I Curved space and curvilinear coordinates

  • Thread starter mertcan
  • Start date
326
6
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

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,930
5,909
An embedding is usually of a lower-dimensional manifold. Curvilinear coordinates are used to describe the Euclidean space itself.
 
You can not introduce Euclidian local coordinates in a curved space
 
326
6
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???
 
why do not you study the definition of the manifold first?
 
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

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,930
5,909
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.
 

Want to reply to this thread?

"Curved space and curvilinear coordinates" You must log in or register to reply here.

Related Threads for: Curved space and curvilinear coordinates

Replies
0
Views
1K
Replies
3
Views
2K
Replies
2
Views
2K
Replies
3
Views
2K
  • Posted
Replies
4
Views
2K
Replies
3
Views
2K
Replies
0
Views
1K
Replies
5
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top