Symmetry of Mixed Partial Derivatives (Clairaut’s Theorem)#
Theorem (Clairaut Schwarz)
Let
Then:
That is, the order of differentiation can be interchanged.
Intuition#
If a function is smooth enough (specifically, if the second-order partial derivatives exist and are continuous), then the “curvature” in the
Proof Sketch#
We will sketch a proof using the mean value theorem and the definition of partial derivatives. Let’s assume that
Define:
Then, as
Step-by-step:#
By the Mean Value Theorem, the numerator of
can be interpreted as a finite difference approximation to a mixed partial derivative.Using Taylor’s Theorem with remainder, or via integral representations of derivatives, one can show that:
and also
due to continuity of the second derivatives.
Hence, the limits agree and the mixed partials are equal.
Therefore:
When Clairaut’s Theorem Does Not Apply#
If the second-order mixed partial derivatives exist but are not continuous, the symmetry may fail. A classic counterexample is:
This function has both mixed partial derivatives at the origin, but they are not equal because they are not continuous there.