The conditions that partial derivatives commute

Question

State the conditions that partial derivatives commute, namely, $D_1D_2f = D_2D_1f$.

I understand how to prove that these partial derivatives are equal but I don't understand what commute means. Please help.

Answer 1

Commute just means that the order of derivatives that we take does not matter. One could take the partials with respect to $x$ and then the partial with respect to $y$ or instead the partial to $y$ and then $x$. The final answer is independent of the order in which we take our derivatives.

Answer 2

Technically, you can think of $D_1$ and $D_2$ as operations on the function $f$. For the operations $D_1$ and $D_2$ to commute we need that after applying $D_1 D_2$ to $f$ we get $D_2 D_1 f$ or $$ D_1 D_2 f(x_1, x_2) = D_2 D_1 f(x_1, x_2) \Leftrightarrow D_1 D_2 f - D_2 D_1 f(x_1, x_2) = 0 \Leftrightarrow (D_1 D_2 - D_2 D_1)f(x_1, x_2) = 0. $$ Yes that last bit means that $D_1 D_2 - D_2 D_1$ is also an operation on $f(x_1, x_2)$ and it will be the zeroing operation. As an exercise you may want to try to rewrite the above tautology in terms of a simple function, say $\exp\{ - (x^2 + y^2) \}.$

We can think of more general operations like, say multiplying $f(x_1,x_2)$ by $x_1$, call this operation $M_1$. That is, $$ M_1 f(x_1, x_2) = x_1 f(x_1, x_2)$$ a new function. Then we see that (this is an exercise) $M_1 D_2 f = D_2 M_1 f$ but $M_1 D_1 f \neq D_1 M_1 f$. In fact $$( M_1 D_1 - D_1 M_1) f = 1 $$ the constant function.

I know this is extra information but I just wanted to point out that this formalism is not arbitrary. It turns out that the certain operations (like differentiation and multiplication by a variable) have a nice algebraic structure and that commutativity is usually not part of it. However, the commutator substructure can be built by considering the bits left over: the operations $$ AB - BA$$ as above. The properties of this substructure are important to many branches of engineering and physics. The example above (with a small but imaginative modification) can be used to describe the uncertainty of position and momentum observations according to the laws of quantum mechanics.