The Lie Derivative
4.1. The Lie Derivative of a Vector Field
Walk one mile east, then north, then west, then south. Have you really returned?
4.1a. The Lie Bracket
Let X and Y be a pair of vector fields on a manifold Mn and let ϕ(t) = ϕt be the local flow generated by the field X (see 1.4a). Then ϕtx is the point t seconds along the integral curve of X, the “orbit” of x, that starts at time 0 at the point x. We shall compare the vector Yϕtx at that point with the result of pushing Yx to the point ϕtx by means of the differential ϕt