Integral Curves and Flows
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 fie
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Integral Curves and Flows
In this chapter we continue our study of vector fields. 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. Any smooth R-action is a flow, for example; but there are flows that are not R-actions because the diffeomorphisms may not be defined on the whole manifold for every t 2 R. The main theorem of the 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. The proof is an application of the existence, uniqueness, and smoothness theorem for solutions of ordinary differential equations (see Appendix D). After proving the fundamental theorem, we explore some of the properties of vector fields and flows. First, we investigate conditions under which a vector field generates a global flow. Then we show how “flowing out” from initial submanifolds along vector fields can be used to create useful parametrizations of larger submanifolds. Next we examine the local behavior of flows, and find that the behavior at points where the vector field vanishes, which correspond to equilibrium points of the flow, is very different from the behavior at points where it does not vanish, where the flow looks locally like translation along parallel coordinate lines. 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. It leads to some deep connections among vector fields, their Lie brackets, and their flows. In particular, we will prove that two vector fields have commuting flows if and only if their Lie bracket is zero. Based on this fact, we can prove a necessary and sufficient condition for a smooth local frame to be expressible as a coordinate frame. We then discuss how some of the results of this chapter can be generalized to time-dependent vector fields on manifolds. In the last section of the chapter, we describe an important application of flows to the study of first-order partial differential equations. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_9, © Springer Science+Business Media New York 2013
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Integral Curves and Flows
Fig. 9.1 An integral curve of a vector field
Integral Curves Suppose M is a smooth manifold with or without boundary. If W J ! M is a smooth curve, then for each t 2 J , the velocity vector 0 .t/ is a vector in T.t/ M . In this section we describe a way to work backwards: given a tangent vector at each point, we seek a curve whose velocity at each point is equal to the given vector there. If V is a vector field on M; an integra
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