Definition:Local Coordinates

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Definition

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

Let $p \in X$, and let $U \subset X$ 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 see


Sources