because (2.11) implies
Similarly, one can find the components of the Lie derivative of any Cr (r > 1) tensor field T of type (r, s) by using Leibniz' rule on
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.
Lx(to ® Y) = L^tti ® Y+to ® LxY (Lie derivative property (3)) to obtain
Lx(T®E°®...®Ed®Ee®...®E(,), and then contracting on all positions. One finds the coordinate components to be
(L*T)ab"de}...0 = (8Tat-def.J^)Xi - Tib-dtf.„e8Xaldxi — (all upper indices) + T®6--dif gdXi/8xt + (all lower indices).
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