Tensor outer product

Multiplication of Am by Bn yields a mixed tensor AmBn = C , called the outer product, which may be formed with tensors of any rank or type,... 

V and vT stands for any element of tensors v v... v (any rank, i.e., any number of factors in this outer product). We may even omit the shift vector... 

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. 

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... 

Traction t is here expressed through the stress tensor by (3.72). We also note that postulating (3.89) for one fixed point y the form (3.89) is valid for arbitrary but fixed point (say yo as follows from the balance of linear momentum (3.74) multiplied by constant (y - yo) a (i.e., as outer product in Rem. 16) and by summation with (3.89), of course all in our inertial frame). For this reason the origin y = o is often used in formulations of this postulate, e.g., [16], without loss of generality. 

The next step is to realise that symmetric tensors can be constmcted from two vectors. In matrix algebra, this is done by forming the outer product of the two vectors. Thus any symmetric tensor f can be constmcted from the proper choice of the two vectors A and B... 

The outer product, which is defined in the previous equation, must not be confused with the inner product of two vectors, which produces a scalar rather than a tensor. 

When only a single scalar product of base vectors is involved, the result of such a product has the added order of the multiplied tensors lowered by two. This contracting or inner product of general tensors thus comprises the scalar product of vectors as a special case. Further on, the tensorial or outer product... 

In this expression A and Q are distance dispersion resulting from electron-vibrational coupling, and frequency tensor (assumed identical in reactant and product states), respectively (work of formation of precursor and successor states is omitted). If we assume that the frequency tensor is diagonal, then we have simply a sum of independent terms for all inner and outer contributing modes. At sufficiently high temperature, the hyperbolic tangents become unity and we obtain the usual (in this approximation) high-temperature expression ... 

Tensors, from the same or different fields, can be combined by outer multiplication, denoted by juxtaposing indices with order preserved on the resultant tensor.33 It is possible that an index is present both in the covariant and contravariant index sets then with the repeated index summation convention, both are eliminated and a tensor of lower rank results. The elimination of pairs of indices is known as contraction, and outer multiplication followed by contraction is inner multiplication.33 In multiplication between tensors, contractions cannot take place entirely within one normal product (i.e., the generalized time-independent Wick theorem see Section IV) hence such tensors are called irreducible. 

The starting point of the investigation is the introduction of a scalar microstruc-tural parameter k which contributes to the total energy E of the body under study as pointed out in Refs. [38] and 39]. In Eq. (1) p, s and x are the mass density, the specific internal energy density and the velocity, respectively. The parameter k in the product pk describes microstructural properties and transfers the square of the rate of k to the dimensions of a specific energy density. In addition, the energy supply Ri and the energy flux R2 are also modified in the form of Eqs. (2) and (3), wherein pb is the body force density, pg is the supply of K, and p r is the heat supply. Further quantities are the stress vector t = T n associated to the Cauchy stress tensor T and to the outer normal n, the microstructural flux s = S n and the heat flux qi = —q n. 

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