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Mathematical Preliminaries

In this chapter the basic definitions of vector and tensor algebra, elements of tensor differential and integral calculus, and concept of a convolutional product for two time-dependent tensor fields are recalled. These concepts are then used to solve particular problems related to the Mathematical Preliminaries.

1.1 Some Formulas in Tensor Algebra A vector will be understood as an element of a vector space V . The inner product of two vectors u and v from V will be denoted by u · v. If Cartesian coordinates are introduced in such a way that the set of vectors {ei } = {e1 , e2 , e3 } with an origin 0 stands for an orthonormal basis, and if u is a vector and x is a point of E 3 , then Cartesian coordinates of u and x are given by u i = u · ei , xi = x · ei

(1.1)

Apart from the direct (vector or tensor) notation we use indicial notation in which subscripts range from 1 to 3 and summation convention over repeated subscripts is observed. For example, u·v =

3 

u i vi = u i vi

(1.2)

i=1

From the definition of an orthonormal basis {ei } it follows that ei · e j = δi j (i, j = 1, 2, 3)

(1.3)

where δi j is called the Kronecker symbol defined by

M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_1, © Springer Science+Business Media Dordrecht 2013

3

4

1 Mathematical Preliminaries

 δi j =

1 if i = j 0 if i = j

(1.4)

We introduce the permutation symbol εi jk , also called the alternating symbol, defined by εi jk

⎧ if (i, j, k) is an even permutation of(1, 2, 3) ⎨1 = −1 if (i, j, k) is an odd permutation of(1, 2, 3) ⎩ 0 otherwise, that is, if two subscripts are repeated

(1.5)

The permutation symbol will be used for the definition of the vector product u × v of two vectors u and v (1.6) (u × v)i = εi jk u j vk We may observe that the following identity holds true εmis ε jks = δm j δik − δmk δi j

(1.7)

An alternative definition of the permutation symbol, given in terms of the vectors ei , is εi jk = ei · (e j × ek )

(1.8)

Using this definition of εi jk , a generalized form of Eq. (1.7) is obtained εi jk ε pqr

   δi p δiq δir    =  δ j p δ jq δ jr   δkp δkq δkr 

(1.9)

Letting k = r in this identity we obtain Eq. (1.7). The permutation symbol εi jk can be also used to calculate a 3 × 3 determinant    a1 a2 a3    εi jk ai b j ck =  b1 b2 b3   c1 c2 c3 

(1.10)

A second-order tensor is defined as a linear transformation from V to V , that is, a tensor T is a linear mapping that associates with each vector v a vector u by u = Tv

(1.11)

The components of T are denoted by Ti j Ti j = ei · Te j

(1.12)

1.1 Some Formulas in Tensor Algebra

5

so the relation (1.11) in index notation takes the form u i = Ti j v j

(1.13)

The Kronecker symbol δij represents an identity tensor that in direct notation is written as 1. A product of two tensors A and B is defined by (AB)v = A(Bv) for every vector v

(1.14)

Thus, in Cartesian coordinates (AB)i j = Aik Bk j

(1.15)

The tr