"Rigorous" definition of Cartesian coordinates I, like most, first learned about Cartesian coordinates very early on in my educational career, and so the most instructional way to think about them was that you place down some perpendicular lines and measure the perpendicular distance from each line to get your coordinates; in other words, an operational definition. Now that I'm much farther along in my education, though, I wonder whether or not there's another, more "rigorous" way to define Cartesian coordinates, perhaps using the idea of a coordinate system as a mapping? Or is my question a bit pointless, since we already have the operational definition?

If you're thinking about manifolds, it's most useful to think about how you can write the metric for the coordinate system: if it can be written as g_ab = diag(+/- 1, +/-1 , ...) then you're in Cartesians. That immediately give you the line element, in the positive case being the familiar Pythagorean addition of ds^2 = dx^2 + dy^2 + ... . The negative case is important in pseudo-Euclidian manifolds.
