Integral Curves and Flows

Abstract

The primary geometric objects associated with smooth vector fields are their integral curves, which are smooth curves whose velocity at each point is equal to the value of the vector field there. The collection of all integral curves of a given vector field on a manifold determines a family of diffeomorphisms of (open subsets of) the manifold, called a flow. The main theorem of this chapter, the fundamental theorem on flows, asserts that every smooth vector field determines a unique maximal integral curve starting at each point, and the collection of all such integral curves determines a unique maximal flow. After proving the fundamental theorem, we show how “flowing out” from initial submanifolds along vector fields can be used to create useful parametrizations of larger submanifolds. We then introduce the Lie derivative, which is a coordinate-independent way of computing the rate of change of one vector field along the flow of another. In the last section, we apply flows to the study of first-order partial differential equations.