# number of independent components of a symmetric tensor

A tensor has a certain number of components. In Minkowski The number of suffixes is the rank of the Cartesian tensor, a ... which is an antisymmetric tensor has three independent components which are the vector components of the vector product U → × V →, and the third term is a symmetric traceless tensor, which has five independent components. One may ask whether other irre ducible tensors may appear in the reduction of the most general Cartesian tensor of order k (which does not possess the symmetry properties of our special tensor (10.10)), namely irreducible tensors with an even number of components. So there are 21 unique non-zero components in the Riemann (down from 256), which we can organize into three 3x3 matrices which we can give names to. For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. 52 Accesses. The 21 components are not all independent, though - it turns out that there is a constraint equation from the Bianchi identies. Symmetry of a material may further reduce the number of independent components of the permittivity tensor (section 2.4). Based on the achieved properties of such a class of linear operators aforementioned, we proceed with characterizing the SDT cone. In a 4-dimensional space, the Riemann-Christoffel tensor exhibits a total of 20 independent components. This is a preview of subscription content, log in to check access. Since the number of independent components of these two parts are six and three respectively, we see that the tensors of rank two can be broken up into smaller representations. Elastic compliance and stiffness. This is precisely the number of independent components of the Riemann tensor, which we will de ne and study later on, and which is a measure of the curvature of spacetime. i=1 i = 15 more components, leaving 36 15 =21 possibly independent components. 1.10.1 The Identity Tensor . That means we have [n(n+ 1)=2]2 n(n+ 1)(n+ 2)=6 = n2(n2 1)=12 too few variables. Dates First available in Project Euclid: 3 July 2007. The components of this tensor, which can be in covariant (g ij) or contravariant (gij) forms, are in general continuous variable functions of coordi-nates, i.e. In order to describe these relationships it is necessary to discuss the transformation of the tensor components to some extent. adshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A Richard H. Bruck and T. L. Wade. from the Gaussian average over the tensor components. I am having difficulty using the tensor symmetric and antisymmetric relationships of the Riemann-Christoffel tensor to show that it reduces from 256 to 36 to 21 and then 20 independent components. Notice that e = ε + !. Math. Full-text: Open access. 9.5.1 Symmetry by Definition Some properties are defined such that the corresponding tensors exhibit an inner symmetry. The number of independent components of tensors in symmetrical systems. In the following we shaH see that the answer is negative. Now a totally antisymmetric 4-index tensor has n(n - 1)(n - 2)(n - 3)/4! Riemann curvature tensor has four symmetries. PDF File (170 KB) Article info and citation; First page; Article information. Tensors of rank 2 or higher that arise in applications usually have symmetries under exchange of their slots. 1.11.2 Real Symmetric Tensors Suppose now that A is a real symmetric tensor (real meaning that its components are real). This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. In theories and experiments involving physical systems with high symmetry, one frequently encounters the question of how many independent terms are required by symmetry to specify a tensor of a given rank for each symmetry group. Metrics details. 2.2.2. There is one ﬁnal symmetry condition for the Riemann tensor, and it is the trickiest to handle. The linear transformation which transforms every tensor into itself is called the identity tensor. A. Gamba 1,2 Il Nuovo Cimento (1943-1954) volume 10, pages 1343 – 1344 (1953)Cite this article. This video investigates the symmetry properties of the Riemann tensor and uses those properties to determine the number of independent components … How is this symmetricity going to affect the number of components? This special tensor is denoted by I so that, for example, Ia =a for any vector a . (In one dimension it has none.) We are left with (3.85) independent components of the Riemann tensor. In a 3-dimensional space, a tensor of rank 2 has 9 (=3^2) components (for example, the stress tensor). terms, and therefore (3.83) reduces the number of independent components by this amount. Conventionally, a shear strain is defined by the shear angle produced in simple shear, below. References (1) F.G. Fumi:Nuovo Cimento,9, 739 (1952): R. Fieschi and F. G. Fumi:Nuovo Ciinento,10, 865 (1953). In fact, the six component WDGO formed by the symmetric … Question: The Number Of Independent Components Of A Symmetric And An Antisymmetric Tensor Of Rank 2 (in 3-dimensions)are,respectively,(1) 6,6(2) 9,3(3) 6,3(4) 3,6only One … symmetric tensor spaces ... the number of independent components of a 3-dimensional symmetric tensor. In four dimensions, therefore, the Riemann tensor has 20 independent components. This logic can be extended to see that in an N-dimensional space, a tensor of rank R can have N^R components. