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Definition 4

[1NX] Let \(I,JβŠ† {\mathbb {R}}\) be intervals. Let \(𝛾:Iβ†’{\mathbb {R}}^ n\) and \(𝛿:Jβ†’{\mathbb {R}}^ n\) be two regular curves. We will write \(π›Ύβ‰ˆπ›Ώ\) if there is a diffeomorphism 1 \(πœ‘:Iβ†’ J\) monotonic increasing, such that \(𝛾=π›Ώβ—¦πœ‘\).

  1. A diffeomorphism is a bijective function \(πœ‘:Iβ†’ J\) of class \(C^ 1\), the inverse of which is class \(C^ 1\); in particular \(πœ‘'\) is never zero, and (when domain and codomain are intervals) it always has the same sign.
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  • curve
  • \(\approx \)
  • diffeomorphism
  • relation, equivalence ---, between curves
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