Smooth function

Definition

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

$\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A}$ such that $f\circ\varphi^{-1}\subseteq\mathbb{R}^n\rightarrow\mathbb{R}^k$ is $C^\infty$/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 $f$ is still smooth.

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 $f(p)=(f_1(p),\cdots,f_k(p))$

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 manifold
• Smoothly compatible charts