Difference between revisions of "Smooth function"
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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))}} | 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))}} | ||
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| + | We can define {{M|f:M\rightarrow\mathbb{R}^k}} as being smooth {{M|1=\iff\forall i=1,\cdots k}} we have {{M|f_i:M\rightarrow\mathbb{R} }} being smooth | ||
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==Notations== | ==Notations== | ||
===The set of all smooth functions=== | ===The set of all smooth functions=== | ||
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==See also== | ==See also== | ||
| + | * [[Smooth map]] | ||
* [[Smooth manifold]] | * [[Smooth manifold]] | ||
* [[Smoothly compatible charts]] | * [[Smoothly compatible charts]] | ||
Latest revision as of 16:16, 14 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} [/ilmath] that satisfies:
- [ilmath]\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A}[p\in U\wedge f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty][/ilmath]
- That is to say [ilmath]f\circ\varphi^{-1} [/ilmath] is smooth in the usual sense - of having continuous partial derivatives of all orders.
Theorem: Any other chart in [ilmath](M,\mathcal{A})[/ilmath] will also satisfy the definition of [ilmath]f[/ilmath] being smooth
Let [ilmath](M,\mathcal{A})[/ilmath] be a given smooth manifold
Let [ilmath](U,\varphi)[/ilmath] be a chart, on which [ilmath]f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R} [/ilmath] is smooth
We wish to show any smoothly compatible chart with [ilmath](U,\varphi)[/ilmath] will support the definition of [ilmath]f[/ilmath] being smooth.
That is to say all other charts in the smooth atlas [ilmath]\mathcal{A} [/ilmath]
Proof:
- Let [ilmath](V,\psi)[/ilmath] be any chart in [ilmath]\mathcal{A} [/ilmath] be given.
- Then [ilmath](U,\varphi)[/ilmath] and [ilmath](V,\psi)[/ilmath] are smoothly compatible
- this means that either:
- [ilmath]U\cap V[/ilmath] is empty - in which case there is nothing to show OR
- [ilmath]\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\varphi(U\cap V)[/ilmath] is a diffeomorphism
- We can compose this with [ilmath]f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R} [/ilmath] as follows
- [ilmath](f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R} [/ilmath] which is smooth as it is a composition of smooth functions
- [ilmath]=f\circ\varphi^{-1}\circ\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R}[/ilmath]
- [ilmath]=f\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R}[/ilmath] - which we know to be smooth as it is equal to [ilmath](f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R} [/ilmath] - which as we've said is smooth
QED
Extending to vectors
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]
We can define [ilmath]f:M\rightarrow\mathbb{R}^k[/ilmath] as being smooth [ilmath]\iff\forall i=1,\cdots k[/ilmath] we have [ilmath]f_i:M\rightarrow\mathbb{R} [/ilmath] being smooth
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
