Smooth chart

Let $M$ be an $n$-dimensional smooth manifold . A (topological) chart on $M$ is a pair $(U,\varphi)$ where $U\subset M$ is open and $\varphi:U\to V$ is a homeomorphism onto an open set $V\subset \mathbb{R}^n$.

A smooth chart (or coordinate chart) on $M$ is a chart $(U,\varphi)$ such that for every chart $(U',\varphi')$ belonging to the smooth structure of $M$, the transition map

is smooth (as a map between open subsets of $\mathbb{R}^n$). Equivalently, $\varphi$ is a diffeomorphism between the open submanifold $U$ (with the induced smooth structure) and the open set $V\subset \mathbb{R}^n$.

Given a smooth chart $(U,\varphi)$ with $\varphi=(x^1,\dots,x^n)$, the functions $x^i:U\to \mathbb{R}$ are called the local coordinates associated to the chart.

Examples

1. Euclidean space. On $M=\mathbb{R}^n$, the pair $(\mathbb{R}^n,\mathrm{id})$ is a smooth chart. Any open set $U\subset\mathbb{R}^n$ with the inclusion $U\hookrightarrow\mathbb{R}^n$ gives a smooth chart $(U,\mathrm{id}_U)$.
2. Stereographic charts on the sphere. On $S^n\subset\mathbb{R}^{n+1}$, stereographic projection from the north pole defines a chart $(S^n\setminus\{N\},\sigma_N)$ with $\sigma_N:S^n\setminus\{N\}\to\mathbb{R}^n$ smooth; similarly from the south pole. The overlap transition map is smooth.
3. Product charts. If $(U,\varphi)$ is a chart on $M$ and $(W,\psi)$ is a chart on $N$, then $(U\times W,\varphi\times\psi)$ is a chart on $M\times N$, giving the standard product coordinate system.