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Encyclopedia results for Tensor product

Tensor product





Encyclopedia results for Tensor product

  1. Tensor product

    In mathematics , the tensor product , denoted by , may be applied in different contexts to vector space ... operation . In some contexts, this product is also referred to as outer product . The term tensor product is also used in relation to monoidal category monoidal categories . Tensor product of vector spaces The tensor product V     W of two vector space s V and W over a field mathematics field ... generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space math V otimes W F V times W R. math It is also called the tensor product space ... . The tensor product of two elements v and w is the equivalence class e sub v , w sub R of e sub ... before v sub i 1 sub     w sub j sub . The dimension of the tensor product therefore is the product ..., an element of the tensor product space is not a pure tensor, but rather a finite linear ... tensors required to express an element of a tensor product is called the tensor rank , not to be confused with tensor order , which is the number of spaces one has taken the product of, in this case ... property The tensor product is universal property universal , in that any bilinear map from V × ... object of that category. And the tensor product V     W can be defined as the image of any initial object , which is isomorphic to . As a functor The tensor product also operates on linear ..., the tensor product of the two linear maps S and T is a linear map math S otimes T V otimes W rightarrow ... of the tensor product of the two corresponding linear maps under a standard choice of bases of the tensor products see Kronecker product Relation to the abstract tensor product the article on Kronecker ... sub , V sub 2 sub , and V sub 3 sub are three vector spaces. The tensor product V sub 1 sub     ... as math F L circ varphi math where L is a linear map. The tensor product is uniquely characterized ... V 3 cong V 1 otimes V 2 otimes V 3. math More generally, the tensor product of an arbitrary indexed ...   more details



  1. Tensor product network

    A tensor product network , in neural network s, is a network that exploits the properties of tensor s to model associative concepts such as variable assignment . Orthonormality Orthonormal vectors are chosen to model the ideas such as variable names and target assignments , and the tensor product of these Euclidean vector vector s construct a network whose mathematical properties allow the user to easily extract the association from it. DEFAULTSORT Tensor Product Network Category Neural networks See also Neural network Neuroscience Sci stub Orphan date December 2009 Unreferenced stub auto yes date December 2009 ...   more details



  1. Tensor product of algebras

    ref improve date May 2010 In mathematics , the tensor product of two algebra ring theory R algebras is also an R algebra. This gives us a tensor product of algebras . The special case R Z gives us a tensor product of rings , since rings may be regarded as Z algebras. Let R be a commutative ring and let A and B be algebra ring theory R algebras . Since A and B may both be regarded as module mathematics R modules , we may form their tensor product of modules tensor product math A otimes R B math which is also an R module. We can give the tensor product the structure of an algebra by defining ref Kassel 1995 , Google books quote id S1KE pToY98C page 32 text we put an algebra structure on the tensor product p. 32 . ref math a 1 otimes b 1 a 2 otimes b 2 a 1a 2 otimes b 1b 2 math and then extending by linearity to all of A sub R sub B . This product is easily seen to be R bilinear, associative ... of A and B . If A and B are both commutative then the tensor product is as well. The tensor product turns the category mathematics category of all R algebras into a symmetric monoidal ... 1 B math math b mapsto 1 A otimes b math These maps make the tensor product a coproduct in the category of commutative R algebras. The tensor product is not the coproduct in the category of all R algebras. There the coproduct is given by a more general free product of algebras . The tensor product ... known as fiber product s. Stuf removed from ring mathematics Since any ring is both a left and right module mathematics module over itself, it is possible to construct the tensor product of rings tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T . See also Extension of scalars Tensor product of modules Tensor product of fields Notes ... 155 series Graduate texts in mathematics publisher Springer isbn 9780387943701 . DEFAULTSORT Tensor Product Of Algebras Category Algebras Category Ring theory Category Commutative algebra Category Multilinear ...   more details



