Smooth

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

Definition

Here $U\subseteq\mathbb{R}^n$ is open, and $V\subseteq\mathbb{R}^m$ is also open, we say a function

$$F:U\rightarrow V$$ is smooth, $$C^\infty$$ or infinitely differentiable if:

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

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

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Warning about diffeomorphisms

A $F$ is a diffeomorphism if it is bijective, smooth and the inverse $F^{-1}$ is also smooth.