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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.2012-03-26
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    I'm confused as to how. I have local charts for $TM$ and $TN$ but I don't know what to do with them.2012-03-26
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    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}$2012-03-26
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    All this chart business is confusing me. Could anyone just write a specific proof so that I can see exactly what goes on?2012-03-27

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