Overview

In the previous chapters, the geometry of surfaces embedded in a three-dimensional Euclidean space was discussed. The basic tool in describing the intrinsic properties of geometric objects is the metric tensor, given in terms of its covariant or contravariant components. The distinction between these components, as well as the concept of invariance, will be explained in this Appendix. In physics, equations which do not change with the transformation of coordinate systems are called invariant. Equations which remain valid because their terms, though not invariant, transform according to identical transformation laws are called covariant [Bergmann, 1976]. An example is provided by Newton's equation of motion F = m a for a particle of mass m. In this equation, both the force F and acceleration a transform as three-dimensional vectors in Cartesian coordinate systems.

In physical applications, as well as in elasticity calculations, the environment where analysis is done is not a linear space. The simplest example is provided by the surface of a sphere, on which one cannot define linear operations acting on vectors referring to different points. The basic tool for dealing with such spaces is the calculus on manifolds. For example, a rigorous description of the stress tensor of an elastic body involves the notion of a vector-valued form on a manifold [Frankel, 1997].

The idea of a function on a manifold is often introduced in terms of the function's graph. The graph of a function may be viewed as a subset of...