Question about curvilinear coordinates

[Lajka]

Just a quick little question.

I was reading a wikipedia article about curvilinear coordinates, as well as some others, and a question popped into my head. Although we take this for granted (at least I do), now I have to ask this.

From what I've seen as an engineer, we always define curvilinear coordinates as some functions of Cartesian coordinates, and we always use Cartesian unit vectors to derive various properties of unit vectors in our new coordinate system. So, in a way, we are always depending on Cartesian coordinate system. It looks like we must define Cartesian coordinates and Cartesian unit vectors first in an affine space we were given, and only then can we start defining some other coordinate system (polar, cylindric, spherical) in there.

Now, I'm no physicist, so I don't know much about manifolds (but I would like to learn, tho), and it seems to me that Cartesian coordinate system cannot be a good choice for some arbitrary manifold, but it also seems to me like that's a mandatory starting point (the thing I explained above).

Is there any way to define curvilinear coordinates without introducing Cartesian coordinates whatsoever? And how do you define inner product, angles, metric etc. in that case?
I hope there's some easy answer for this.

Thanks in advance.

 

[HallsofIvy]

I really do not understand your question. The standard defintions of, say, polar coordinates, and spherical coordinates do not use Cartesian Coordinates. Of course, the formulas connecting them are typically given for ease in going form one to the other.

If we pick a point in the plane to be the "origin", and choose a direction to correspond to theta= 0, then we can define the polar coordinates of any point in the plane without any reference to Cartesian Coordinates. Given the two points (r1, theta1) and (r2,theta2), the lines connecting them to the origin, together with the line between them gives a triangle in which I know that two of the sides have lengths r1 and r2 and the angle between them is theta2- theta1. I can then use the "cosine law" to find the distance between the two points, the "metric". Angles can be defined geometrically and calculated with trigonometry. And, of course, the "inner product" of two vectors u and v is |u||v|cos(theta)- that is, the product of the lengths of the two vectors multiplied by the cosine of the angle between them. No Cartesian Coordinates need there.

 

[Lajka]

Hm, okay, you have a point there. But riddle me http://planetmath.org/?method=l2h&from=collab&id=83&op=getobj".

What we have here is a derivation of unit vectors in curvilinear coordinates, pretty standard stuff I'd say. And this is the starting point, which is essential for the rest of the text:

As you can see, the use of Cartesian unit vectors is obligatory here, they must figure somehow in formulas for expressing curvilinear unit vectors. Personally, I've never seen any different derivations, other than the ones like these, for curvilinear unit vectors.
This makes me think that, if I don't have Cartesian unit vectors (therefore, Cartesian coordinate system), I cannot express curvilinear unit vectors (therefore, curvilinear coordinate system), which returns me to my point that Cartesian coordinate system is necessary as a starting point.

I also believe that I'm mistaken :D But alas, I also cannot find where exactly, in my reasoning, is the error.