I was asked to show that is diffeomorphic to .
But why can I not just use the map given by , since each ?
I know this cannot be the correct map, but I am not sure why. Any help would be appreciated.
TM cannot always be covered by a single chart. Your approach only proved that in that single chart the two are diffeomorphism. This is obvious since the bundle is patched together in such a way. Your argument doensn't state anything globally, which is needed.