# Definition:Local Coordinates

Jump to navigation
Jump to search

## Definition

There is believed to be a mistake here, possibly a typo.In particular: This is garbage. "Also known as" contains correct definitionYou can help ProofWiki by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mistake}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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): Entry:**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): Entry:**local coordinate**