  1. Tensor product of quadratic forms

    Unreferenced date February 2008 The tensor product of quadratic form s is most easily understood when one views the quadratic forms as quadratic spaces . So, if V, q sub 1 sub and W, q sub 2 sub are quadratic spaces, which V,W vector spaces, then the tensor product is a quadratic form q on the Tensor product Tensor product of vector spaces tensor product of vector spaces math V otimes W math . It is defined in such a way that for math v otimes w in V otimes W math we have math q v otimes w q 1 v q 2 w math . In particular, if we have diagonalizations of our quadratic forms which is always possible when the characteristic algebra characteristic is not 2 such that math q 1 cong langle a 1, ... , a n rangle math math q 2 cong langle b 1, ... , b m rangle math then the tensor product has diagonalization math q 1 otimes q 2 q cong langle a 1b 1, a 1b 2, ... a 1b m, a 2b 1, ... , a 2b m , ... , a nb 1, ... a nb m rangle. math Category Quadratic forms Category Tensors math stub ...   more details



  1. Tensor product of graphs

    Image Graph tensor product.svg thumb 360px The tensor product of graphs. In graph theory , the tensor product G H of graphs G and H is a graph such that the vertex set of G H is the Cartesian product V ... is adjacent with v . The tensor product is also called the direct product , categorical product , cardinal product , relational product , Kronecker product , weak direct product , or conjunction . As an operation on binary relations, the tensor product was introduced by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica 1912 . It is also equivalent to the Kronecker product ... sometimes used to represent another construction known as the Cartesian product of graphs , but more commonly refers to the tensor product. The cross symbol shows visually the two edges resulting from the tensor product of two edges. Examples The tensor product G K sub 2 sub is a bipartite graph , called ... graph a complete bipartite graph K sub n , n sub minus a perfect matching . The tensor product ... row or column of the grid. Properties The tensor product is the Product category theory category theoretic product in the category of graphs and graph homomorphism s. That is, there is a homomorphism ... through the homomorphisms to G and H . The adjacency matrix of G H is the Kronecker product tensor product of the adjacency matrices of G and H . If a graph can be represented as a tensor product, then there may be multiple different representations tensor products do not satisfy unique factorization ... time algorithm for recognizing tensor product graphs and finding a factorization of any such graph. If either G or H is bipartite , then so is their tensor product. G H is connected if and only if both .... The Hedetniemi conjecture gives a formula for the chromatic number of a tensor product. References cite journal author Imrich, W. title Factoring cardinal product graphs in polynomial time journal ... Product Graphs Structure and Recognition publisher Wiley year 2000 isbn 0 471 37039 8 cite journal ...   more details



  1. Topological tensor product

    In mathematics , there are usually many different ways to construct a topological tensor product of two ... theory of tensor product s see Tensor product of Hilbert spaces , but for general Banach space s or locally ... to avoid it if possible. Fact date January 2008 Tensor products of Hilbert spaces Main Tensor product of Hilbert spaces The algebraic tensor product of two Hilbert spaces A and B has a natural positive ... A B , called the Hilbert space tensor product of A and B . If the vectors a sub i sub and b sub j ... basis of A B . Cross norms and tensor products of Banach spaces We shall use the notation from harv Ryan 2002 in this section. The obvious way to define the tensor product of two Banach spaces A and B is to copy the method for Hilbert spaces define a norm on the algebraic tensor product, then take ... a norm on the tensor product. If A and B are Banach spaces the algebraic tensor product of A and B means the tensor product of A and B as vector spaces and is denoted by math A otimes B math . The algebraic tensor product math A otimes B math consists of all finite sums math x Sigma i 1 n a i otimes ... tensor product math A otimes B math is a norm satisfying the conditions math p a otimes ... tensor product in these two norms are called the projective and injective tensor products, and are denoted ... space tensor product is not equal to either of these norms in general. Some authors denote it by , so the Hilbert space tensor product in the section above would be math A hat otimes sigma ... cross norm then &alpha defines a reasonable cross norm on the algebraic tensor product math A otimes ... A otimes B math and on the completed tensor product math A hat otimes alpha B math for an element x ... space , Frechet space , locally convex topological vector space , Nuclear space Tensor product of Hilbert ... , math alpha u sup alpha Q E otimes Q F u X E otimes Y F dim X E, dim Y F infty . math A tensor norm ... cross norm math varepsilon math defined above are tensor norms and they are called ...   more details



