3/14/2024 0 Comments Mount whitney mx mx sim![]() 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. ![]() 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. 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 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.
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