Definition:Local Coordinates

Definition

Let $M$ be an $n$-dimensional manifold.

Let $p \in X$, and let $U \subseteq M$ be a neighbourhood of $p$.

Let $\phi$ be the local coordinate map of $M$ such that:

$\map \phi p = \tuple {\map {x_1} p, \map {x_2} p, \ldots \map {x_n} p}$

The component mappings $\tuple {x_1, x_2, \ldots x_n}$ of $\phi$ are known as the local coordinates around $p$.

Also see

• Results about local coordinates can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): local coordinates
• 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1.1$: Smooth Manifolds. Topological Manifolds
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): local coordinates