Curved space and curvilinear coordinates

[mertcan]

hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...

 

[Orodruin]

An embedding is usually of a lower-dimensional manifold. Curvilinear coordinates are used to describe the Euclidean space itself.

 

[wrobel]

You can not introduce Euclidian local coordinates in a curved space

 

[mertcan]

I think curvilinear coordinates generally define tangent space, but in curved space also defines normal component besides the tangent space. Am I right? I saw some close definition like this. Is it true?

 

[wrobel]

why do not you study the definition of the manifold first?

 

[wrobel]

By definition the Euclidean coordinates are the local coordinates $x^i$ on a manifold such that $\nabla_i\equiv \frac{\partial }{\partial x^i}$

 

[Orodruin]

I think curvilinear coordinates generally define tangent space, but in curved space also defines normal component besides the tangent space. Am I right? I saw some close definition like this. Is it true?

No. You have several threads hinting that you are reading a text which presents manifolds in general through their embedding into a higher dimensional Euclidean space. The general definition of a manifold and its tangent space does not require this. I suggest that you pick up a reference where manifolds are treated properly.