8 - COVARIANT DIFFERENTIATION AND THE STRESS EQUATIONS OF MOTION

Abstract

Most of the tensor operations have one-point operations whose coordinate representations do not involve differentiation or integration with respect to coordinates. This chapter discusses operations involving differentiation with respect to coordinates. The analysis is developed in three stages: (1) operations with general tensors, not requiring covariant derivatives; (2) covariant differentiation in a Euclidean manifold; and (3) covariant differentiation in an affinely connected Riemannian manifold. For general vector and tensor fields, it is necessary to select combinations of derivatives and other terms in such a way that the unwanted second derivatives do not appear in the final transformation equations. The chapter presents a method based on the curvature of a surface in a three-dimensional Euclidean manifold to derive formulas for the principal radii of curvature in a form convenient for use with the stress equations of motion.