The Wave Equation, Mixed Partial Derivatives, and Fubini's Theorem

Abstract

In a recent paper [1] the two authors of this note have shown that Fubini's theorem on changing the order of integration and Schwarz's lemma on the equality of mixed partial derivatives are equivalent when standard assumptions of continuity and differentiability are made. The proof relies heavily upon the fundamental theorem of integral calculus as usually presented in calculus textbooks. Fubini's theorem is regarded as intuitive and easy to prove. Therefore, the equivalence established in [1] with a straightforward argument provides a simple proof of Schwarz's lemma. However, there are many cases in which the standard assumptions from which the equality of the mixed partials is derived do not hold. A typical scenario is offered by some cases of the one-dimensional wave equation.