Theorem 1: Equality of Mixed Partials

Discusson

This is a surprising theorem. ∂²z/∂y ∂x is the rate of change with respect to y of the slope ∂z/∂x, while ∂²z/∂x ∂y is the rate of change with respect to x of the slope ∂z/∂y. There is no simple intuitive way to see that these should be equal.

As a matter of fact, there are functions f(x, y) whose mixed second partial derivatives exist but are not equal. One such example is the function

We have left the computation of the second partials of f(x,y) as a problem. It turns out that at (0, 0),

How can this be in view of Theorem 1? The answer is that in this example the second partial derivatives exist but are not continuous at (0, 0), so the theorem does not apply. We shall only rarely encounter functions whose second partial derivatives are not continuous, so in all ordinary problems it is true that the mixed partials are equal. We shall prove the theorem later. We now turn to some applications. Our first application concerns mixed third partial derivatives.

If the third partial derivatives of z = f(x, y) are continuous, then

so we write for each of them. Similarly,

and we write for each of them.

We prove the first equation as an illustration.

Example 2: Third Partial Derivatives

If a function has continuous second partial derivatives we may apply the Chain Rule to the first partial derivatives. For one independent variable,

Example 3: Continuous Second Partials

By holding one variable fixed in Theorem 1, we get equalities of mixed partials for functions of three or more variables.