related computations as if, is initialized, the default spacetime metric is of Minkowski type. The type is the data type of the tensor, e.g. {\displaystyle \mathbf {e} _{j}} j T [34] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. But the theory is then less geometric and computations more technical and less algorithmic. [Note 3] The contemporary usage was introduced by Woldemar Voigt in 1898. be a representation of GL(n) on W (that is, a group homomorphism 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. {\displaystyle \delta _{j}^{k}} or related to the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" - see references at the end. ) The stresses inside a solid body or fluid are described by a tensor field. T Equivariance here means that, When V [6] In this approach, a type (p, q) tensor T is defined as a multilinear map. {\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}} On components, the effect is to multiply the components of the two input tensors pairwise, i.e. The multidimensional array of components of T thus form a tensor according to that definition. 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. 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, . ). is an equivariant map The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Such a tensor is said to be of order or type (p, q). In modern mathematical terminology such an object is called a tensor field, often referred to simply as a tensor.[1]. where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. 1 j 3 ′ Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. This transformation does not depend on the path taken through the space of frames. 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. 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). Then a tensor of type By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗. These come from the rational representations of the general linear group. Contraction of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table. 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. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. What kind of issue would you like to report? m Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. , Cambridge Monographs on Mathematical Physics, second edition. Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. T Two examples, together with the vectors they operate on, are: The stress tensor 1 [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]. [36] Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations. This implementation is based on density functional theory (DFT) and the use of gauge-including atomic orbitals (GIAO). j A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:[19]. a iand B jk i) are used to denote tensors of rank >0 in their explicit tensor form (index notation). You can only create a new copy with the edits. The transformation law may then be expressed in terms of partial derivatives of the coordinate functions. As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array •, 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. Question:is there a definition on how to integrate a tensor? y 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. {\displaystyle (Tv)^{i}} This discussion motivates the following formal definition:[3][4], Definition. 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. d T Definition of Tensors: What is a Tensor? It could be either covariant, contra variant or mixed tensors? f Cambridge University Press, 2003. {\displaystyle {\hat {T}}=R^{-1}TR} {\displaystyle \rho :{\text{GL}}(n)\to {\text{GL}}(W)} {\displaystyle T_{i}^{i}} ρ Compare this to the array representing Elsevier, 1975. On components, these operations are simply performed component-wise. For example, a (2, 0)-tensor j i Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E. Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. . 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. ), 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. Orientation defined by an ordered set of vectors. This notation captures the expressiveness of indices and the basis-independence of index-free notation. {\displaystyle T:F\to W} 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 … 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. which again produces a map that is linear in all its arguments. 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. a (p + q)-dimensional array of components can be obtained. Objects that tensors may map between include vectors and scalars, and even other 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 more on the intrinsic meaning, see Density on a manifold. {\displaystyle R=(R_{j}^{i})} 3 In $${\displaystyle n}$$ dimensions (of arbitrary signature): 3 [9] For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product). : 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 ∗ Tensors can have spacetime and space indices at the same time. For example, tensors are defined and discussed for statistical and machine learning applications . The reader must be prepared to do some mathematics and to think. -symbol, mapping two vectors to one vector, would have order 2 + 1 = 3. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors), where the new basis vectors This table shows important examples of tensors on vector spaces and tensor fields on manifolds. i × j the place where most texts on tensor analysis begin. {\displaystyle {v}^{i}\,\mathbf {e} _{i}} 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 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. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold. The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as. [37] Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s. 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. T i . A basic knowledge of vectors, matrices, and physics is assumed. This can be done directly from, ; it automatically implies on setting spherical coordinates as the differentiation variables for, Or, for the matrix components of the all contravariant, , which as a matrix is equal to the inverse of the all-covariant, in different ways. 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 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 n One, for instance, is via the tensor product of Hilbert spaces. The force's vector components are also three in number. A Visualization of Rank-3 Tensors (Figure by Author) Tensors are TensorFlow’s multi-d imensional arrays with uniform type. as. F 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. 3 General relativity is formulated completely in the language of tensors. 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. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. ( A different choice of basis will yield different components. This is called a contravariant transformation law, because the vector components transforms by the inverse of the change of basis. I would like to report a problem with this page, • Student Licensing & Distribution Options. What is a Tensor(A Simple Definition) Tensors are a type of data structure used in machine learning to represent various kinds of objects including scalars, vectors, arrays, matrices and other tensors. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type. © Maplesoft, a division of Waterloo Maple Inc. 2020. i 1 Where the summation is again implied. W Element-Wise Tensor Operations 4. R an algebraic expression involving tensors (e.g. However, the space of frames is not simply connected (see orientation entanglement and plate trick): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. They are denoted by indices giving their position in the array, as subscripts and superscripts, following the symbolic name of the tensor. k In the same way as a dot product, metric tensors are used to define the length of and … k 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. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor. For some mathematical applications, a more abstract approach is sometimes useful. Some define tensors as multidimensional arrays. returns unevaluated, after normalizing its indices taking into account that the spacetime metric is symmetric. For example, a (1, 1)-tensor Then F is a principal homogeneous space for GL(n). For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above: where ^ T 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. Tensor definition is - a muscle that stretches a part. {\displaystyle \mathbf {\hat {e}} _{i}} The notion of a tensor can be generalized in a variety of ways to infinite dimensions. n Some automatic checking and normalization are carried out each time you enter, . See the, section. ⊗ 2. i [20][21] An example of a tensor density is the current density of electromagnetism. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. and the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" (second edition). A tensor is a linear mapping of a vector onto another vector. i {\displaystyle \rho } i Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. 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.

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