Many of you may be familiar with the equation for the magnetic force between two thin wire elements carrying electric currents:
$$d^2\vec{F} = \frac{\mu_0}{4\pi} \frac{I_1 \, d\vec{\ell}_1 \times \left( I_2 \, d\vec{\ell}_2 \times \hat{r} \right)}{r^2}$$
or, regarding the currents and separation as constants,
$$d^2 F \propto d\ell_1 \, d\ell_2 \tag{1}$$
Similarly the magnetic flux through a surface is modeled by
$$\Phi = \iint \vec B \cdot d\vec A$$
where it is understood that
$$dA = dx \, dy \tag{2}$$
My question is: Why is the left-hand side of equation $\left( 1 \right)$ marker with “$d^2$” while the left-hand side of equation $\left( 2 \right)$ is marked with just “$d$”?