- #1

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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##.

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- Thread starter kent davidge
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- #1

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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,

- #2

fresh_42

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The

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.

- #3

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But do the polar coordinates uniquely specify a point in ##\mathbb R^2##? I think there's a issue when ##r = 0##.In general, coordinates are any system which allows to uniquely specify a point in some space.

- #4

fresh_42

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

- #5

WWGD

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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.

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