Higher-Order Symmetry of Partials¶

Theorem¶

Let $f$ have continuous partial derivatives up to order $k$ on an open neighbourhood of $\boldsymbol{a}$ . Then any two $k$ th-order partial derivatives that differentiate with respect to the same variables the same number of times are equal at $\boldsymbol{a}$ .

For third-order derivatives this means, for example,

\begin{aligned} f_{xyz}(\boldsymbol{a})&=f_{xzy}(\boldsymbol{a})\\ &=f_{yxz}(\boldsymbol{a})=f_{yzx}(\boldsymbol{a})\\ &=f_{zxy}(\boldsymbol{a})=f_{zyx}(\boldsymbol{a}) \end{aligned}

whenever the required third-order partial derivatives are continuous near $\boldsymbol{a}$ .

Interpretation¶

When the function is smooth enough, only the number of times each direction is used matters, not the order in which the differentiations are performed.

The eight third-order partials of f(x, y) grouped by colour into four distinct values, for example f xxy, f xyx and f yxx all being equal for a C^3 function.

The top row shows the eight formal third-order partials in two variables. For a $C^3$ function, they fall into four groups according to how often each variable is used, for example $f_{xxy}=f_{xyx}=f_{yxx}$ $f_{y xx}$ . More generally, symmetry reduces the $n^k$ ordered partials of a $C^k$ function to at most $\binom{n+k-1}{k}$ distinct values.

The differentiability class $C^k$ packages the hypothesis that all partial derivatives through order $k$ exist and are continuous. This regularity permits repeated interchange of the order of differentiation; existence of derivatives alone is insufficient.

Key Results¶

The second-order case is the usual symmetry of mixed partial derivatives.
For a sufficiently smooth scalar function, the third derivative tensor is symmetric under permutations of its indices.
Without the needed continuity assumptions, mixed partials can fail to agree.

Examples¶

For $f(x,y,z)=x^2yz^3$ $x^{2} y z^{3}$ , the derivatives $f_{xyz}$ , $f_{xzy}$ , and $f_{zyx}$ agree because $f$ is a polynomial.
For $f(x,y)=\mathrm{e}^{xy}$ $e^{x y}$ , $f_{xxy}=f_{xyx}=f_{yxx}$ at every point because all partial derivatives are continuous.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 2, Lecture 1

Aliases: clairaut theorem higher order Keywords: symmetry of higher-order partials, equality of mixed partials

Last updated: 17 October 2026 21:53