Why is $\frac{\partial^2 f}{\partial x\partial x}$ equal to $\frac{\partial^2 f}{\partial x^2}$ and not $\frac{\partial^2 f}{(\partial x)^2}$?

Question

I know that this is a borderline pedantic question, but is there any other reason than a convention why usual calculus and differential equation texts say that $\frac{\partial^2 f}{\partial x\partial x} = \frac{\partial^2 f}{\partial x^2}$ and not $\frac{\partial^2 f}{\partial x\partial x} = \frac{\partial^2 f}{(\partial x)^2}$? Or are there some meaningful set of rules with which you can manipulate the differential forms $\partial x$ so that the $\frac{\partial}{\partial x}\frac{\partial f}{\partial x} = \frac{\partial^2 f}{\partial x^2}$ doesn't feel that arbitrary (although I suppose that this is a question of preference)?

Answer 1

One formalism that can be used to make sense of various a-priori-meaningless-nonethless-highly-suggestive symbolic manipulations (e.g. proving the chainrule via "cancelling" $\partial y$ in the $\frac{\partial z}{\partial x} = \frac{\partial z }{\partial y} \frac{\partial y}{\partial x}$ ) is nonstandard calculus, which is in turn a component of nonstandard analysis.

As far as notation is concerned, I would err on the side of $\frac{\partial^2}{(\partial x)^2}$ so as to avoid confusion with expressions like $\frac{\partial}{\partial (x^2) }$ (which could be syntactically meaningful, albeit clumsy)

Answer 2

It's just concise notation. Since $\partial$ is never separate from the $x$ in the "denominator" of the partial derivative expression, there's no ambiguity: $\partial x$ is a single, indivisible expression.

From a different perspective, there's nothing intrinsically wrong with writing $$ \frac{\partial^2 f}{(\partial x)^2}. $$ It just muddies up your notation with unnecessary extra parentheses.

Answer 3

It's amusing to read in a comment that $\partial^2f/(\partial x)^2$ is "meaningless" because the standard notation without parentheses is a kind of simplification of the intuitive notation that would use parentheses: $d^2f/dx^2$ means $(d/dx)(d/dx)f = (d^2/(dx)^2)(f) = d^2f/(dx)^2 = d^2f/dx^2$. We just need to remember that the notation $dx$ is a single thing that stays together.

In arc length formulas, where we see things like $ds^2$ written all over the place, it is a kind of simplification of $(ds)^2$, which in more careful modern notation might be expressed as $(ds)^{\otimes 2}$.

Answer 4

It's just a matter of convention; and yes your proposed formulation is probably more technically correct if one wants to be pedantic about removing potential ambiguities.

However, the thing to keep in mind is that no one would ever want to use your version because it makes for an absolutely horrendous shorthand notation!

$$ \frac{\partial}{\partial x}\frac{\partial}{\partial x}f(x)=\frac{\partial^2}{(\partial x)^2}f(x) $$

Writing the partial derivatives out in full actually takes fewer characters! (At least for the low-integer exponents one commonly-encounters.)

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Presumably, the convention $\frac{\partial}{\partial x}\frac{\partial}{\partial x}f(x)=\frac{\partial^2}{\partial x^2}f(x)$ carries on (even though it might be considered an abuse of notation) simply because the ambiguities that it potentially introduces are such edge-cases that it's not worth losing the handy shorthand. After all, you're only really losing the ability to distinguish the following: $$ \frac{\partial^2}{\partial x^2}f(x)= \begin{cases} \frac{\partial}{\partial x}\frac{\partial}{\partial x}f(x)\\ \frac{\partial}{\partial x^2}\frac{\partial}{\partial 1}f(x) \end{cases} $$ The second of which is probably better off simply using a different notation if it ever comes up at all.