Multivariable Calculus (Calc III) - Complete Semester Course - Proof of Clairaut's Theorem, Real Analysis II

We prove Clairaut’s theorem, a central result in multivariable calculus that tells us the mixed second-order partial derivatives of a sufficiently smooth (C^2) function are equal, regardless of the order of differentiation. In other words, for a scalar-valued function z=f(x,y), we can say f_xy = f_yx if the function’s second partial derivatives exist and are continuous. We then work through a step-by-step derivation, starting with the limit definitions of partial derivatives. Changing the order of two iterated limits must be justified rigorously; we do so by employing mean value theorem arguments and constructing auxiliary functions to maneuver through the difference quotients. We visualize the arguments in terms of a rectangle formed by (a, b), (a + h, b), (a, b + k), and (a + h, b + k), highlighting how function values at these four points let us track changes in both x and y directions. Finally, by shrinking that rectangle to a single point (a, b) and invoking continuity of partial derivatives, we conclude that the two mixed partial derivatives coincide at every point in the domain. Our thorough, careful approach underlines the geometry of Clairaut’s theorem. #mathematics #maths #MultivariableCalculus #partialderivatives #advancedmathematics #meanvaluetheorem #calculustutorial #calculus3 #realanalysis