« Analysis on Manifolds

Definition. smooth map between manifolds [009G]

Let be a map between smooth manifolds and . We say that is smooth if for all charts of and of the composition is a smooth map between subsets of .

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Lemma. [009H]

Smoothness is independent of the choice of atlas.

TODO: show this

Defintion. Differential of a smooth map between manifolds [009L]

Let be a smooth map between manifolds. For a poing in and a chart around and a chart around we define the differential of at as

TODO: Show that this is well defined and independent of the choice of charts. TODO: Show the differential is functorial.