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.