1. Tensor Component and Matrix Notations

Derivations and presentations of results in this book will appear in the tensor components, or in the related matrix notation. In the tensor component or subscript notation, vectors or first-order tensors are denoted by lower case italics with a single letter subscript, such as \( {n_i} \) or \( {\nu_j} \), while second, third and fourth-order tensors are written as \( {\varepsilon_{\textit{ij}}},\,\,{ \in_{\textit{ijk}}},\,\,{L_{\textit{ijkl}}} \), with the number of subscripts indicating the order or rank R of the tensor. The subscripts have a certain assigned range of values, which is i, j,… = 1, 2, 3, or \( \rho = 3 \) for tensorial quantities in the Cartesian coordinates \( {x_i} \). The number of tensor components is \( N = {R^{\rho }} \). It is then convenient to write the components of a first, second or fourth order tensors as \( {(3} \times {1),}\,\,{(3} \times {3)}\,\,{\text{\; or (9}} \times {9)} \) arrays, which need not conform to the rules of matrix algebra. The third order tensor can be displayed in three \( {(3} \times {3)} \) arrays.