Math 2.0

Global analysis, analysis on manifolds Definition

Diffeomorphism

Builds on C^k function characterised by continuous derivativesDifferentiable function of several variablesInverse function theorem

Definition

Let U,V⊆ℝnU, V \subseteq \mathbb{R}^n be open, and let k≥1k \ge 1 be an integer or k=∞k = \infty. A map f ⁣:U→Vf \colon U \to V is a CkC^k diffeomorphism if ff is a bijection of class C^k on UU. In addition, f−1 ⁣:V→Uf^{-1} \colon V \to U must be of class CkC^k on VV. The sets UU and VV are CkC^k diffeomorphic if there is a CkC^k diffeomorphism from UU to VV.

Remarks

If ff is a diffeomorphism, the chain rule applied to f−1∘f=idf^{-1} \circ f = \mathrm{id} shows that Df(x)Df(x) is invertible for each x∈Ux \in U. Conversely, a CkC^k bijection f ⁣:U→Vf \colon U \to V with Df(x)Df(x) invertible at each point is a CkC^k diffeomorphism, by the inverse function theorem. The map x↦x3x \mapsto x^3 is a C∞C^\infty bijection of ℝ\mathbb{R}, but it is not a diffeomorphism, because its inverse x↦x1/3x \mapsto x^{1/3} is not differentiable at 00. Polar coordinates (r,θ)↦(rcos⁡θ,rsin⁡θ)(r, \theta) \mapsto (r \cos \theta, r \sin \theta) give a C∞C^\infty diffeomorphism from (0,∞)×(0,2π)(0, \infty) \times (0, 2\pi) onto the plane without the closed half-line {(x,0):x≥0}\{(x, 0) : x \ge 0\}.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementDiffeomorphism, definition, no evidence levelDiffeomorphismdefinition
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  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Jacobian matrix definition
  12. Chain rule in several variables statement
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. Jacobian determinant definition
  16. Derivative of the inverse function statement
  17. C^k function characterised by continuous derivatives statement
  18. Inverse function theorem statement
  19. Diffeomorphism definition

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Diffeomorphism, Definition
  2. Web https://en.wikipedia.org/wiki/Diffeomorphism, Local description
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Kinddefinition
WritingW1 stated
MSC58C25
DefinesDiffeomorphism
UsesC^k function characterised by continuous derivatives, Differentiable function of several variables, Inverse function theorem

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