Style when typesetting functions and operators I've been making an effort to type all function names and operators in roman font. For example
$$\int \operatorname{f}(x) \, \operatorname{d}\!x$$
This was all well and good until I tried to write out the rule for integration by parts. The usual formula comes from the product rule. If $\operatorname{u}$ and $\operatorname{v}$ are functions of $x$ then
$$\frac{\operatorname{d}}{\operatorname{d}\!x}(\operatorname{uv}) = \frac{\operatorname{du}}{\operatorname{d}\!x}\operatorname{v}+\operatorname{u}\frac{\operatorname{dv}}{\operatorname{d}\!x}$$
Indeed, we can even write the rule for integration by parts:
$$\int \operatorname{u}\frac{\operatorname{dv}}{\operatorname{d}\!x}\, \operatorname{d}\!x = \operatorname{uv}-\int\operatorname{v}\frac{\operatorname{du}}{\operatorname{d}\!x}\, \operatorname{d}\!x$$
The problem comes when we try to write this in the form that I know it, i.e. by "cancelling the $\operatorname{d}\!x$". Without using any roman letters at all, I need to write:
$$\int u \, dv = uv-\int v\, du$$
It's tempting to write the first integrand as $\operatorname{u} \, \operatorname{d}\!v$ and the second as $\operatorname{v} \, \operatorname{d}\!u$. However, both $u$ and $v$ and $\operatorname{u}$ and $\operatorname{v}$ appear, meaning that they are sometimes functions and sometimes variables.
Does this mean that the roman system is doomed, or that the short-hand version is nonsensical? I always thought it held as an expression in terms of differential forms.
 A: To answer your first question: Not at all, as far as I'm concerned!
In fact, it can be argued that your method, which would yield:
$$\int {\rm u \, dv} = {\rm uv} - \int {\rm v\,du}$$
draws attention to the fundamental difference between the natures of both $\rm d$s. 
For, the $\rm d$ in ${\rm d}x$ is not to be separated from the $x$. It is just a notation that makes the $\rm d$ look like an operator, while it is not. We could just as well have used a symbol like $\mu_x$ or something else expressing "an infinitesimal quantity in the variable $x$".
On the other hand, the $\rm d$ in $\rm du$ is of a fundamentally different nature: it is, in the language of differential geometry, an operation ${\rm d}: \wedge^0 T^*\Bbb R\to \wedge^1 T^*\Bbb R$. As such, it takes functions (like $\rm u$) to so-called $1$-forms (roughly "things with respect to which you can integrate"). In the one-dimensional case, this proceeds via the well-known rule $ {\rm du} = \dfrac{\rm du}{{\rm d}x}{\rm d}x$ (where both expressions on the right-hand side cannot be decomposed in any way). Any course or book on differential geometry will explain how to generalise these concepts from $\Bbb R$ to sufficiently well-behaved manifolds.

The notation for the expression defining $\rm du$ is thus certainly very convenient and reminiscent of ordinary numbers. Nonetheless, $\rm du$ and ${\rm d}x$ are (or should be) conceptually distinct, and I think your notation does a nice job of reflecting this distinction.
The supposed defect of the notation is thus not due to the notation itself, but due to the obfuscation of the distinction between the operator $\rm d$ and the use of $\rm d$ in denoting the basis element ${\rm d}x$ of $T^*\Bbb R$ that is very ubiquitous in mathematical literature.
