Tensor skew-symmetric

This matrix describes the transformation from x y z to xyz as a rotation about the z axis over angle a, followed by a rotation about the new y" axis over angle /), followed by a final rotation over the new z " axis over angle y (Watanabe 1966 148). Formally, the low-symmetry situation is even a bit more complicated because the nondiagonal g-matrix in Equation 8.11 is not necessarily skew symmetric (gt] -g. Only the square g x g is symmetric and can be transformed into diagonal form by rotation. In mathematical terms, g x g is a second-rank tensor, and g is not. 

If the components of a tensor satisfy the relation Amn = Anm, such a tensor is called symmetric. If Amn — —Anm, the tensor is skew-symmetric. 

There is an important relationship between vectors and skew-symmetric tensors. Suppose A and B are two vectors in a three-dimensional rectangular coordinate system whose components are connected by... 

If the coefficients were components of a skew-symmetric tensor, = —a3i 0, then... 

The nonsymmetrical tensor S can be written as the sum of a symmetric tensor with elements (Sfj = (Sy -I- Sjt)/2 and a skew-symmetric tensor with elements = (Sfj — Sji)/2. Expressed in terms of principal axes, Ss consists of three principal screw correlations Positive and negative screw correlations... 

The spatial velocity gradient a = grad va can be decomposed into symmetric and skew-symmetric parts as la = sym a + skw 1 = dQ + wQ., where da and wa are the deformation rate and the spin tensors, respectively. 

It should be noted that curl W(x) is, in general, a skew-symmetric tensor T(x) of rank 2, whose elements Ttj are defined by [29]... 

The curl in Eq. (20) is the skew-symmetric tensor of rank 2, whose k, I element... 

As is well known, explicit expressions for the polarization fields can be given such that equation (3) has the requisite properties these expressions, involving multipole series or line integrals, are by no means tmique. That this must be so can be seen from at least two levels of theory. We noted earlier that the notion of an electric field (or a magnetic field) is not invariant with respect to Lorentz transformations under such transformations however V should be an invariant scalar and this implies a definite transformation law for (P(x), M(x) that mixes them, and mirrors that for E(x), B(x) classically both pairs can be shown to be components of skew-symmetric second-rank tensors [7]. Although... 

This follows from the definition of rigid motion (3.11)-(3.13) and from the very definition of material derivative (quadratic form with skew-symmetric tensor W is zero) ... 

Generalizations oftensors for nonCartesian coordinates see, e.g., [7, 16, 17] and Appendix A.4. Similar to matrices (3 x 3), tensors maybe symmetric, skew-symmetric, etc., about vector and outer products. See Rems. 6, 16. 

Lemma (equivalency of skew-symmetric tensors with axial vectors) For every skew-symmetric tensor (of second order) W it is possible to define tm axitil vector w (both contain three (independent) components) and vice versa by... 

Axiality of w is automatically achieved by the usual transformation ((c) in Rem. 4) of tensor W. Therefore the skew-symmetric tensors instead of axial vectors and outer product (see Rem. 16) may be used and we do it this way at the moment of momentum balances in the Sects. 3.3,4.3, cf. [7, 8, 14, 27]. Generalization of this Lemma to third-order tensors, made by M. Silhav, is published in Appendix of [28]. 

We use the outer product A defined for two vectors a, basaAb = a b — b a, i.e. (a A h)d = a b-i — a-ib . This product is obviously the skew-symmetric tensor which, using the results from Rem. 10, is equivalent to the axial vector created by the vector product of these vectors, see Rem. 6... 

Therefore, an admissible thermodynamic process exists and is caused by thermokinetic process (3.133)-(3.135) by the admissibility principle the inequality (3.132) must be satisfied. Then this inequality must be satisfied at (arbitrarily chosen) particle X and instant t by arbitrarily chosen values of mutually independent F, T, g (or GradT) and W, D, t, (GradT) (this follows from the independence of derivatives in expansions (3.133), (3.134) and (3.14), (3.15) note that T (and p) are positive scalars and D, W are symmetric or skew-symmetric tensors, respectively). But the inequality (3.132) depends only linearly on the latter values and therefore members containing them must be zero because of Lemmas A.5.1, A.5.2 from Appendix 5. 

Further, we use Lemma A.5.2 (with consequences for symmetric and skew-symmetric tensors). Namely, if we choose D, (GradF), f zero at arbitrary F, T, g, the first member in (3.132) must be zero at any skew-symmetric tensor W, i.e., the following tensor in this member must be symmetric... 

By analogical arguments, we obtain from Lemma A.5.2 that the tensor standing at D in (3.132) must be skew-symmetric. But it is at the same time symmetric (see (3.138), (3.93)) and therefore it is zero. ... 

The last definition of function / of 9 variables (allowed by symmetry of D) permits to employ the customary tensor (or matrix) descriptions, e.g. the summation convention in component form. This is the reason for using this definition of / in (3.146), (3.147) and other formulae in this book (similar definitions may be used for skew-symmetric tensor and vector and tensor functions [7, 14, 79]). As may be seen from the definition above, the main property of / is (when D is symmetrical and this is just such a case) that is indeed symmetrical, e.g. 

Using (4.19) in the left-hand side (p are components of (skew symmetric) tensor) and (4.69) in the right-hand side of (4.72) we obtain... 

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

This theorem is valid for any tensors but in applications there are tensors A or Y/j often symmetric or skew-symmetric. The most important case used in our treatise is that from i tensors Y/j the first h tensors (77 = 1,..., /i) are symmetric and the... 

Similarly, if A is a skew-symmetric tensor M given as a function (A.59), then skew-symmetrization of (A.59) (not changing M) gives... 

Therefore, Eqs. (A.57)-(A.59), (A.67)-(A.69) express the representation theorems of isotropic vector, scalar, and tensor (even symmetric or skew-symmetric) functions linear in vectors and (possibly symmetric or skew-symmetric) tensors. Of course, special cases of these representations follow, e.g., (A.68) is a representation theorem of the isotropic symmetric tensor function linear in symmetric tensors (this was used in Sects. 3.7,4.5) or (A.34) is a special case of (A.59) as was noted above, etc. 

We see that the definition of sgrad / copies the definition of grad /. The only difference is that, instead of the symmetric tensor gij, one considers the skew-symmetric tensor Wij. 

When the displacement gradient is split into its symmetric and skew symmetric portions, the infinitesimal strain tensor of Eq. (3.18) is identified to be the former, while the latter represents infinitesimal rotations that do not... 

The stress tensor for the SmA phase can be obtained from the two director theory for the SmC phase by elimination of all terms containing the c director. The sum of the symmetric and the skew symmetric part becomes... 

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