Let’s go back to the way a vector field on a manifold gives us a “derivative” of smooth functions . If is a smooth vector field it has a maximal flow which gives a one-parameter family of diffeomorphisms, which we can think of as “moving forward along by .
Now given a smooth function we use this as if we were taking a derivative from all the way back in single-variable calculus: measure at , flow forward by and measure at , take the difference, divide by , and take the limit as approaches zero:
Note that even if is not complete we do always have some interval around on which is defined and this difference quotient makes sense.
So far this is just a complicated (but descriptive!) way of restating something we already knew about. But now we can take this same approach and apply it to other vector fields. So if is another smooth vector field, we define the “Lie derivative” of by as:
Again we evaluate at both and , but here’s where a trick comes in: we can’t compare these two vectors directly, since they live at different points on , and thus in different tangent spaces. So in order to compensate we use the flow itself to move backwards from back to , and use the derivative to carry along the vector .
We can come up with an alternate version of this formula by using similar techniques to those above:
That is, if we define the curve in the tangent space then . This would seem to make it live in the tangent space to — that is, in — but remember that since is a vector space we identify it with all of its tangent spaces. Thus , just like is.