## What Does the Bracket Measure? (part 2)

Today we’ll prove the assertions we made last time: if and are vector fields with flows and , respectively in some neighborhood of some point , we define the curve

We assert that for any smooth we have

To show the first, we define the three “rectangles”

Notice that , , and . The chain rule lets us then calculate:

as asserted. As for the other assertion, we start by observing

Using the fact that we can turn the first term on the right into

Now, similar tedious calculations that make the big one above look like idle doodling give us two more identities:

and we conclude that

as asserted.

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