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?
Differential map between smooth manifolds is smooth
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differential-geometry
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4Express it locally, as usual. – 2012-03-26
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0I'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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0if $x$ 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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0All 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