Curved space and curvilinear coordinates

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  • #1
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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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  • #2
Orodruin
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An embedding is usually of a lower-dimensional manifold. Curvilinear coordinates are used to describe the Euclidean space itself.
 
  • #3
wrobel
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You can not introduce Euclidian local coordinates in a curved space
 
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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???
 
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wrobel
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why do not you study the definition of the manifold first?
 
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wrobel
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By definition the Euclidean coordinates are the local coordinates ##x^i## on a manifold such that ##\nabla_i\equiv \frac{\partial }{\partial x^i}##
 
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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.
 

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