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 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.

See also

• Smooth manifold
• Smoothly compatible charts