Difference between revisions of "Smooth"

Latest revision as of 06:38, 7 April 2015

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^[1] $$F:U\rightarrow V$$ 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

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

Warning about diffeomorphisms

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

References

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