I am trying to phrase Maxwell's differential equations in terms of the de Rham complex. The equations are
$$ \nabla \cdot E = \frac{\rho}{\epsilon_0}$$
$$ \nabla \cdot B = 0$$
$$ \nabla \times E = - \frac{\partial B}{\partial t}$$
$$ \nabla \times B = - \mu_0 \left( J + \epsilon_0 \frac{\partial E}{\partial t} \right) $$
I am confused, because the first equation (Gauss's law) suggests that the electric field $E$ is a $2$ form and $\nabla \cdot E = dE$ is a $3$-form. But the third equation (the Maxwell-Faraday equation) suggests that $E$ is a $1$ form, and that $\nabla \times E$, the curl, is $d E$. I was thinking the hodge-star operation must be involved here somehow, since it turns $1$ forms into $2$ forms here and vice versa.
Can someone point me to a source which phrases things this way?
Thanks so much!