Ivancevic_Applied-Diff-Geom

Chapter 2Technical Preliminaries: Tensors,Actions and Functors2.1 Tensors: Local Machinery of Differential GeometryPhysical and engineering laws must be independent of any particular coordinatesystems used in describing them mathematically, if they are to bevalid. In other words, all physical and engineering equations need to betensorial or covariant. Therefore, for the reference purpose, in this section,we give the basic formulas from the standard tensor calculus, which is usedthroughout the text. The basic notational convention used in tensor calculusis Einstein’s summation convention over repeated indices. More onthis subject can be found in any standard textbook on mathematical methodsfor scientists and engineers, or mathematical physics (we recommend[Misner et al. (1973)]).2.1.1 Transformation of Coordinates and ElementaryTensorsTo introduce tensors, consider a standard linear nD matrix system, Ax = b.It can be rewritten in the so–called covariant form asa ij x j = b i , (i, j = 1, ..., n). (2.1)Here, i is a free index and j is a dummy index to be summed upon, so theexpansion of (2.1) givesa 11 x 1 + a 12 x 2 + ... + a 1n x n = b 1 ,a 21 x 1 + a 22 x 2 + ... + a 2n x n = b 2 ,...a n1 x 1 + a n2 x 2 + ... + a nn x n = b n ,51