We often write $d^3 r$ to denote differential volume element. But I never us writing $d^2 r$. We normally write $dA$ instead. Is there a reason why we don't see $d^2 r$?
I'm just curious as in solid-state physics I would like to write the differential number, $dN$, of allowed wavevectors in 2 dimensional $\vec k$-space as $$dN=\frac{A}{\left(2\pi\right)^2}d^2\vec k$$ I can't write $dA$ above at it is misleading, since it conflicts with the real-space area, $A$.
Extending the $d$ to powers $d^2$, $d^3$ under an integral sign means higher differences, or higher order differentials, for me, and not higher dimensional measures.
It is true that we don't have universally accepted designers for area or volume "elements". Therefore we write ${\rm d}A$ or ${\rm dvol}$. On the other hand euclidean Lebesgue measure is standard in ${\mathbb R}^n$. Therefore you could write ${\rm d}(x)$ for the $n$-dimensional Lebesgue measure in an integral over a domain $A\subset{\mathbb R}^n$, like $$\int_Af(x)\>{\rm d}(x)\ ,\quad{\rm or}\quad\int_Af({\bf x})\>{\rm d}({\bf x})\ .$$