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I have started reading about vector calculus and I noted that with other coordinate systems (spherical/polar etc), when you differentiate a vector in this coordinate system you have to differentiate the basis vectors as well. This was at first confusing to me as I thought that, since the vectors are orthogonal, there should not be a need to do this as you could place the origin anywhere. But then I realised that the origin would then be moving (say, with time) too. I haven't quite figured out how/why that applies yet...

So with regards to the cartesian coordinate system, I found this link. It is a bit too advanced for me but the first answer mentions that the derivatives of the bases are also orthogonal. I am guessing this is not so with coordinate systems that are not cartesian?

Essentially, to what extent can the cartesian coordinate system be said to be 'special', if at all? Is there a coordinate system which better describes the real world? I have a feeling this will have to do with general relativity and curved spacetime...

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The reason differentiating in non-rectangular coordinates requires additional work is because the basic vectors at any given point in space may be orthogonal to each other, but they aren't orthogonal to the basic vectors at any other given point in space. They don't need to vary with time; the geometry of the situation need not be dependent on any non-spatial variables.

For example, consider $\hat{{r}}$. The $\hat{r}$ you have at, say $\theta = \pi, ~ \phi = \pi / 4$ is not the same $\hat{r}$ you have at other points in space. In other words, the basis vectors in spherical coordinates are not constants; they are functions of each other.

Compare that with $\hat{i}, \hat{j}, \hat{k}$. These bases are invariant. $\hat{x}$ is $\hat{x}$ anywhere you go.

So are rectangular coordinates "special"? Not in any meaningful way. Everything that's true in rectangular coordinates is true in spherical; you just need to be conscientious about what $\hat{r}$ really means.

Also, just to clarify in a somewhat non sequitur fashion, remember that basis vectors aren't coordinates; it would be somewhat inconvenient to specify a point in space as a linear combination of non rectangular bases. In fact, what is often done in practice (in basic electrodynamics, for example) is to use rectangular basis vectors but spherical coordinates. Coordinates tell you the address of a point in space, vectors $\neq$ coordinates.

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The Cartesian base vectors are the same at each point, while non-Cartesian vary over space and thus feature only local bases.