 Symmetry of second derivatives

When the second partial derivatives are continuous in a neighborhood, the order of the mixed partial derivatives can be interchanged. Expressing a rectangular finite difference as either iterated integral by the fundamental theorem of calculus and shrinking the rectangle proves the equality. Mere existence of both derivatives does not imply this symmetry.

The symmetry of second derivatives refers to a result in multivariable calculus often associated with functions of several variables. Specifically, if a function \( f \) has continuous second partial derivatives, then the mixed second derivatives are equal.