As usual, we’re going to want our objects of study — smooth (or differentiable) manifolds — to be objects in a category. And a category means we need morphisms. The morphisms between smooth manifolds are smooth maps.
Given two smooth manifolds, and , a continuous map is smooth at a point if we can find a coordinate patch containing and a coordinate patch containing the image so that the composite function is smooth as a function from into . We say it’s smooth if it’s smooth at all points.
But wait, maybe we just got lucky when we picked these coordinate patches. Well, it actually doesn’t matter. If is smooth according to one pair of coordinate patches, it’s smooth according to any other pair. Indeed, if we take another set of coordinates around then the compatibility condition says that the transition function is smooth. And then so is the composite:
But this is just the smoothness condition in terms of and . Similarly, if we change coordinates in the target from to , the compatibility condition says that is smooth, and so the composite
is smooth as well. The condition for smoothness at a point, therefore, really only depends on the behavior of the function near that point, and not on what particular coordinates we use to attest to its smoothness.
In particular, let’s consider what it means for a function to be smooth from a manifold to the real space . In this case, we can choose the entire space with the identity function as the coordinate patch on the target manifold. Thus a function is smooth at a point if there is some containing with coordinate map such that the composite is smooth.
Finally, just like we have the fancy word “homeomorphism” for isomorphisms of topological spaces, we have the fancy word “diffeomorphism” for isomorphisms of differentiable manifolds.