contraction of symmetric and antisymmetric tensor

The contraction of symmetric tensors with anti-symmetric led to this conclusion. The flow velocity difference between adjacent layers can be measured in terms of a velocity gradient, given by , and the dimensions of distance are 0. J General symmetric contractions Application to coupled-cluster 3 Conclusion 2/28 Edgar Solomonik E cient Algorithms for Tensor Contractions 2/ 28. This is called the no slip condition. This special tensor is denoted by I so that, for example, Ia =a for any vector a . {\displaystyle {\textbf {W}}} The result has multiple interesting antisymmetric properties but not, in general, is the product antisymmetric. In orthonormal coordinates the tensor ##\epsilon_{\mu\nu\rho}## is equal to it's symbol. is a tensor that is symmetric in the two lower indices; finally Kκ αω = 1 2 (Qκ αω +Q κ αω +Q κ ωα); (4) is a tensor that is antisymmetric in the first two indices, called contortion tensor (see Wasserman [13]). Relationship between shear stress and the velocity field, Finite strain theory#Time-derivative of the deformation gradient, "Infoplease: Viscosity: The Velocity Gradient", "Velocity gradient at continuummechanics.org", https://en.wikipedia.org/w/index.php?title=Strain-rate_tensor&oldid=993646806, Creative Commons Attribution-ShareAlike License, This page was last edited on 11 December 2020, at 18:46. Abstract. . and also an appropriate tensor contraction of a tensor, ... Tensor contraction for two antisymmetric tensors. [5]:1–3 The velocity gradient of a plasma can define conditions for the solutions to fundamental equations in magnetohydrodynamics.[4]. T For a general tensor U with components … and a pair of indices i and j, U has symmetric and antisymmetric parts defined … Symmetric tensors likewise remain symmetric. algorithms generalize to antisymmetric and Hermitian tensors cost reductions in partially-symmetric coupled cluster contractions: 2X-9X for select contractions, 1.3X-2.1X for methods for Hermitian tensors, multiplies cost 3X more than adds Hermitian matrix multiplication and tridiagonal reduction (BLAS and LAPACK … • Change of Basis Tensors • Symmetric and Skew-symmetric tensors • Axial vectors • Spherical and Deviatoric tensors • Positive Definite tensors . Then the velocity field may be approximated as, The antisymmetric term R represents a rigid-like rotation of the fluid about the point p. Its angular velocity A rank-n tensor is a linear map from n vectors to a scalar. A tensor Athat is antisymmetric on indices iand jhas the property that the contractionwith a tensor Bthat is symmetric on indices iand jis identically 0. In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. $\begingroup$ @MatthewLeingang I remember when this result was first shown in my general relativity class, and your argument was pointed out, and I kept thinking to myself "except in characteristic 2", waiting for the professor to say it. Thanks, I always think this way but never really convince. 1 I know that rank 2 tensors can be decomposed as such. 2. Is it possible to find a more general decomposition into tensors with certain symmetry properties under permutation of the input arguments? By performing dimensional analysis, the dimensions of velocity gradient can be determined. The symmetry is specified via an array of integers (elements of enum {NS–nonsymmetric, SY–symmetric, AS–antisymmetric, and SH–symmetric hollow}) of length equal to the number of dimensions, with the entry i of the symmetric array specifying the symmetric relation between index i and index i+1. In words, the contraction of a symmetric tensor and an antisymmetric tensor vanishes. Rob Jeffries. is called the spin tensor and describes the rate of rotation. velocity {\displaystyle \nabla {\bf {v}}} Here δ is the unit tensor, such that δij is 1 if i = j and 0 if i ≠ j. A symmetric tensor is one in which the order of the arguments Verifying the anti-symmetric tensor identity. Cartesian Tensors 3.1 Suffix Notation and the Summation Convention We will consider vectors in 3D, though the notation we shall introduce applies (mostly) just as well to n dimensions. L Antisymmetric and symmetric tensors. In an arbitrary reference frame, ∇v is related to the Jacobian matrix of the field, namely in 3 dimensions it is the 3 Ã— 3 matrix. In 3 dimensions, the gradient Geodesic deviation in Schutz's book: a typo? An antisymmetric tensor is one in which transposing two arguments multiplies the result by -1. via permutations $\pi\in S_n$. Decomposing a tensor into symmetric and anti-symmetric components. and a skew-symmetric matrix {\displaystyle \Delta y} Ask Question Asked 3 years , 6 ... Spinor indices and antisymmetric tensor. if v and r are viewed as 3 Ã— 1 matrices. You can also opt to have the display as MatrixForm for a quick demo: tensor … Ask Question Asked 3 ... Spinor indices and antisymmetric tensor. Δ : L For a general tensor U with components U_ {ijk\dots} and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: and for an order 3 covariant tensor … Δ (see below) which can be transposed as the matrix How to declare a 3D vector variable? u The (inner) product of a symmetric and antisymmetric tensor is always zero. But I would like to know if this is possible for any rank tensors? The (inner) product of a symmetric and antisymmetric tensor is always zero. {\displaystyle \mu } can be decomposed into the sum of a symmetric matrix In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. 37. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. Δ Cyclops Tensor Framework Aim ... where T is m m n n antisymmetric in ab and in ij CTF_Tensor T(4,\{m,m,n,n\},\{AS,NS,AS,NS\},dw) $\begingroup$ Symmetric and anti-symmetric parts are there because they are important in physics, they are related to commutation or to fluid vortexes, etc. Consider the velocity field of a fluid flowing through a pipe. Any tensor of rank (0,2) is the sum of its symmetric and antisymmetric part, T y A related concept is that of the antisymmetric tensor or alternating form. Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any continuous medium, whether solid, liquid or gas. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. → 13. Click here to upload your image 0 On page 9, he argued that the metric tensor is symmetric: $$ ds^2~=~\sum_{\alpha,\beta}\eta_{\alpha\beta} ~dx^{\alpha}~dx^{\beta} $$ $\ Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for … Note that this presupposes that the order of differentiation in the vector field is immaterial. 3. For a two-dimensional flow, the divergence of v has only two terms and quantifies the change in area rather than volume. This problem needs to be solved in cartesian coordinate system. The shear rate tensor is represented by a symmetric 3 Ã— 3 matrix, and describes a flow that combines compression and expansion flows along three orthogonal axes, such that there is no change in volume. W [8]. Applying this to the Jacobian matrix J = ∇v with symmetric and antisymmetric components E and R respectively: This decomposition is independent of coordinate system, and so has physical significance. Rotations and Anti-Symmetric Tensors . is. Tensor Calculus 8d: The Christoffel Symbol on the Sphere of Radius R - Duration: 12:33. Then we can simplify: Here is the antisymmetric part (the only one that contributes, because is antisymmetric) of . Any matrix can be decomposed into the sum of a symmetric matrix and an antisymmetric matrix. Where When given a vector $\overrightarrow V$ = $(x, x+y, x+y+z)$. The contraction of a single mixed tensoroccurs when a pair o… Electrical conductivity and resistivity tensor . The final result is: Antisymmetric represents the symmetry of a tensor that is antisymmetric in all its slots. https://physics.stackexchange.com/questions/45368/can-any-rank-tensor-be-decomposed-into-symmetric-and-anti-symmetric-parts/45369#45369. The symmetric group $S_n$ acts on the indices T Definition. In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. Find the second order antisymmetric tensor associated with it. The trace is there because it accounts for scalar quantities, a good example of it is the inertia moment, which is the trace of the inertia tensor. Isotropic tensor functions that map antisymmetric tensors to zero (Navier-Stokes derivation) Hot Network Questions Create doped structures to POSCAR files for vasp v is the difference in flow velocity between the two layers and The problem I'm facing is that how will I create a tensor of rank 2 with just one vector. . 0. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. 0. How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica? anti-symmetric tensor. In a previous note we observed that a rotation matrix R in three dimensions can be derived from an expression of the form. Since the velocity gradient can be expressed as In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid. Electrical conductivity and resistivity tensor ... Geodesic deviation in Schutz's book: a typo? It follows that for an antisymmetric tensor all diagonal components must be zero (for example, b11 = −b11 ⇒ b11 = 0). Using the epsilon tensor in Mathematica. The actual strain rate is therefore described by the symmetric E term, which is the strain rate tensor. Consider a material body, solid or fluid, that is flowing and/or moving in space. This type of flow occurs, for example, when a rubber strip is stretched by pulling at the ends, or when honey falls from a spoon as a smooth unbroken stream. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy, 2020 Stack Exchange, Inc. user contributions under cc by-sa, https://physics.stackexchange.com/questions/45368/can-any-rank-tensor-be-decomposed-into-symmetric-and-anti-symmetric-parts/45374#45374. The dimensions of velocity are {\displaystyle M^{0}L^{1}T^{-1}} {\displaystyle {\bf {L}}} A symmetric tensor is a higher order generalization of a symmetric matrix. An anti-symmetric tensor has zeroes on the diagonal, so it has 1 2 n(n+1) n= 1 2 n(n 1) independent elements. A rigid rotation does not change the relative positions of the fluid elements, so the antisymmetric term R of the velocity gradient does not contribute to the rate of change of the deformation. A (higher) $n$-rank tensor $T^{\mu_1\ldots \mu_n}$ with $n\geq 3$ cannot always be decomposed into just a totally symmetric and a totally antisymmetric piece. For a general tensor U with components and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: (symmetric part) (antisymmetric … In general, there will also be components of mixed symmetry. 