with functions of multiple variables. We will also discuss Clairaut's Theorem to help with some of the work in finding higher order derivatives. Clairaut's theorem is a general mathematical law giving the surface gravity on a viscous rotating ellipsoid in equilibrium under the action of its gravitational field and centrifugal force.‎History · ‎Formula · ‎Somigliana equation. Requirement of continuity. The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not continuous).‎Schwarz's theorem · ‎Sufficiency of twice · ‎History · ‎Requirement of continuity.


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Since the surface of the Earth is closer to its center at the poles than at the equator, gravity is stronger there.


Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth. Newton pushed for scientists to look further into the unexplained variables. For clairaut s theorem, in the counterexample above, we can randomly choose any line in the plane that intersect the surface defined by f.

It can be proven using an epsilon-delta argument that for any such line, f is continuous at 0. Otherwise, we need to use a trial and clairaut s theorem method to find if the limit does not exist.

If it doesn't, we should be able to pick 2 lines of approach where we get 2 different one-sided limits of f at 0,0.

Clairaut's theorem

We will also be dropping it for the first order derivatives in most cases. This is not by coincidence. The following theorem tells us.

In pretty much every example in clairaut s theorem class if the two mixed second order partial derivatives are continuous then they will be equal. Finally, six years later Hermann Schwarz gave the first satisfactory proof. The statement can be generalized in two ways: We can generalize it to higher-order partial derivatives.

Clairaut's theorem on equality of mixed partials - Calculus

We can generalize it to functions of more than two variables. The general version states the following. Suppose is a function of variables defined on an open subset of. Suppose all mixed partials with a certain clairaut s theorem of differentiations in each input variable exist and are continuous on.