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Note: there are many definitions of smooth and it changes a lot between books - I shall be consistent in this wiki and mention the others


Here [ilmath]U\subseteq\mathbb{R}^n[/ilmath] is open, and [ilmath]V\subseteq\mathbb{R}^m[/ilmath] is also open, we say a function[1] [math]F:U\rightarrow V[/math] is smooth, [math]C^\infty[/math] or infinitely differentiable if:

  • Each component function ([ilmath]F^i[/ilmath] for [ilmath]1\le i\le m\in\mathbb{N} [/ilmath] has continuous partial derivatives of all orders

TODO: Expand this to a more formal one - like the one from Loring W. Tu's book

Warning about diffeomorphisms

A [ilmath]F[/ilmath] is a Diffeomorphism if it is bijective, smooth and the inverse [ilmath]F^{-1} [/ilmath] is also smooth.


  1. John M Lee - Introduction to smooth manifolds - Second Edition