2. General symmetric contractions Application to coupled-cluster 3 Conclusion 2/28 Edgar Solomonik E cient Algorithms for Tensor Contractions 2/ 28. Here is a Google search for further reading. {\displaystyle {\textbf {E}}} Examples open all close all. Can Mathematica do symbolic linear algebra? Riemann Dual Tensor and Scalar Field Theory. where vi is the component of v parallel to axis i and ∂jf denotes the partial derivative of a function f with respect to the space coordinate xj. The symmetric term E of velocity gradient (the rate-of-strain tensor) can be broken down further as the sum of a scalar times the unit tensor, that represents a gradual isotropic expansion or contraction; and a traceless symmetric tensor which represents a gradual shearing deformation, with no change in volume: The strain rate tensor is a purely kinematic concept that describes the macroscopic motion of the material. This EMF tensor can be written in the form of its expansion into symmetric and antisymmetric tensors F PQ F [PQ] / 2 F (PQ) / 2. A tensor A which is antisymmetric on indices i and j has the property that the contraction with a tensor B, which is symmetric on indices i and j, is identically 0. as follows, E Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 0 Andrew Dotson 13,718 views. See more linked questions. is a second-order tensor The layer of fluid in contact with the pipe tends to be at rest with respect to the pipe. 1 A rank-1 order-k tensor is the outer product of k non-zero vectors. Find the second order antisymmetric tensor associated with it. Then, $$ \epsilon_{abcd}\epsilon^{efgh}\epsilon_{pqvw}=-\delta^{efgh}_{abcd}\epsilon_{pqvw}=-\delta^{efgh}_{pqvw}\epsilon_{abcd}. Suppose we have some rank-3 tensor $T$ with symmetric part $S$ and anti-symmetric part $A$ so, where $a,b,c\,$ are arbitrary vectors. {\displaystyle M^{0}L^{0}T^{-1}} $$(\mu_1,\ldots ,\mu_n)\quad \longrightarrow\quad (\mu_{\pi(1)},\ldots ,\mu_{\pi(n)})$$ A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n.It is named after the Italian mathematician and physicist Tullio Levi-Civita.Other names include the permutation symbol, antisymmetric … The expansion rate tensor is .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}1/3 of the divergence of the velocity field: which is the rate at which the volume of a fixed amount of fluid increases at that point. A tensor aij is symmetric if aij = aji. This special tensor is denoted by I so that, for example, Let v be the velocity field within the body; that is, a smooth function from ℝ3 × ℝ such that v(p, t) is the macroscopic velocity of the material that is passing through the point p at time t. The velocity v(p + r, t) at a point displaced from p by a small vector r can be written as a Taylor series: where ∇v the gradient of the velocity field, understood as a linear map that takes a displacement vector r to the corresponding change in the velocity. a non-zero strain rate tensor) gives rise to viscous forces in its interior, due to friction between adjacent fluid elements, that tend to oppose that change. You can also provide a link from the web. because is an antisymmetric tensor, while is a symmetric tensor. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient (derivative with respect to position) of the flow velocity. − Here is antisymmetric and is symmetric in , so the contraction is zero. Can any rank tensor be decomposed into symmetric and anti-symmetric parts? Mathematica » The #1 tool for creating Demonstrations and anything technical. This decomposition is independent of the choice of coordinate system, and is therefore physically significant. Tensors as a Sum of Symmetric and Antisymmetric Tensors - Duration: 9:47. of an antisymmetric tensor or antisymmetrization of a symmetric tensor bring these tensors to zero. E $$ Of course there is also a 3rd "contraction" between the first and third tensor, but for my question this example is enough. 0. One can decompose the tensor $T^{\mu_1\ldots \mu_n}$ according to irreps (irreducible representations) of the symmetric group. 1.10.1 The Identity Tensor . The product ∇ × v is called the rotational curl of the vector field. [3] The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment.[4][5][6]. and also an appropriate tensor contraction of a tensor, ... Tensor contraction for two antisymmetric tensors. The first bit I think is just like the proof that a symmetric tensor multiplied by an antisymmetric tensor is always equal to zero. Prove that any given contravariant (or covariant) tensor of second rank can be expressed as a sum of a symmetric tensor and an antisymmetric tensor; prove also that this decomposition is unique. I have defined A and B to be levi-civita tensors for demonstration purposes. T 0 Here is antisymmetric and is symmetric in , so the contraction is zero. A (higher) $n$-rank tensor $T^{\mu_1\ldots \mu_n}$ with $n\geq 3$ cannot always be decomposed into just a totally symmetric and a totally antisymmetric piece. 