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Exercise 1080Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Clairaut's theorem

#Clairaut#Schwarz#mixed partials

Statement

State Clairaut's theorem (Schwarz) and give a sufficient condition for fxy=fyxf_{xy}=f_{yx}.

Hint

Continuity of second partials is sufficient.

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Solution

Theorem. If ff has continuous second partial derivatives in a neighbourhood of a point, then fxy=fyxf_{xy}=f_{yx} at that point.

Sufficient condition. ff is C2C^2.

Conclusion. Mixed partials commute under C2C^2 regularity.

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