Math 2.0

Real functions Functions of several real variables Theorem

Symmetry of second derivatives

Builds on Iterated derivativeHigher partial derivativesDifferentiable function of several variables

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝmf \colon U \to \mathbb{R}^m, and let a∈Ua \in U. Suppose that ff is differentiable at each point of UU, and that the derivative Df ⁣:U→L(ℝn,ℝm)Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m) is differentiable at aa. Then the second derivative D2f(a)D^2 f(a) is a symmetric bilinear map: D2f(a)(u,v)=D2f(a)(v,u)D^2 f(a)(u, v) = D^2 f(a)(v, u) for all u,v∈ℝnu, v \in \mathbb{R}^n. In particular, ∂2f∂xi ∂xj(a)=∂2f∂xj ∂xi(a)\dfrac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \dfrac{\partial^2 f}{\partial x_j \, \partial x_i}(a) for all i,ji, j.

Remarks

Here D2f(a)D^2 f(a) is the iterated derivative of order 22, and D2f(a)(ei,ej)D^2 f(a)(e_i, e_j) is the mixed partial derivative ∂i∂jf(a)\partial_i \partial_j f(a). The hypothesis is that ff is twice differentiable at aa. It holds when the second partial derivatives are continuous near aa, which gives Schwarz's theorem. The existence of the second partial derivatives at aa is not enough: there is a function with unequal mixed partial derivatives.

History

Euler published a proposed proof in 1740, and many incomplete proofs followed over more than a century. In 1867 Lindelöf analysed the earlier proofs and gave a counterexample to the general claim. Schwarz gave the first rigorous proof six years later, and Jordan gave in 1883 the general version that most textbooks use.

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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…statement
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  1. Euclidean space object
  2. Function concept
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  4. Function of several variables concept
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  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

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9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives, Sufficiency of twice-differentiability
  2. Web https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives, Requirement of continuity
  3. Web https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives, History
About this page
Tag01V9FPBBVK
URIm2://01V9FPBBVK
Aliasesequality-of-mixed-partials
Kindstatement, theorem
WritingW1 stated
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 01V9FPBBVK.
UsesIterated derivative, Higher partial derivatives, Differentiable function of several variables
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