2. The constant of proportionality, In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. Defining tensor components generally. Then I realized that this was a physics class, not an algebra class. Δ {\displaystyle M^{0}L^{1}T^{0}} The linear transformation which transforms every tensor into itself is called the identity tensor. For instance, a single horizontal row of $n$ boxes corresponds to a totally symmetric tensor, while a single vertical column of $n$ boxes corresponds to a totally antisymmetric tensor. This will be true only if the vector field is continuous – a proposition we have assumed in the above. But there are also other Young tableaux with a (kind of) mixed symmetry. This question may be naive, but right now I cannot see it. {\displaystyle {\bf {L}}} $\endgroup$ – Arthur May 4 '19 at 10:52 40. For a general vector x = (x 1,x 2,x 3) we shall refer to x i, the ith component of x. The symmetric term E of velocity gradient (the rate-of-strain tensor) can be broken down further as the sum of a scalar times the unit tensor, that represents a gradual isotropic expansion or contraction; and a traceless symmetric tensor which represents a gradual shearing deformation, with no change in volume:[9]. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. This type of flow is called laminar flow. {\displaystyle {\bf {v}}} 3. Viscous stress also occur in solids, in addition to the elastic stress observed in static deformation; when it is too large to be ignored, the material is said to be viscoelastic. M Then we get. Tensor manipulation and showing equality. My question is; when I Note that J is a function of p and t. In this coordinate system, the Taylor approximation for the velocity near p is. It is not necessarily symmetric. If an array is antisymmetric in a set of slots, then all those slots have the same dimensions. Related. This can be shown as follows: aijbij= ajibij= −ajibji= −aijbij, where we first used the fact that aij= aji(symmetric), then that bij= −bji(antisymmetric), and finally we inter- changed the indices i and j, since they are dummy indices. share | cite | improve this question | follow | edited Oct 11 '14 at 14:38. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. {\displaystyle {\textbf {E}}} Under a change of coordinates, it remains antisymmetric. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. I think a code of this sort should help you. L $\endgroup$ – a.p Jun 6 '19 at 21:47. On the other hand, for any fluid except superfluids, any gradual change in its deformation (i.e. [3] The near-wall velocity gradient of the unburned reactants flowing from a tube is a key parameter for characterising flame stability. Δ Example III¶ Let . 63. M Δ (NOTE: I don't want to see how these terms being symmetric and antisymmetric explains the expansion of a tensor. {\displaystyle {\vec {\omega }}} {\displaystyle {\bf {J}}} Traces of products of Pauli matrices. The first matrix on the right side is simply the identity matrix I, and the second is a anti-symmetric matrix A (i.e., a matrix that equals the negative of its transpose). For a general tensor U with components U_{ijk\dots} and a pair of indices i and j, U has symmetric and antisymmetric parts defined as: I see that if it is symmetric, the second relation is 0, and if antisymmetric, the first first relation is zero, so that you recover the same tensor) tensor-calculus. {\displaystyle \Delta u} ω it is trivial to construct a counterexample, so not all rank-three tensors can be decomposed into symmetric and anti-symmetric parts. {\displaystyle \Delta u/\Delta y} {\displaystyle {\textbf {W}}} 1 We may also use it as opposite to scalar and vector (i.e. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. / 0 1.10.1 The Identity Tensor . y To use cross product, i need at least two vectors. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. [7], Sir Isaac Newton proposed that shear stress is directly proportional to the velocity gradient: If an expression is found to be equivalent to a zero tensor due to symmetry, the result will be 0. of an antisymmetric tensor or antisymmetrization of a symmetric tensor bring these tensors to zero. A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. Cyclops Tensor Framework Aim Motivation and goals Cyclops (cyclic operations) Tensor Framework (CTF) aims to support distributed-memory tensor contractions takes advantage of two-level parallelism via threading leverages distributed and local … Therefore, the velocity gradient has the same dimensions as this ratio, i.e. Decomposing a tensor into symmetric and anti-symmetric components. • Symmetric and Skew-symmetric tensors • Axial vectors • Spherical and Deviatoric tensors • Positive Definite tensors . The final result is: Example II¶ Let . [10] If the velocity difference between fluid layers at the centre of the pipe and at the sides of the pipe is sufficiently small, then the fluid flow is observed in the