Math 2.0

Real functions Theorem

Schwarz's theorem for C^2 functions

Builds on Symmetry of second derivativesHigher partial derivativesC^k function characterised by continuous derivatives

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open and let f ⁣:U→ℝf \colon U \to \mathbb{R} be of class C^2 on UU. Then for all i,j∈{1,…,n}i, j \in \{1, \dots, n\} and each a∈Ua \in U, ∂2f∂xi ∂xj(a)=∂2f∂xj ∂xi(a).\frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \frac{\partial^2 f}{\partial x_j \, \partial x_i}(a).

Remarks

A function of class C2C^2 is twice differentiable at each point, so this is a case of the symmetry of second derivatives. The result is also called Clairaut's theorem. By induction, for a function of class CkC^k the partial derivatives of order at most kk do not depend on the order of differentiation. So for CkC^k functions one can use multi-index notation ∂αf\partial^\alpha f without ambiguity.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet 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…conceptDifferentiable 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 …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementSchwarz's theorem for C^2 functions, statement, no evidence levelSchwarz's theorem for…statement
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  1. Euclidean space object
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  4. Function of several variables concept
  5. Limit of a function of several variables concept
  6. Differentiable function of several variables definition
  7. Partial derivative definition
  8. Differentiable on an open set definition
  9. Higher partial derivatives definition
  10. Iterated derivative definition
  11. Symmetry of second derivatives statement
  12. C^k function characterised by continuous derivatives statement
  13. Schwarz's theorem for C^2 functions statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives, Schwarz's theorem
About this page
Tag0NNQY8WCW6
URIm2://0NNQY8WCW6
Aliasesclairaut-theorem, schwarz-theorem
Kindstatement, theorem
WritingW0 stub
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 0NNQY8WCW6.
UsesSymmetry of second derivatives, Higher partial derivatives, C^k function characterised by continuous derivatives

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