Difference between revisions of "Smooth function"

Latest revision as of 16:16, 14 April 2015

Definition

A smooth function on a smooth $n$ -manifold, $(M,\mathcal{A})$ , is a function^[1] $f:M\rightarrow\mathbb{R}$ that satisfies:

∀p∈M ∃ (U,φ)∈A[p∈U∧f∘φ−1:φ(U)⊆Rn→R∈C∞]
- That is to say $f\circ\varphi^{-1}$ is smooth in the usual sense - of having continuous partial derivatives of all orders.

[Expand]
Theorem: Any other chart in $(M,\mathcal{A})$ will also satisfy the definition of $f$ being smooth

Let $(M,\mathcal{A})$ be a given smooth manifold

Let $(U,\varphi)$ be a chart, on which $f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R}$ is smooth

We wish to show any smoothly compatible chart with $(U,\varphi)$ will support the definition of $f$ being smooth.

That is to say all other charts in the smooth atlas $\mathcal{A}$

Proof:

Let

$(V,\psi)$ be any chart in

$\mathcal{A}$ be given.

Then

$(U,\varphi)$ and

$(V,\psi)$ are smoothly compatible

this means that either:

$U\cap V$ is empty - in which case there is nothing to show OR
$\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\varphi(U\cap V)$ is a diffeomorphism

We can compose this with

$f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R}$ as follows

$(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R}$ which is smooth as it is a composition of smooth functions

$=f\circ\varphi^{-1}\circ\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R}$

$=f\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R}$ - which we know to be smooth as it is equal to

$(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R}$ - which as we've said is smooth

QED

Extending to vectors

Note that given an $f:M\rightarrow\mathbb{R}^k$ this is actually just a set of functions, $f_1,\cdots,f_k$ where $f_i:M\rightarrow\mathbb{R}$ and $f(p)=(f_1(p),\cdots,f_k(p))$

We can define $f:M\rightarrow\mathbb{R}^k$ as being smooth $\iff\forall i=1,\cdots k$ we have $f_i:M\rightarrow\mathbb{R}$ being smooth

Notations

The set of all smooth functions

Without knowledge of smooth manifolds we may already define $C^\infty(\mathbb{R}^n)$ - 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 $n$ -manifold, $M$ , we now know what it means for a function to be smooth on it, so:

Let

$f\in C^\infty(M)\iff f:M\rightarrow\mathbb{R}$ is smooth

References

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

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

[1]

@@ Line 1: / Line 1: @@
 ==Definition==
-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} }} 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}[p\in U\wedge f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]}}
+** That is to say {{M|f\circ\varphi^{-1} }} is [[Smooth|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.
+{{Begin Theorem}}
+Theorem: Any other chart in {{M|(M,\mathcal{A})}} will also satisfy the definition of {{M|f}} being smooth
+{{Begin Proof}}
+Let {{M|(M,\mathcal{A})}} be a given [[Smooth manifold|smooth manifold]]
+Let {{M|(U,\varphi)}} be a chart, on which {{M|f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R} }} is [[Smooth|smooth]]
+We wish to show any [[Smoothly compatible charts|smoothly compatible chart]] with {{M|(U,\varphi)}} will support the definition of {{M|f}} being smooth.
+That is to say all other charts in the [[Smooth atlas|smooth atlas]] {{M|\mathcal{A} }}
+'''Proof:'''
+: Let {{M|(V,\psi)}} be any chart in {{M|\mathcal{A} }} be given.
+:: Then {{M|(U,\varphi)}} and {{M|(V,\psi)}} are smoothly compatible
+:: this means that either:
+::* {{M|U\cap V}} is empty - in which case there is nothing to show OR
+::* {{M|\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\varphi(U\cap V)}} is a [[Diffeomorphism|diffeomorphism]]
+::: We can compose this with {{M|f\circ\varphi^{-1}:\varphi(U)\rightarrow\mathbb{R} }} as follows
+:::: {{M|(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R} }} which is [[Smooth|smooth]] as it is a composition of smooth functions
+:::: {{M|1==f\circ\varphi^{-1}\circ\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R} }}
+:::: {{M|1==f\circ\psi^{-1}:\psi(U\cap V)\rightarrow\mathbb{R} }} - which we know to be smooth as it is ''equal to'' {{M|(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R} }} - which as we've said is smooth
+QED
+{{End Proof}}
+{{End Theorem}}
+===Extending to vectors===
+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))}}
+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
+==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==
+* [[Smooth map]]
 * [[Smooth manifold]]
 * [[Smoothly compatible charts]]

Difference between revisions of "Smooth function"

Latest revision as of 16:16, 14 April 2015

Contents

Definition

Extending to vectors

Notations

The set of all smooth functions

The set of all smooth functions on a manifold

See also

References

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