. T( M 2) That is, (f)*X = T«(f») 0 X 0 (f)-i. This clearly means that (f)* turns the vector fields in just the same way as (f) turns the curves that define the direction. 22) 1. M = IRn, (f): x -+ x + a, T«(f»: (x, u) -+ (x + a, u), X: x -+ (x, vex»~, *X: x -+ (x, vex - a».
A Euclidean chart may well suffice for an open subset of ~n, but this coordinate system is not to be preferred to the others, which may require many charts (as with polar coordinates), but are more convenient in other respects. When depicted in a new chart, a given neighborhood in M will appear distorted. " As we have seen (Examples 6 and 7), not all subsets of ~n may be used as manifolds. They need not necessarily be open subsets of ~n, but one should at least be able to define differentiation on them.
A Course in Mathematical Physics 1 and 2: Classical Dynamical Systems and Classical Field Theory by Walter Thirring, E.M. Harrell