- #1
Lajka
- 68
- 0
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.
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.