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Given a smooth map $f:M\to N$ between smooth manifolds how do you show that the differential map $df:TM\to TN$ is smooth?

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Express it locally, as usual. –  Mariano Suárez-Alvarez Mar 26 '12 at 16:58
I'm confused as to how. I have local charts for $TM$ and $TN$ but I don't know what to do with them. –  09867 Mar 26 '12 at 17:16
if $x$ is a chart for $M$, $y$ one for $N$, then you have $Tx$ a chart for $TM$, $Ty$ a chart for $TN$. Untangle the definitions and write down $Ty \circ df \circ (Tx)^{-1}$ –  Blah Mar 26 '12 at 20:07
All this chart business is confusing me. Could anyone just write a specific proof so that I can see exactly what goes on? –  09867 Mar 27 '12 at 15:15

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