Differential Geometry - Question about Limit

Hello everyone, I had a question I was hoping I could get some help with. Let be a smooth manifold and let be vector fields. Let be the flow of . So in my notation I have and for a smooth function then . Let , so that pointwise.

In a proof in my course there is a statement

.

Why can we just take limits inside? It is not obvious to me that the map is smooth.

Thanks for any help.

Re: Differential Geometry - Question about Limit

Quote:

Originally Posted by

**slevvio** Hello everyone, I had a question I was hoping I could get some help with. Let

be a smooth manifold and let

be vector fields. Let

be the flow of

. So in my notation I have

and for a smooth function

then

. Let

, so that

pointwise.

In a proof in my course there is a statement

.

Why can we just take limits inside? It is not obvious to me that the map

is smooth.

Thanks for any help.

No, but do you disagree that it's not *continuous*?

Re: Differential Geometry - Question about Limit

I can't really see why it's continuous in t :(

Re: Differential Geometry - Question about Limit

have you any ideas as to why this might be continuous in t?

Thanks very much