Definition of coordinate system

[kent davidge]

In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to $\mathbb R^m$, can we say that the polar coordinates on $\mathbb R^2$ are a coordinate system?

I was thinking about this and the answer sounds to be a no, because the polar coordinates are not everywhere bijective to the cartesian coordinates, which we know, is a coordinate system that spans $\mathbb R^2$.

 

[fresh_42]

Your definition on a manifold is called a chart ($f(U)$) or a coordinate chart $(U,f)$.
The usual coordinates, are the Cartesian coordinates.
Polar, or cylindrical coordinates are also coordinates, even though not Cartesian.

In general, coordinates are any system which allows to uniquely specify a point in some space.

 

[kent davidge]

In general, coordinates are any system which allows to uniquely specify a point in some space.

But do the polar coordinates uniquely specify a point in $\mathbb R^2$? I think there's a issue when $r = 0$.

 

[fresh_42]

But do the polar coordinates uniquely specify a point in $\mathbb R^2$? I think there's a issue when $r = 0$.

Yes, the origin has to be assigned separately by a definition. $r=0$ is o.k. but it has no angle, but we can simply require $0:=(0,0)$ and have a unique system again. $(0,\varphi)$ with $\varphi > 0$ will then be undefined. But this is more of a debate for logicians (or linguists), and I'm neither.

 

[WWGD]

Yes, coordinate systems can be locally- or globally- defined. In the Polar case, they are defined only locally. EDIT: Most coord systems are locally, otherwise the manifold is isomorphic to the space where it is embedded, i.e., local homeos become global ones.