  1. Tensor product of Hilbert spaces

    In mathematics , and in particular functional analysis , the tensor product of Hilbert space s is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert space is another Hilbert space. Roughly speaking, the tensor product is the complete metric space completion of the ordinary tensor product. This is a special case of a topological tensor product ..., and therefore a topology, on the tensor product that arise naturally from those of the factors. Let  ..., cdot rangle 1 math and math langle cdot, cdot rangle 2 math , respectively. Construct the tensor product ... product s. We can turn this vector space tensor product into an inner product space by defining math ... tensor product. Finally, take the complete space Completion completion under this inner product. The resulting Hilbert space is the tensor product of   H sub 1 sub and  H sub 2 sub . Explicit construction The tensor product can also be defined without appealing to the metric space completion. If H sub 1 sub and H sub 2 sub are two Hilbert spaces, one associates to every simple tensor product ... tensor product of H sub 1 sub and H sub 2 sub , that is isometrically and linearly isomorphic to HS H sub 1 sub sup &lowast sup , H sub 2 sub . Universal property The Hilbert tensor product math H H ... the tensor product H up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. Infinite tensor ... of unit vectors in these Hilbert spaces then the incomplete tensor product or Guichardet tensor product ... math . Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set ... of the tensor product is the product as cardinal number s of the Hilbert dimensions. Examples ... in the construction of the Hilbert space tensor product. Similarly, we can show that L sup 2 sup ... of both particles is described by the tensor product of  H sub 1 sub and  H sub 2 sub . For example ...   more details



  1. Tensor product of modules

    In mathematics , the tensor product of modules is a construction that allows arguments about bilinear ... s . The module construction is analogous to the construction of the tensor product of vector space ... of bilinear or multilinear operations via linear operator linear operations . The tensor product ... product of modules can be iterated to form the tensor algebra of a module, allowing one to define ... M , N Z . Definition Let M , N and R be as in the previous section. The tensor product over R math M .... ref Image Tensor product of modules.png 200px right For every abelian group Z and every bilinear ... universal properties , the above property defines the tensor product uniquely up to a unique isomorphism .... The tensor product can also be defined as a representable functor representing object for the functor ..., then we see that the properties for the tensor product are satisfied. Therefore, the tensor product ... PA97 p. 97 , Ex. 4.5.1. ref Construction To construct the tensor product, we can proceed ... product of M and N is rarely isomorphic to the tensor product of M and N . When R is not commutative, then the tensor product requires that M and N be modules on opposite sides, while the direct product ... pair as input, and assigns them to the tensor product in the category of abelian groups . By fixing ... the definition to a tensor product of any number of spaces. For example, the universal property ... tensor product is associative M sub 1 sub M sub 2 sub M sub 3 sub is naturally isomorphic to M sub 1 sub M sub 2 sub M sub 3 sub . The tensor product of three modules defined by the universal property ... a commutative ring, then their tensor product is again an R module. If R is a ring, sub R sub M is a left ... of R on M factors through an action of a quotient commutative ring. In this case the tensor product .... See also Tor functor Tensor product of algebras Tensor product of fields Notes references References ... ring , with result an abelian group . Tensor products are important in areas of abstract algebra , homological ...   more details



  1. Tensor product of fields

    s of a smaller field N for example a prime field . The tensor product of fields is the best available ... M . Correspondingly their tensor product will in that case be the trivial ring collapse of the construction ... we can identify K . L as a vector space tensor product , taken over the field N that is the intersection ... Cyclotomic field ref The tensor product as ring To get a general theory, we need to consider a ring structure on math K otimes mathrm N L math . We can define the product math a otimes b c otimes d math ... a ring structure on the tensor product, making math K otimes mathrm N L math into a commutative algebra over a field N algebra , called the tensor product of fields . Analysis of the ring structure ... b . math The kernel of will be a prime ideal of the tensor product and conversely any prime ideal of the tensor product will give a homomorphism of N algebras to an integral domain inside a field ... can speak of the product of all embeddings of K and L in various M , over N . In case K and L are finite extensions of N, the situation is particularly simple, since the tensor product is of finite dimension ... K otimes mathbf Q K math is the product of a copy of K , and a splitting field of X sup 3 sup &minus 2, of degree 6 over Q . One can prove this by calculating the dimension of the tensor product ... one, of products of fields alone. See also Extension of scalars tensor product of a field ...In mathematics , the theory of field mathematics fields in abstract algebra lacks a direct product the direct product of two fields, considered as a ring mathematics ring is never itself a field. On the other ... a field, and often a direct product of fields it can, though, contain non zero nilpotents see radical ... math K otimes mathrm N mathrm L mathrm R math a direct product of finitely many fields. Each such field .... Classical theory of real and complex embeddings In algebraic number theory , tensor products ... otimes mathbb Q mathbb R math is always a product of fields isomorphic to R or C . The totally real ...   more details



