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 part
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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
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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
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