Vector Analysis on Manifolds

Abstract

Two central ideas of this chapter are orientation and vector field. When we studied integrals of real-valued functions over manifolds, neither of these ideas were used. Yet orientations and vector fields often play important roles in integrals over curves, surfaces and higher dimensional manifolds. For example, when Computing work done by a particle moving along a curve C through a potential field Ø, we have

$$\int\limits_c {(\nabla \phi ) \cdot T = \phi (terminal point)} - \phi (initial point)$$

where T is a unit tangent vector to C. Or perhaps the reader is familiär with the classical theorems of vector analysis, Green’s theorem, Gauss’ divergence theorem, and Stokes’ theorem. He or she perhaps knows something of their importance in such fields as fluid mechanics and electromagnetism.