Smooth chart - Knowlpedia

Let $(M,\mathcal A)$ be a smooth manifold, with chosen maximal smooth atlas $\mathcal A$ . A smooth chart on $M$ is a coordinate chart $(U,\varphi)$ belonging to $\mathcal A$ .

Local coordinates

Writing $\varphi=(x^1,\ldots,x^n)$ , the component functions $x^i:U\to\mathbb R$ are the associated local coordinates.

Compatibility

Equivalently, a coordinate chart is smooth when its transition maps with every chart in $\mathcal A$ are $C^\infty$ in both directions. Requiring only one direction to be smooth is insufficient.

The coordinate map of a smooth chart is a diffeomorphism from its domain, with the induced smooth structure, onto its Euclidean image. This is a consequence of compatibility, rather than a prerequisite for constructing the smooth structure.