Clairaut’s Theorem (Equality of Mixed Partials)

Summary

If the mixed second partial derivatives of $f$ are continuous in a neighborhood of a point, then the order of differentiation does not matter: $f_{xy}=f_{yx}$ at that point.

Prerequisites

Partial Derivatives, Higher-Order Derivatives

Theorem

Let $f$ be defined on an open set containing $(a,b)$ . If $f_{xy}$ and $f_{yx}$ exist on a neighborhood of $(a,b)$ and are continuous at $(a,b)$ , then

Conditions / Assumptions

• Continuity of the mixed partials is a standard sufficient condition.
• There exist pathological examples where mixed partials exist but are unequal when continuity fails; such examples are rare in applications.

Worked Example

For $f(x,y)=x^{3}y+2xy-5$ ,

The mixed partials agree (and are continuous) on ${\mathbb{R}}^2$ .

Common Mistakes

• Assuming $f_{xy}=f_{yx}$ with no regularity hypothesis.
• Mixing subscript order conventions; here $f_{xy}=(f_x{)}_y$ .

Connections

• Used when forming the Hessian in Maxima and Minima
• Related to exact differentials and conservative fields (vector calculus)

References

Clairaut’s theorem is stated in OpenStax Calculus Volume 3.^[1]