Tangent space - Wikipedia: "A real-valued function {\displaystyle f:M\to \mathbf {R} } is said to belong to {\displaystyle {C^{\infty }}(M)} if and only if for every coordinate chart {\displaystyle \varphi :U\to \mathbf {R} ^{n}} , the map {\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbf {R} ^{n}\to \mathbf {R} } is infinitely differentiable. "
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