The discussion clarifies the distinction between curvilinear coordinates in Euclidean space and the embedding of curved spaces into Euclidean space. Curvilinear coordinates are utilized to describe the Euclidean space itself, while embeddings typically involve lower-dimensional manifolds. The conversation emphasizes that Euclidean local coordinates cannot be introduced in a curved space and that curvilinear coordinates define both tangent and normal components in curved spaces. Participants recommend studying the definition of manifolds to gain a clearer understanding of these concepts.
PREREQUISITESStudents and professionals in mathematics, physics, and engineering who are interested in understanding the geometric concepts of curvilinear coordinates and manifolds.
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.mertcan said: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?