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 49: / Line 49: @@
 ==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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