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I am confused about the definition of "differential".

Sometimes I see it is a pushforward mapping $df:TM\rightarrow TN$, which gives another tangent vector when acting on a tangent vector.

https://en.wikipedia.org/wiki/Pushforward_(differential)

Sometimes it acts as a cobasis in contangent space, which contracts with tangent vectors and gives a real number, $df:TM\rightarrow \mathcal{R}$

https://en.wikipedia.org/wiki/Cotangent_space

How are the two definition compatible with each other?

Thank you very much.

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marked as duplicate by Anthony Carapetis, Community May 4 '16 at 10:24

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  • $\begingroup$ oh, thank you very much! $\endgroup$ – mmssm May 4 '16 at 10:24