Doubt on electric field $1$-form definition This seems to be a math problem, but since involve physical quantities and is presented generally in physics literature, I've posted here anyway.
Most books that use differential forms to talk about electrodynamics define electric field $E$ as a one-form over $\mathbb{R}^3$, i.e.
$$E = E_1 (t,x,y,z) dx + E_2 (t,x,y,z) dy + E_3 (t,x,y,z) dz .\tag1$$
This seems to be a problem to me because as far as I understand, one-forms written on a basis of  forms $(dx,dy,dz)$ constitutes a module over the commutative ring of real smooth functions on $\mathbb{R}^3$.But if so, how can the components of electric field dependent of time coordinate $t$? Should not $E$ be written as a one-form on the $4$-dimensional Minkowski space $\mathbb{M}$?
Remark: the same problem arise of I treat the magnetic field $B$ as a $2$-form on $\mathbb{R}^3$ but still consider it's components to depend on a time coordinate so that expressions like "$\partial_t E$" or "$\partial_t B$" makes sense.
 A: You're right that electromagnetism most naturally happens in $\mathbb{M}$. However, once you choose a reference frame, you can foliate $\mathbb{M}$ into spacelike hypersurfaces given by the level sets of $t$, which has the obvious identification with $\mathbb{R}^3$. Pulling $E$ back onto each hypersurface for time $t$ gets you a time dependent 1-form on $\mathbb{R}^3$. The same goes for the magnetic $2$-form.
A: Your observations and your concern are well founded. As electrodynamics is a relativistic theory, it can, of course, be formulated in manifestly covariant form. The electric and the magnetic field are contained in the two-form
$F= F_{\alpha \beta} dx^\alpha \wedge dx^\beta$ (summation convention for indices $\alpha, \beta= 0, \ldots 3)$. In cartesian coordinates $c t = x^0, x= x^1, y=x^2, z=x^3$. The components of the electric field $\vec{E}$ and the magnetic field $\vec{B}$ are related to the componets of the (antisymmetric) field strength tensor $F_{\alpha \beta}$ by $F_{01}=-F_{10}=E_x$, $F_{02}=-F_{20}=E_y$, $F_{03}=-F_{30}=E_z$, $F_{12}=-F_{21}=-B_z$, $F_{13}=-F_{31}=B_y$ and $F_{23}=-F_{32}= -B_x$. The two homogeneous Maxwell equations are now immediately obtained from $d F =0$.
The formulation of the two inhomogeneous equations requires in addition the introduction of the one-form $j =j_\alpha dx^\alpha$ with $j^\alpha = (c \rho, \vec{j})$. (See e.g. W. Thirring, Classical Mathematical Physics for further details).
