Smooth function

Smooth functions on and between manifolds

Given a smooth manifold${\displaystyle M}$, of dimension ${\displaystyle m,}$ and an atlas${\displaystyle {\mathfrak {U}}=\{(U_{\alpha },\phi _{\alpha })\}_{\alpha },}$ a map ${\displaystyle f:M\to \mathbb {R} }$ is smooth on ${\displaystyle M}$ if, for every ${\displaystyle p\in M}$, there is a chart ${\displaystyle (U,\phi )\in {\mathfrak {U}},}$ with ${\displaystyle p\in U,}$ such that ${\displaystyle f\circ \phi ^{-1}:\phi (U)\to \mathbb {R} }$ is a smooth function from the open subset ${\displaystyle \phi (U)}$ of ${\displaystyle \mathbb {R} ^{m}}$ to ${\displaystyle \mathbb {R} }$. Similarly, ${\displaystyle f}$ is of class ${\displaystyle C^{k}}$ if these coordinate representations are of class ${\displaystyle C^{k}}$. Smoothness can be checked with respect to any chart of the atlas that contains ${\displaystyle p,}$ since the smoothness requirements on the transition functions between charts ensure that if ${\displaystyle f}$ is smooth near ${\displaystyle p}$ in one chart it will be smooth near ${\displaystyle p}$ in any other chart.