1 Antisymmetric and symmetric tensors. For a general tensor Uwith components and a pair of indices iand j, Uhas symmetric and antisymmetric parts defined as: (symmetric part) (antisymmetric part). But I would like to know if this is possible for any rank tensors? Consider the velocity field of a fluid flowing through a pipe. 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. {\displaystyle {\textbf {W}}} 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. 3. Antisymmetric and symmetric tensors. Electrical conductivity and resistivity tensor ... Geodesic deviation in Schutz's book: a typo? Then we can simplify: Here is the antisymmetric part (the only one that contributes, because is antisymmetric) of . Decomposing a tensor into symmetric and anti-symmetric components. This will be true only if the vector field is continuous – a proposition we have assumed in the above. 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. 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. 0. 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. Andrew Dotson 13,718 views. . of an antisymmetric tensor or antisymmetrization of a symmetric tensor bring these tensors to zero. and a skew-symmetric matrix Here is antisymmetric and is symmetric in , so the contraction is zero. Under a change of coordinates, it remains antisymmetric. 2. A tensor aij is symmetric if aij = aji. The layer of fluid in contact with the pipe tends to be at rest with respect to the pipe. {\displaystyle \Delta u/\Delta y} This problem needs to be solved in cartesian coordinate system. You can also provide a link from the web. Δ (see below) which can be transposed as the matrix 1.10.1 The Identity Tensor . , and the dimensions of distance are In words, the contraction of a symmetric tensor and an antisymmetric tensor vanishes. : L tensor … 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. https://physics.stackexchange.com/questions/45368/can-any-rank-tensor-be-decomposed-into-symmetric-and-anti-symmetric-parts/45369#45369. A rank-n tensor is a linear map from n vectors to a scalar. The final result is: Example II¶ Let . is. Click here to upload your image Tensors as a Sum of Symmetric and Antisymmetric Tensors - Duration: 9:47. 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. The final result is: This special tensor is denoted by I so that, for example, Therefore, the velocity gradient has the same dimensions as this ratio, i.e. • Change of Basis Tensors • Symmetric and Skew-symmetric tensors • Axial vectors • Spherical and Deviatoric tensors • Positive Definite tensors . We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. [1] Though the term can refer to the differences in velocity between layers of flow in a pipe,[2] it is often used to mean the gradient of a flow's velocity with respect to its coordinates. 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