now therefore

One can rewrite this in the form

for all C2 functions/. We shall sometimes denote Z^Y by [X, Y], i.e.

If the Lie derivative of two vector fields X, Y vanishes, the vector fields are said to commute. In this case, if one starts at a pointy, goes a parameter distance t along the integral curves of X and then a parameter distance s along the integral curves of Y, one arrives at the same point as if one first went a distance s along the integral curves of Y and then a parameter distance t along the integral curves of X (see figure 7). Thus the set of all points which can be reached along integral curves of X and Y from a given point p will then form an immersed two-dimensional submanifold through p.

Figure 7. The transformations generated by commuting vector fields X, Y move a point p to points <j>n[p), <P,t(P) respectively. By successive applications of these transformations, p is moved to the points of a two-surface.

The components of the Lie derivative of a one-form o> may be found by contracting the relation

(by property (2) of Lie derivatives), where X, Y are arbitrary C1 vector fields, and then choosing Y as a basis vector E4. One finds the coordinate components (on choosing E{ = d/dx1) to be

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