The Wave Equation, Mixed Partial Derivatives, and Fubini's Theorem
In a recent paper 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 deriatives 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 with a straightforward argument provides a simple proof of Schwarz's lemma.
© 2004 Mathematical Association of America
Aksoy, A. Martelli, M., “Wave Equation, Mixed Partial Derivatives and Fubini’s Theorem” the American Mathematical Monthly, Vol. 111, 2004, N.4, 2004. doi: 10.2307/4145243