Smooth function

Definition

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

• $\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A}[p\in U\wedge f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]$
  • That is to say $f\circ\varphi^{-1}$ is smooth in the usual sense - of having continuous partial derivatives of all orders.

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

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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

See also

• Smooth map
• Smooth manifold
• Smoothly compatible charts