APPENDIX 1 - Orthogonal Curvilinear Coordinate Systems

Summary

Introduction

Rectangular Cartesian coordinate systems are well suited for presenting the basic concepts of continuum mechanics since many mathematical complications that arise with other coordinate systems, for example, convected systems, are avoided. However, for the solution of specific problems, the use of rectangular Cartesian coordinate systems may result in considerable difficulties, and the use of other systems may be desirable in order to take advantage of symmetry aspects of the problem. For example, in the analysis of the expansion of a thick-walled cylindrical tube, the use of cylindrical polar coordinates has an obvious advantage. We introduce a simple theory of curvilinear coordinates in this appendix and specialize it for orthogonal curvilinear systems, in particular cylindrical and spherical. We avoid the use of the general theory of tensor components referred to curvilinear coordinates by considering what are known as the physical components of tensors that are derived for orthogonal coordinate systems. Superscripts are used to denote curvilinear coordinates.