Partial derivative condition

Question

Does $\frac{\partial^2{z} }{\partial{x} \partial{y} }$ always equal $\frac{\partial^2{z} }{\partial{y} \partial{x}}$?

I find myself in situations where using one will be easier and faster than the other.

Answer 1

Since you are probably dealing with elementary functions, this will hold true. But I will point out some caveats. Let's not forget the obvious: the second partial derivatives must exist at the given point you are evaluating the expression.

Additionally, if the partials are continuous at that point, and analogously, if the $n^{\text{th}}$ partials are continuous at the point of evaluation, the $n^{\text{th}}$ degree mixed partials will commute. However, if this is not the case (it is most of the time), then you will have to resort to other methods to determine the truth of this statement.

Here's an example where Clairaut's Theorem fails.