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You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no idea what the term actually means.

Is there a rigorous definition of "coordinate system"?

In particular, if I were to write "Let $c : C \rightarrow X$ denote a coordinate system for $X$," what kind of objects are $C$ and $X$ (affine spaces? topological spaces? something else?), and what kind of entity is $c$ (a surjective function? a continuous mapping? something else?)

goblin GONE
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A "coordinate system" or more precisely a "local coordinate system" in this context is also known as a chart. A chart for a topological space $X$ is a continuous map $\phi:U\to X$ which is a homeomorphism onto its image, where $U$ is an open subset of $\mathbb{R}^n$. The coordinate functions $x_1,\ldots,x_n$ on $U\subset\mathbb R^n$ then give rise to coordinate functions $x_i\circ\phi^{-1}$ on $\phi(U)\subset X$.

bradhd
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    Depending on the context, one might want this to be a diffeomorphism. – wckronholm Mar 16 '14 at 03:53
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    Okay, but the function $f : \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R}$ given by $f(r,\theta) = (r\sin \theta,r\cos\theta)$ fails to be a coordinate system under this definition. Is this actually desirable? – goblin GONE Mar 16 '14 at 03:54
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    Sorry, accidentally deleted my comment: if we restrict the polar coordinates map $f$ to $\mathbb (0,\infty)\times (0,2\pi)$ then we get a chart. It's true that the image of this map isn't $\mathbb R^2$, but rather $\mathbb R^2$ with the nonnegative part of the $x$-axis removed. Whether this is a problem depends on what you're trying to do. For example, if you're using these coordinates to simplify some function you're trying to integrate over a region, there's no problem -- the $x$-axis has measure zero and so removing it from the region won't change the outcome of the integral. – bradhd Mar 16 '14 at 04:07
  • Could you add an example ? I have seen the general definition of a chart but i am unable to apply it to an explicit example myself. What would be $x_i \circ \phi^{-1}$ in polar coordinates ? What is $\phi: U \rightarrow X$ in this context. Likewise what would be $X,U$. – Hans Wurst Jun 01 '20 at 08:27