  1. Tensor product model transformation

    Orphan date May 2010 The tensor product TP model transformation originally proposed for control design by Baranyi ref name compind cite journal author P. Baranyi and D. Tikk and Y. Yam and R. J. Patton title From Differential Equations to PDC Controller Design via Numerical Transformation journal Computers in Industry, Elsevier Science year 2003 volume 51 pages 281&ndash 297 ref ref name Baranyi04 cite journal author P. Baranyi title TP model transformation as a way to LMI based controller design journal IEEE Transaction on Industrial Electronics year 2004 volume 51 number 2 month April pages 387&ndash 400 ref is capable of numerically reconstructing the higher order singular value decomposition HOSVD of continuous multi variable functions. The HOSVD of functions is defined by Baranyi et al. in ref name canon1 cite book author P. Baranyi and L. Szeidl and P. V rlaki and Y. Yam title Definition of the HOSVD based canonical form of polytopic dynamic models booktitle 3rd International Conference ... details could be found in ref name JGDC cite journal author P. Baranyi title Tensor Product Model Based ... of elements. The given function can be a matrix or tensor, and it is irrelevant whether it is given ... is based on the analogy of tensor HOSVD . The key difference is that instead of singular matrices ... of tensor HOSVD can be carried over to function HOSVD . In the same way the main component analysis ..., in the same manner as the tensor HOSVD is used for complexity reduction. The TP model transformation ... R N math . Note that math mathbf F mathbf x math can be matrix or even tensor. In the followings ... tensor notation math mathbf F mathbf x mathcal S boxtimes n 1 N mathbf w n x n , math where the N 2 dimensional coefficient tensor math mathcal S in mathcal R I 1 times I 2 times ldots I N times ... math are respectively assigned to the elements of the parameter vector math mathbf x math . For tensor ... S math stored in tensor math mathcal S math , for all math mathbf x in Omega math . The TP model transformation ...   more details



  1. Tensor

    in terms of elements of tensor product s of vector spaces, which in turn are defined through a universal property . A type n , m tensor is defined in this context as an element of the tensor product ... m , n tensor. Moreover, the universal property of the tensor product gives a bijection 1 to 1 correspondence ... form is the same thing as a 0, 2 tensor an inner product is an example of a 0, 2 tensor, but not all ... functional , 1 form linear transformation m 2 bilinear form , e.g. inner product , metric tensor ..., however there also exist operations that change the type of the tensors. Tensor product Main Tensor product The tensor product takes two tensors, S and T , and produces a new tensor, S T , whose ... n , m , then the tensor product S T has type k n , l m . Contraction Main Tensor contraction ... as the trace linear algebra trace . The contraction is often used in conjunction with the tensor product ... of a tensor as an element of a tensor product of copies of the space V with the space V sup ... Fibre bundle One form Tensor field Tensor product Tensor product of modules Multilinear subspace ...About a modern but abstract treatment Tensor intrinsic definition other uses Tensor disambiguation Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . Image Components stress tensor cartesian.svg 300px right thumb Stress, a second order tensor. The tensor s components ... curvature tensor . ref name Kline cite book title Mathematical thought from ancient to modern ... theorem stress tensor stress tensor T takes a direction v as input and produces the stress T sup v sup ... independent of a particular choice of coordinate system . It is possible to represent a tensor by examining ... of numerical values. The coordinate independence of a tensor then takes the form of a covariant transformation ... to that computed in another one. The order or degree of a tensor is the dimensionality of the array ..., a 0th order tensor. A coordinate vector , or 1 dimensional array, can represent a vector, a 1st order ...   more details



