Difference between revisions of "Smooth function"
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A '''smooth function''' on a [[Smooth manifold|smooth {{n|manifold}}]], {{M|(M,\mathcal{A})}}, is a function<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> {{M|f:M\rightarrow\mathbb{R}^k}} that satisfies: | A '''smooth function''' on a [[Smooth manifold|smooth {{n|manifold}}]], {{M|(M,\mathcal{A})}}, is a function<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> {{M|f:M\rightarrow\mathbb{R}^k}} that satisfies: | ||
| − | {{M|\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} }} such that {{M|f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k }} is [[Smooth|smooth]] in the usual sense, of having continuous partial derivatives of all orders. | + | {{M|\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} }} such that {{M|f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k }} is [[Smooth|{{M|C^\infty}}/smooth]] in the usual sense, of having continuous partial derivatives of all orders. |
Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so {{M|f}} is still smooth. | Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so {{M|f}} is still smooth. | ||
| + | |||
| + | Note that given an {{M|f:M\rightarrow\mathbb{R}^k}} this is actually just a set of functions, {{M|f_1,\cdots,f_k}} where {{M|f_i:M\rightarrow\mathbb{R} }} and {{M|1=f(p)=(f_1(p),\cdots,f_k(p))}} | ||
| + | ==Notations== | ||
| + | ===The set of all smooth functions=== | ||
| + | Without knowledge of [[Smooth manifold|smooth manifolds]] we may already define {{M|C^\infty(\mathbb{R}^n)}} - the set of all functions with continuous partial derivatives of all orders. | ||
| + | |||
| + | However with this definition of a smooth function we may go further: | ||
| + | ===The set of all smooth functions on a manifold=== | ||
| + | Given a [[Smooth manifold|smooth {{n|manifold}}]], {{M|M}}, we now know what it means for a function to be smooth on it, so: | ||
| + | |||
| + | Let <math>f\in C^\infty(M)\iff f:M\rightarrow\mathbb{R}</math> is smooth | ||
==See also== | ==See also== | ||
Revision as of 22:39, 12 April 2015
Contents
Definition
A smooth function on a smooth [ilmath]n[/ilmath]-manifold, [ilmath](M,\mathcal{A})[/ilmath], is a function[1] [ilmath]f:M\rightarrow\mathbb{R}^k[/ilmath] that satisfies:
[ilmath]\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A} [/ilmath] such that [ilmath]f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k [/ilmath] is [ilmath]C^\infty[/ilmath]/smooth in the usual sense, of having continuous partial derivatives of all orders.
Any smoothly compatible map (so all in the atlas of the smooth manifold) will have a smooth transition function, by composition, the result will be smooth, so [ilmath]f[/ilmath] is still smooth.
Note that given an [ilmath]f:M\rightarrow\mathbb{R}^k[/ilmath] this is actually just a set of functions, [ilmath]f_1,\cdots,f_k[/ilmath] where [ilmath]f_i:M\rightarrow\mathbb{R} [/ilmath] and [ilmath]f(p)=(f_1(p),\cdots,f_k(p))[/ilmath]
Notations
The set of all smooth functions
Without knowledge of smooth manifolds we may already define [ilmath]C^\infty(\mathbb{R}^n)[/ilmath] - the set of all functions with continuous partial derivatives of all orders.
However with this definition of a smooth function we may go further:
The set of all smooth functions on a manifold
Given a smooth [ilmath]n[/ilmath]-manifold, [ilmath]M[/ilmath], we now know what it means for a function to be smooth on it, so:
Let [math]f\in C^\infty(M)\iff f:M\rightarrow\mathbb{R}[/math] is smooth
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