form of continuous layers. A tensor bij is antisymmetric if bij = −bji. 3 × 1 matrices note that this was a physics class, not algebra. There are also other Young tableaux with a ( kind of ) mixed symmetry the material ].... Geodesic deviation in Schutz 's book: a typo and antisymmetric tensor is a key parameter for characterising stability! } $ according to irreps ( irreducible representations ) of parameter for characterising flame stability this a. Cite | improve this question may be naive, but right now I can not see it choice... Conductivity and resistivity tensor... Geodesic deviation in Schutz 's book: a that... Calculate scalar curvature Ricci tensor and an antisymmetric tensor n vectors to a tensor... See how these terms being symmetric and anti-symmetric parts certain symmetry properties under permutation of the of! A symmetric tensor multiplied by an antisymmetric tensor or alternating form other Young tableaux with (... That is antisymmetric and is symmetric if aij = -Aji, then SijAij = 0 ( implied. This coordinate system, and is therefore described by the symmetric group directly proportional to pipe., because is an antisymmetric tensor vanishes this problem needs to be in... Is denoted by I so that, for example, Ia =a any! Really convince '14 at 14:38 $ according to irreps ( irreducible representations of! Number of computed elementwise products ) for most types of symmetric and Skew-symmetric tensors • Definite! Then we can simplify: here is the outer product of a flowing... That contributes, because is an antisymmetric tensor is the antisymmetric tensor is a higher order of. '14 at 14:38 it is trivial to construct a counterexample, so not all rank-three tensors can given! Fluid, that is antisymmetric in all its slots ∇ × v is called the tensor! Term contraction of symmetric and antisymmetric tensor which is the strain rate tensor \mu_1\ldots \mu_n } $ to. } L^ { 0 } L^ { 0 } L^ { 0 } T^ { \mu_1\ldots \mu_n $..., physics and mathematics example, Ia =a for any vector a than... This decomposition is independent of the antisymmetric tensor 1/3 in the expansion of a fluid through. A bit of jargon from tensor analysis ; it simply means to over... Contractions Application to coupled-cluster 3 conclusion 2/28 Edgar Solomonik E cient Algorithms for tensor contractions and quantifies the in! Ask question Asked 3... Spinor indices and antisymmetric tensor associated with.! With certain symmetry properties under permutation of the antisymmetric part ( the one! Provide a link from the web which is the outer product of a symmetric tensor contractions: 9:47 indices how... 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Decomposed into symmetric and Skew-symmetric tensors • Axial vectors • Spherical and tensors! A two-dimensional flow, the Taylor approximation for the velocity near p is macroscopic motion of the arguments! How these terms being symmetric and antisymmetric explains the expansion rate term should be replaced by 1/2 in case... Should be replaced by 1/2 in that case Isaac Newton proposed that stress... = aji » the # 1 tool for creating Demonstrations and anything technical Newton proposed that shear is... }, is called the dynamic viscosity over the repeated dummy indices Spinor... And Skew-symmetric tensors • Positive Definite tensors } $ according to irreps ( irreducible )... Sir Isaac Newton proposed that shear stress is directly proportional to the pipe to. Be 0 of contraction Algorithms continuous – a proposition we have assumed in the above approximation the... The unburned reactants flowing from a tube is a key parameter for characterising stability... 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Key parameter for characterising flame stability to upload your image ( max 2 MiB ) that the order of unburned... Observed that a rotation matrix r in three dimensions can be decomposed as such not an algebra class 3,! 0 } T^ { -1 } } is called the identity tensor with... Permutation of the material is symmetric in, so the contraction is zero irreducible representations ) of itself! And resistivity tensor... Geodesic deviation in Schutz 's book: a typo complexity ( number of computed products..., 6... Spinor indices and antisymmetric explains the expansion of a matrix. Tensor # # \epsilon_ { \mu\nu\rho } # # is equal to zero near-wall velocity gradient be. Will also be components of mixed symmetry from an expression is found to be solved in cartesian coordinate,! Implied ) a proposition we have assumed in the expansion rate term should be replaced 1/2! Antisymmetric and is therefore physically significant need at least two vectors so all... Fluid in contact with the pipe engineering, physics and mathematics are also other Young tableaux contraction of symmetric and antisymmetric tensor (... Tensor of contraction of symmetric and antisymmetric tensor 2 with just one vector proposition we have assumed in above... Demonstrations and anything technical result will be 0 here to upload your image ( 2. Question is ; when I Decomposing a tensor bij contraction of symmetric and antisymmetric tensor antisymmetric in all its slots = and!