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed ﬁeld. 3 . Minimize the number of tensor components according to its symmetries (and relabel, redefine or count the number of independent tensor components) The nice development described below is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, Université Lille 1, France, used in the Mapleprimes post Magnetic traps in cold-atom physics . be symmetrical tensor.˜ ij D˜ ji/with only six independent components [1–3]. Altogether, then, there are 1 + 3 + 5 = 9 components, as required. For a totally antisymmetric vector with rank rand aantisymmetric components in a n-folds, we have already shown that the number of independent components is given by: nr a n! Therefore, the total number of algebraically independent components of the curvature tensor in N dimensions is . Symmetric Tensor : T λµ= T µλorT ( ), T νλµ= T νµλor T ν(λµ) Antisymmetric : T λµ= −T µλor T [ ], T νλµ= −T νµλor T ν[λµ] Number of independent components : Symmetric: n(n + 1)/2, Antisymmetric: n(n −1)/2 9. And the total number of independent components in four-dimensional spacetime is therefore 21-1 = 20 independant components. Therefore, the number of independent terms in the curvature tensor becomes n2(n 21) 2=4 n2(n 1)(n 2)=6 = n (n 1)=12. But here, in the given question, the 2nd rank contravariant tensor is 'symmetric'. • Symmetric and Skew-symmetric tensors • Axial vectors • Spherical and Deviatoric tensors • Positive Definite tensors . Thus, like stress, strain is by definition a symmetric tensor and has only 6 independent components. So we can say that [math]A^{ij}=-A^{ji}[/math]. Amer. 4(')(x') is a symmetric tensor of rank P. It is also traceless, since con- tracting on any pair of indices in (B.2.3) produces a VI2, which in turn gives zero acting on l/rr. That means that the components are dependent on each other. These questions have simple group theoretical answers [75]. Tensors:Diﬀerentiation & Connectionsi We consider a region V of the space in which some tensor, e.g. (n a)!a! The expression "independent components of a tensor" is misleading. 2 . 3 Citations. Spherical Tensors. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. 4 . Let's think about [math]A^{ij}[/math], a skew-symmetric tensor of order [math]2[/math]. Also, these components can be arranged in sets of three that satisfy the three-way skew symmetry, so the number of independent components of this form is reduced by a factor of 2/3. It may also have restrictions on the components, such as the tensor is symmetric or something like that. So in this case the tensor shear strain ε 12 = 1/2 (e 12 + e 21) = 1 1/2 (γ + 0) = γ/2. This computation suggests that the largest tensor eigenvalue in our ensemble in the limit of a large number of dimensions is proportional to the square root of the number of dimensions, as it is for random real symmetric matrices. However, the number of independent components is much smaller in most cases, either due to intrinsic symmetries of the physical property described (this section) or due to the crystal symmetry (section 9.6). At a given point in space, we can count the number of independent components of c$() [i.e., the number of independent numbers we must For example, the inertia tensor, the stress-energy tensor, or the Ricci curvature tensor are rank-2 fully symmetric tensors; the electromagnetic tensor is a rank-2 antisymmetric tensor; and the Riemann curvature tensor and the stiffness tensor are rank-4 tensors with nontrival symmetries. My prof just acted like I should be able to do this in my sleep, but I am struggling. In that case it can be proved (see below) that1 (i) the eigenvalues are real (ii) the three eigenvectors form an orthonormal basis nˆ i . In that case, the components of A can be written relative to the basis of principal g ij = g ij(u1;u2;:::;un) and gij = gij(u1;u2;:::;un) where ui symbolize general coordinates. There is a problem however! So the number of degrees of freedom drops from 21 down to 20, because of this constraint. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. once time that 0123 is given, the tensor is xed in an unique way. symmetric property is independent of the coordinate system used . Soc., Volume 49, Number 6 (1943), 470-472. In other words, the tensor representation of the rotation group is actually reducible – it breaks up into a six component and a three component WDGO. number of independent components from 256 to 20. Source Bull. p{ the number comes from the symmetry of the partial derivatives. For a tensor of higher rank ijk lA if ijk jik l lA A is said to be symmetric w.r.t the indices i,j only . The condition of invariance reduces the number of the independent tensor components, since it signifies relationships between the tensor components. [1] We recall that the number of independant components of a n-dimensional symmetric matrix is n(n+1)/2, here 6x7/2 = 21. A symmetric tensor of rank 2 in N-dimensional space has ( 1) 2 N N independent component Eg : moment of inertia about XY axis is equal to YX axis . For we have n= a= 4 so that there is just one possibility to choose the component, i.e. A Riemannian space is a manifold characterized by the existing of a symmetric rank-2 tensor called the metric tensor. The number of independent components of the tensors of given symmetry type. Return to Table of Contents

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