I recently learned the notion of coordinate chart: If $M$ is a manifold and $U\subseteq M$ is an open set in $M$ then a coordinate chart would be a smooth homeomorphism $\varphi : U \to V \subseteq \mathbb R^n$.

So far so good. Now let $M$ be the smooth manifold $\mathbb R^3$. We can specify a point on $M$ by coordinates of the form

$$ x = r \sin \theta \cos \varphi$$

$$ y = r \sin \theta \sin \varphi$$

$$ z = r \cos \theta $$

where $r \in [0,\infty), \theta \in [0,\pi], \varphi \in [0, 2\pi )$. So the map

$$ \phi (r,\theta, \varphi) = (x,y,z)$$

would be a (global) chart. Except: its domain is not an open subset of $\mathbb R^3$ so it is not a chart. It is however a local chart: for $r \in (0,\infty), \theta \in (0,\pi), \varphi \in (0, 2\pi )$.

Yet, I have seen that $\phi$ is commonly referred to as coordinates which I understand to be a synonym for "coordinate chart".

My question: Is $\phi$ (the spherical coordinates) a coordinate chart or not?


Yeah, it should really by the corresponding open intervals to be a genuine 'chart'. Moreover, all sorts of things are bad if we allow the boundary -- e.g. $\phi$ isn't bijective anymore since as soon as $r=0$, it no longer matters what the angle coordinates are.

I think they're just doing a little abuse of language, since it's convenient to allow the boundary. These are still informally 'coordinates' since they're, you know, numbers that specify points on the manifold.


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