Are the following two derivatives same?

[sgsawant]

1. f_xy
2. f_yx


Are the above 2 derivatives equal, in general. Please explain if you know the answer.

Regards,

-sgsawant

 

[phyzguy]

Yes, in general if these two derivatives exist and are continuous, then they are equal. This is called "equality of mixed partial derivatives". I have also seen it called Clairaut's theorem, although supposedly it was first proved by Euler (like so much of the rest of mathematics).

 

[HallsofIvy]

As long as $f_{xy}$ and $f_{yx}$ are continuous in some neighborhood of a point, then, at that point, they are equal.

 

[Landau]

Even better:

Let U in R^2 be open, $f:U\to\matbb{R}$ partial differentiable w.r.t. both variables, and $D_1f$ partial differentiable w.r.t the second variable. Suppose further that (x,y) is in V, and $D_2D_1f$ is continuous at (x,y). Then $D_2f$ is partial differentiable w.r.t the first variable at (x,y), and

$$D_1D_2f(x,y)=D_2D_1f(x,y)$$.

i.e. we only need $D_2D_1f$ to exist and be continuous at some point in the interioir, this already implies that $D_1D_2f$ exists at that point and the two are equal.