  1. Tensor (disambiguation)

    Tensor may refer to Mathematics Tensor , geometric entities introduced into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices Antisymmetric tensor Bach tensor Completely antisymmetric tensor Cotton tensor Diffusion tensor Dyadic tensor Einstein tensor Electrogravitic tensor Electromagnetic stress energy tensor Electromagnetic tensor Four tensor Gyration tensor Inertia tensor Lanczos tensor Maxwell stress tensor Metric tensor general relativity Metric tensor Mixed tensor Polder tensor Ricci tensor Riemann curvature tensor Scalar tensor theory Schouten tensor Strain tensor Stress energy tensor Structure tensor Symmetric tensor Tensor intrinsic definition Tensor algebra Tensor bundle Tensor category Tensor contraction Tensor density Tensor field Tensor product of algebras Tensor product of modules Tensor product Topogravitic tensor Topological tensor product Torsion tensor Weyl tensor Anatomy Tensor veli palatini muscle Tensor fasciae latae muscle Tensor tympani muscle Other Tensor Trucks , skateboarding truck company See also Glossary of tensor theory Curvature tensor disambiguation Stress tensor disambiguation Tense disambiguation disambig ...   more details



  1. Schouten tensor

    In mathematics , more specifically Riemannian geometry , the Schouten tensor is for n 3 dimensions, math P frac 1 n 2 left Ric frac R 2 n 1 g right , math where Ric is the Ricci tensor , R is the scalar curvature , g is the Riemannian metric and n is the dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni&ndash Nomizu product of the Schouten tensor with the metric. Category Riemannian geometry Category Tensors geometry stub ru ...   more details



  1. Product

    wiktionarypar product Product may mean Business Product business , an item that ideally satisfies a market s want or need Product breakdown structure Product project management , a deliverable or set of deliverables that contribute to a business solution Sciences Product biology , something manufactured by an organelle Product chemistry , a substance found when a chemical reaction ends Product mathematics , the result of multiplying Product category theory , a more abstract product Arts and entertainment Product album , a musical recording by Norwegian group De Press Product, a three CD compilation set by Buzzcocks .the .product, a notable 64K demo by the demogroup Farbrausch disambig bs Proizvod vor ca Producte cs Produkt da Produkt de Produkt et Toode es Producto fr Produit ia Producto disambiguation it Prodotto nl Product no Produkt pl Produkt pt Produto ro Produs ru simple Product sk Produkt sv Produkt th tr r n uk ur ...   more details



  1. By-product

    refimprove article date March 2011 A by product is a secondary or incidental product deriving from a manufacturing process, a chemical reaction or a biochemical pathway, and is not the primary product or service being produced. A by product can be useful and market able, or it can be considered waste . Water can also be a by product when a reaction causes carbon dioxide. IEA offers the following definition for the purpose of life cycle assessment ref http www.ieabioenergy task38.org systemdefining biomitre technical manual.pdf BIOMITRE Technical Manual, Horne, R. E. and Matthews, R., November 2004 ref ... main products, co products which involve similar revenues to the main product , by products which result in smaller revenues , and waste products which provide little or no revenue . Major by products Animal sources blood meal from slaughterhouse operations poultry by product meal clean parts of the carcass of slaughtered poultry, such as necks, feet, undeveloped eggs, and intestines chrome shavings from a stage of leather manufacture collagen and gelatin from the boiled skin and other parts of slaughtered livestock feather s from poultry processing feather meal from poultry processing fetal pig s lanolin from the cleaning of wool manure from animal husbandry meat and bone meal from the rendering food processing rendering of animal bones and offal poultry litter swept from the floors of chicken coops whey from cheese manufacturing Vegetation acidulated soap stock from the refining of vegetable oil bagasse the fibrous residue remaining after sugarcane or sorghum stalks are crushed to extract their juice black liquor from the production of Pulp paper cellulose pulp using the Kraft ... extraction Other Motor vehicle emissions sludge from wastewater treatment See also By product synergy Co product References Reflist Category Manufacturing Category Recycling ca Subproducte de Nebenprodukt ... Subproduto simple By product sv Biprodukt th ...   more details



