Definition:Local Coordinates
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Definition
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Let $X$ be an $n$-dimensional manifold.
Let $p \in X$, and let $U \subseteq X$ be a neighbourhood of $p$.
Then a set of mappings $x_i: U \to \R$, $1 \le i \le n$, satisfying:
- $a = b \iff \forall i: \map {x_i} a = \map {x_i} b$
is called a set of local coordinates.
When the neighbourhood $U$ is to be stressed, one may also say local coordinates for $U$.
Similarly, when the element $p$ is to be stressed, one may also say local coordinates around $p$.
Also known as
Local coordinates $\tuple {x_1, x_2, \ldots x_n}$ are also know as the component functions of the local coordinate map $\phi$ defined by:
- $\map \phi p = \tuple {\map {x_1} p, \map {x_2} p, \ldots \map {x_n} p}$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): local coordinate
- 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 coordinate