Symmetry of second derivatives

For a function with continuous second partial derivatives, the mixed partial derivatives are equal: differentiating first with respect to either variable gives the same result.

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Clairaut's theorem: A theorem stating that continuous mixed second partial derivatives are equal. It gives the standard continuity condition behind this symmetry.
Partial derivative: A derivative of a multivariable function with respect to one variable while holding others fixed. Mixed derivatives are built by applying partial differentiation successively.
Clairaut's theorem counterexample: A function whose mixed partial derivatives exist at a point but are unequal there. It demonstrates that existence of both mixed derivatives alone does not guarantee symmetry.
Hessian matrix: The square matrix of a function's second partial derivatives. Symmetry makes the Hessian symmetric when the mixed partials are continuous.
Schwarz theorem: A theorem equating mixed partial derivatives under suitable continuity conditions. This is another common name for the result and its broader formulations.
Multivariable calculus: The study of functions, derivatives, and integrals involving multiple variables. The theorem is a basic structural result about derivatives in this subject.
Schwarz's theorem counterexample: A differentiable-function example where mixed partial derivatives fail to agree without suitable regularity. It highlights the role of regularity assumptions in versions of the theorem.
Conservative vector field: A vector field equal to the gradient of a scalar potential. Equality of cross-partials is a necessary condition for a sufficiently smooth field to be a gradient.
Mixed partial derivative: A second partial derivative formed by differentiating with respect to two variables. The symmetry compares the two possible orders of this operation.
Differentiability: The property that a function admits a suitable linear approximation near a point. Existence of second partial derivatives alone does not ensure their symmetry.