  1. The Product

    citation style date February 2011 Infobox musical artist Name The Product Background group or band Origin Detroit, Michigan Genre Alternative Rock URL URL productrock.com Years active Start date 2009 Present Current members B.J. Perry br R.J. Perry br Rich Bennett br Charlie Jewell br Past members Jerome Reilly The Product is an Alternative Rock band based out of Detroit, Michigan . The Product formed in early 2009 with the original line up of B.J. Perry, R.J. Perry, Rich Bennett and Jerome Reilly. Several months later the band parted ways with Jerome, and picked up former band mate and high school friend Charlie Jewell. They have released one E.P. titled Break The Silence along with several singles. History Break The Silence 2009 Their debut EP, Break The Silence, was self released in 2009 and generated an instant buzz throughout the Midwest. ref name Biography http restlessmanagement.com ?p 106 , Biography ref The track Nightmare stayed at Number 1 on Alternative Addiction s top ten Unsigned Bands list for several months. The band went on to tour the Midwest extensively and sold over 4,000 copies of BTS. Also, the track from the E.P. Better Off This Way had a clip appear in the hit MTV show Jersey Shore ref name Jersey Shore http www.mtv.com videos jersey shore season 2 ep 18 drunk punch love 1656631 playlist.jhtml series 2211&seriesId 29241&channelId 1 , Jersey Shore ref One ... name TNA http www.tnastars.com 2010 11 01 the product debuts official theme song for tna final resolution ...?id 85011 , Red Wings. ref The song is expected to be re released on The Product s follow up release So Alive expected in 2011. The Product was also named the number one unsigned band of the year by voters ... 2010 12 01 the product , Daily Unsigned. ref So Alive 2011 Present The band is currently ... ref name Producer http www.alternativeaddiction.com musicnews article 1911 The Product Ready To Make Their Move , Greg Archilla ref Tour The Product has toured throughout the Midwest gaining a loyal ...   more details



  1. Weyl tensor

    n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor , s is the scalar curvature , and h small O small k denotes the Kulkarni&ndash Nomizu product ...In differential geometry , the Weyl curvature tensor , named after Hermann Weyl , is a measure of the curvature ... tensor , the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic . The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how ... force. The Ricci curvature , or trace linear algebra trace component of the Riemann tensor contains ... tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace free Tensor contraction Metric contraction ... tensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero. If the Weyl tensor ... system in which the metric tensor is proportional to a constant tensor. This fact was a key component ... . Definition The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a 0,4 valence tensor by contracting with the metric . The 0,4 valence Weyl tensor is then Harv Petersen 2006 p 92 NumBlk math W R ... 1,3 valent Weyl tensor is then given by contracting the above with the inverse of the metric. The decomposition EquationNote 1 expresses the Riemann tensor as an orthogonal direct sum of vector bundles ... decomposition , expresses the Riemann curvature tensor into its irreducible representation irreducible ... tensor further decomposes into invariant factors for the action of the special orthogonal group , the self dual and antiself dual parts W sup sup and W sup &minus sup . The Weyl tensor can also be expressed using the Schouten tensor , which is a trace adjusted multiple of the Ricci tensor, math ... math R abcd math is the Riemann tensor, math R ab math is the Ricci tensor, math R math is the Ricci ...   more details



  1. Tensor density

    In differential geometry , a tensor density transforms as a tensor when passing from one coordinate system to another see classical treatment of tensors , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the coordinate transition function or its absolute value. For example, a mixed order 2 tensor density of weight W transforms as math bar mathfrak A alpha beta det left frac partial x iota partial bar x gamma right W , frac partial bar x alpha partial ... A math is the order 2 tensor density in the x coordinate system, math bar mathfrak A math is the transformed tensor density in the math bar x math coordinate system and we use the Jacobian determinant . A distinction is made between odd tensor densities, in which as here the term attributable to the determinant may be negative, and even tensor densities which have a power of the absolute value of the determinant, or an even power of it, in the transformation rule. A tensor density of weight zero is an ordinary tensor. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle . General relativity If one transforms from a locally inertial coordinate system where the metric tensor general relativity metric is the Minkowski metric , diag ... sqrt g math where math g , det , g mu nu , math is the determinant of the metric tensor, which is negative ... if , g alpha beta eta alpha beta ,. math Consequently, an even tensor density, math mathfrak ... math p mu dots , math is a tensor which everywhere has the same values as the tensor density in a locally inertial coordinate system. The covariant derivative of an even tensor density is defined as math ..., namely math W , Gamma delta delta alpha , mathfrak p mu dots ,. math Or equivalently, the product ... derivative of g , the determinant of the metric tensor, is always zero math g alpha 0 ,. math ... tensoriale pt Densidade de tensor ...   more details



