I 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....

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Orodruin

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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

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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.

"Curved space and curvilinear coordinates"

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