# Fundamental Theorem on Flows

Let V be a smooth vector field on a smooth manifold M. There is a unique smooth maximal flow whose infinitesimal generator is V. The flow has the following properties:

(a) For each point on the manifold, there is curve theta that is the unique maximal integral curve of V starting at said point.

(b) If there is another point in the flow domain… (need to reinterpret what this means, actually, as I can’t parse the proof as I’m writing this).

(c) The integral curves are diffeomorphisms, whose inverses can be computed by reversing time.

…we’ll get back to this and elaborate on what it means with respect to existence and uniqueness of solutions to ODEs. I just wanted you to see where my wordplay came from. Btw, integral curves are smooth immersions.
🙂

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