Re: Derivations on manifolds. J. Lee, p.65

Thanks, everyone. A follow-up if I may:

I am trying to show that differentiability in functions

f:M->IR is a local property: f is differentiable if

for every m in M there is a 'hood ('hood=neighborhood)

U , such that f|_U is differentiable (i.e, (foPhi^-1)

is differentiable , etc.).

Would you check my proof:


Assume for every m in M there is a 'hood U

such that f|_U is differentiable. Then there

is a chart (V, h), with m in V and V< U ,

with foh^-1 is differentiable. Then (V,h)

is also a chart in M for m , and f:M-->IR^n

is differentiable.



Other side looks a little more difficult:

Let f:M-->IR^n be differentiable. Then, for

m in M, there is a chart (W,g), with m in W

and fog^-1 is differentiable. Now I need to

show the restriction of f to any open set O

in M , is differentiable.

Then we take o in O . Then take a chart (Z,j)

intersecting O and containing O. Then (O/\Z,j|_(O/\Z))

is a chart in O (i.e, M|_O) where f is differentiable.


Is this correct?. Sorry for the notational mess.

Thanks in Advance.
.