  1. Nonmetricity tensor

    In mathematics , the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor . It is therefore a tensor field of Tensor Tensor rank rank three. It vanishes for the case of Riemannian geometry . Category Differential geometry Geometry stub ...   more details



  1. Tensor contraction

    is the linear transformation from the Component free treatment of tensors Definition Tensor Product of Vector Spaces tensor product of these two spaces to the field k math C V otimes V rightarrow ..., in the presence of an inner product also known as a Metric tensor metric g , such contractions ... of tensors T and U . The tensor product math T otimes U math is a new tensor, which, if it has at least ... of the same term. This implements the tensor product, yielding a composite tensor. Contracting two ... make sense in this context. See also Tensor product Partial trace Interior product Raising and lowering ...In multilinear algebra , a tensor contraction is an operation on one or more tensor s that arises from ... vector space dual . In components, it is expressed as a sum of products of scalar components of the tensor ... to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices one a subscript, the other a superscript of the tensor are set equal to each other and summed ... tensor with Tensor Tensor rank rank or order reduced by 2. Tensor contraction can be seen as a generalization ... is in V . The map C defines the contraction operation on a tensor of type 1,1 , which is an element ..., a tensor of type m , n with m 1 and n 1 is an element of the vector space math V otimes cdots ..., defines the k , l contraction operation, which is a linear map which yields a tensor of type m   ... of f in the corresponding dual basis . Since a general mixed dyadic tensor is a linear combination ... case follows let math mathbf T T i j mathbf e i e j math be a mixed dyadic tensor. Then its contraction ... tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type 2,2 on the second and third indices to create a new tensor U of type 1,1 is written as math ... e i e j math be an unmixed dyadic tensor. This tensor does not contract if its base vectors are dotted the result is the contravariant metric mathematics metric tensor , math g ij mathbf e i cdot ...   more details



  1. Tensor field

    the tensor product concept is independent of any choice of basis, taking the tensor product of two ... of W , the tensor product of V with L sup s sup , and consider tensor density fields with weight s . Half ...Unreferenced date August 2008 In mathematics , physics and engineering , a tensor field assigns a tensor to each point of a mathematical space typically a Euclidean space or manifold . Tensor fields are used ... physics stress and strain tensor strain in materials, and in numerous applications in the physical sciences and engineering. As a tensor is a generalization of a scalar physics scalar a pure number representing a value, like length and a Euclidean vector vector a geometrical arrow in space , a tensor ... tensor fields , that is, fields defined over a manifold which define a tensor at every point of the manifold. An example is the Riemann curvature tensor . Geometric introduction Intuitively, a vector ... velocity at each point of the Earth s surface. The general idea of tensor field combines the requirement ... tensor &mdash with the idea that we don t want our notion to depend on the particular method ... explanation The contemporary mathematical expression of the idea of tensor field breaks it down into a two ... of tensor field , namely as a section fiber bundle section of some tensor bundle . There are vector bundles which are not tensor bundles the M bius band for instance. This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space math V otimes cdots otimes V otimes ... . See also tangent bundle and cotangent bundle . Given two tensor bundles E M and F M , a map A E F from the space of sections of E to sections of F can be considered itself as a tensor section ..., where f is a smooth function on M . Thus a tensor is not only a linear map on the vector space ... tensor torsion and Affine connection curvature tensors built from them are. Notation The notation ...   more details



  1. Tensor algebra

    Unreferenced date March 2009 In mathematics , the tensor algebra of a vector space V , denoted T V or T sup sup V , is the algebra over a field algebra of tensor s on V of any rank with multiplication being the tensor product . It is the free algebra on V , in the sense of being left adjoint to the forgetful ... integer k , we define the k sup th sup tensor power of V to be the tensor product of V with itself ... by the canonical isomorphism math T kV otimes T ell V to T k ell V math given by the tensor product, which is then extended by linearity to all of T V . This multiplication rule implies that the tensor ... of the corresponding universal property see below . The tensor algebra also has two Coalgebra ... consists of all tensors on V of Tensor Tensor rank rank k . By convention T sup 0 sup V is the ground ... generalizes in straightforward manner to the tensor algebra of any module mathematics ... because the iterated tensor products cannot be formed. Adjunction and universal property The tensor ..., the tensor algebra satisfies the following universal property , which formally expresses the statement ... by the following commutative diagram Image TensorAlgebra 01.png center Universal property of the tensor ..., define the tensor algebra T V as the unique algebra satisfying this property specifically, it is unique ... exists. The above universal property shows that the construction of the tensor algebra is functorial ... dimension n , another way of looking at the tensor algebra is as the algebra of polynomials over K in n ... the given coordinate of the vector . Quotients Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing ... algebra s. Coalgebra structures The tensor algebra has two coalgebra structures one simple one ... structure on the tensor algebra is given as follows. The coproduct is defined by math Delta ... to bigoplus i j m T iV otimes T jV math and is also compatible with the grading. The tensor algebra ...   more details



