# I Curved space and curvilinear coordinates

1. Jun 11, 2016

### 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....

2. Jun 11, 2016

### Orodruin

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

3. Jun 11, 2016

### wrobel

You can not introduce Euclidian local coordinates in a curved space

4. Jun 11, 2016

### 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???

5. Jun 11, 2016

### wrobel

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

6. Jun 11, 2016

### wrobel

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

7. Jun 11, 2016

### Orodruin

Staff Emeritus
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.