Changing order of partial derivatives

Question

Why does $\displaystyle\frac{\partial^2 u}{\partial x \partial y}=\frac{\partial^2 u}{\partial y \partial x}$?

Is there a simple proof of this property?

Answer 1

Use first principles so $\frac {\partial f}{\partial x}=\lim _ { h\to 0}\frac {f (x+h, y )-f (x, y )}{h}$ and $\frac {\partial f}{\partial y}=\lim _ { h\to 0}\frac {f (x, y +h)-f (x, y )}{h}$, apply these again and you will see that either order is equivalent, but it only holds if both the $x $ and $y $ second derivatives are continuous.

I.e $\frac {\partial^2f}{\partial x\partial y}=\lim _ { h\to 0}\frac {f (x+h, y+h )-f (x+h, y )- f (x, y+h )+ f (x, y ) }{h^2}=\frac {\partial^2f}{\partial y\partial x}. $

Answer 2

For any region $D$ in the $xy$-plane, Green's theorem and the gradient theorem imply,

$$\iint_D\left(\frac{\partial^2 u}{\partial x \partial y}-\frac{\partial^2 u}{\partial y \partial x}\right)dxdy=\oint_{\partial D}\left(\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy\right)=0.$$

If the integral $\iint_D\left(\frac{\partial^2 u}{\partial x \partial y}-\frac{\partial^2 u}{\partial y \partial x}\right)dxdy$ vanishes for any region of integration $D$, then the integrand itself must be zero:

$$\frac{\partial^2 u}{\partial x \partial y}-\frac{\partial^2 u}{\partial y \partial x}=0.$$

Answer 3

If you assume that $f=f(x,y)$ is writable as sum of powers \begin{equation} f(x,y) = \sum_{n=0}^\infty \sum_{m=0}^\infty a_{nm} x^n y^m \tag{1} \end{equation} then we have $ \frac{\partial f}{\partial x} = \sum_{n,m} n a_{nm} x^{n-1} y^m $ and $ \frac{\partial f}{\partial y} = \sum_{n,m} m a_{nm} x^n y^{m-1}$. Deriving again, with respect the other variable, we get in both cases the same thing: $\sum_{n,m} nm a_{nm} x^{n-1} y^{m-1}$. Of course assumption (1) is not completely general, but you can guess that you can approximate every (not pathological) function $f=f(x)$ with a polynomial (eventually with infinite terms): the same you can do with $f=f(x,y)$ with (1), so it is not so restrictive and almost every function you can find in physics and engineering is in principle writable in that way. If you assume that the approximating polynomial (eventually infinite) can be identified with the original function, then you can see why exchanging derivatives is lawful. I found this simple proof in a thermodynamics book. I think someone else will like it.