  1. Symmetric tensor

    r math a tensor of order r . Then T is a symmetric tensor if math tau sigma T T math for the Tensor product Tensor powers and braiding braiding maps associated to every permutation &sigma on the symbols ... anti symmetrization. If T is a simple tensor, given as a pure tensor product math T v 1 otimes v 2 otimes cdots otimes v r math then the symmetric part of T is the symmetric product of the factors ...In mathematics , a symmetric tensor is tensor that is invariant under a permutation of its vector arguments. Thus a r sup th sup order symmetric tensor represented in coordinates as a quantity with r indices ... concept is that of the antisymmetric tensor or alternating form . Symmetric tensors occur widely ... of V , any symmetric tensor T of rank r can be written as math T sum i 1, dots,i r 1 N T i 1i 2 dots ... T i 1i 2 dots i r math the components of the tensor in the basis that are symmetric on the indices. That is to say ... coefficient . Symmetric part of a tensor If math scriptstyle T in V otimes r math is a tensor of order r , then the symmetric part of T is the symmetric tensor defined by math operatorname ... cdots otimes e i r . math The components of the tensor appearing on the right are often denoted ... used in physics and engineering can be represented as symmetric tensor fields for example, stress physics stress , strain tensor strain , and anisotropic conductivity disambiguate thermal or electrical ... tensor of order 2 can be diagonalized . More precisely, for any tensor T   &isin   Sym sup 2 ... number n for which such a decomposition is possible is the rank of a tensor rank of T . The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the ellipsoid ... varieties , and are often studied as such. See also antisymmetric tensor transpose symmetric polynomial ... sym trique nl Symmetrische tensor pt Tensor sim trico ru ...   more details



  1. Stress tensor

    Stress tensor may refer to Stress mechanics , in classical physics Stress energy tensor , in relativistic theories Maxwell stress tensor , in electromagnetism See also Stress disambiguation Tensor disambiguation disambig de Spannungstensor he ...   more details



  1. Electromagnetic tensor

    product of the field strength tensor a Lorentz invariant is formed math F mu nu F mu nu 2 left B 2 frac E 2 c 2 right mathrm invariant math The product of the tensor math F mu nu , math with its dual tensor gives the pseudoscalar invariant math frac 1 2 epsilon alpha beta gamma delta F alpha beta ... tensor or electromagnetic field tensor sometimes called the field strength tensor , Faraday tensor or Maxwell bivector is a mathematical object that describes the electromagnetic field of a physical system in Maxwell s theory of electromagnetism . The field tensor was first used after the 4 dimensional tensor formulation of special relativity was introduced by Hermann Minkowski . The tensor ... In this article, the abstract index notation will be used. The electromagnetic tensor starts with the Electromagnetic ... nabla phi , math math vec B vec nabla times vec A , math By definition, the electromagnetic tensor ... z c & B y & B x & 0 end bmatrix math Properties From the matrix form of the field tensor, it becomes clear that the electromagnetic tensor satisfies the following properties antisymmetric antisymmetry ... change of the magnetic field with respect to time . With the electromagnetic tensor, the equations ... F gamma alpha 0 math The field tensor and relativity The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law , this general property ... systems this led to the introduction of tensor s. The tensor formalism also leads to a mathematically ... may be written using the field tensor as math F alpha beta, gamma , 0 math and math F alpha beta ... tensor, so this finally simplifies to cellpadding 2 style border 2px solid ccccff math partial ... for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED. See also Application of tensor theory in physics Classification ... Relativity Category Tensors Category Tensors in general relativity ca Tensor electromagn tic de Elektromagnetischer ...   more details




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