In this section we'll discuss about natural physical quantity representation in 3D. I used two main notations.
Abstract index : $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ (LINK) Component index : $\alpha$, $\beta$, $\gamma$
I also used $[\cdot]$ to indicate totally anti-commuting indices and $(\cdot)$ to indicate totally commuting indices for convention.
(1) Magnetic flux density $\underline{\underline{B}}$ is naturally a 2-form. Think about 4D field-strength tensor $\underline{\underline{F}}=\underline{\nabla} \wedge \underline{A}$. Traditionally used magnetic flux density $\vec{B}$ is defined volume dual ($\star$) of $\underline{\underline{B}}$. Here I used $\star ^{-1}=\star$ in 3D.
$$\begin{align} \vec{B}=\star \underline{\underline{B}} = \underline{\underline{B}} \cdot \vec{\vec{\vec{\epsilon}}}{}^{-1} & = \frac{1}{2!} B_{[\mathbf{ab}]} {\epsilon^{-1}}^{[\mathbf{abc}]} & \;\;\;\;\;\text{(abstract)} \\ &= \left( \frac{1}{2!} B_{\alpha \beta} \underline{e}^{\alpha} \wedge \underline{e}^{\beta} \right) \left( \frac{1}{3! \epsilon_{\alpha \beta \gamma}}\vec{e}_{\alpha} \wedge \vec{e}_{\beta} \wedge \vec{e}_{\gamma} \right) & \\ & = B_{23} \vec{e}_1 + B_{31} \vec{e}_2 + B_{12} \vec{e}_3 & \;\;\;\;\;\text{(component)}\end{align}$$
(2) Magnetic field $\underline{H}$ is naturally a 1-form. It is connected to the magnetic flux density $\underline{\underline{B}}$ by
$$\underline{H}=\underline{\vec{\vec{\mu}}_0}{}^{-1} \cdot \underline{\underline{B}}-\underline{M} = \mu_0{}^{-1}* \underline{\underline{B}}-\underline{M}$$
where $*=\star \diamond$ is Hodge dual, which gives a mapping between $n$-forms and $(3-n)$-forms.
$$\mu_0 {}^{-1}{}_{\mathbf{a}}^{[\mathbf{bc}]} = \mu_0{}^{-1} \epsilon_{[\mathbf{ade}]} g^{\mathbf{db}} g^{\mathbf{ec}}=\mu_0{}^{-1} *$$
(3) Covector potential $\underline{A}$ is naturally a 1-form. Since its exterior derivative produces the magnetic flux density.
$$\underline{\nabla} \wedge \underline{A} = \underline{\underline{B}}$$
Note the identities below :
$$\underline{\nabla} \wedge w_{\text{n-form}} = \frac{1}{n!} \partial_{\alpha} w_{\beta_1 \cdots \beta_n} \underline{e}^{\alpha} \wedge \underline{e}^{\beta_1} \wedge \cdots \wedge \underline{e}^{\beta_n}$$
$$(\underline{e}^{\alpha_1} \wedge \cdots \wedge \underline{e}^{\alpha_m}) \cdot (\vec{e}_{\beta_1} \wedge \cdots \wedge \vec{e}_{\beta_n}) = \begin{cases} \frac{1}{(n-m)!} \delta^{\alpha_1 \cdots \alpha_m \gamma_{m+1} \cdots \gamma_n}_{\beta_1 \cdots \beta_n} \vec{e}_{\gamma_{m+1}} \wedge \cdots \wedge \vec{e}_{\gamma_n} & \;\;\;\; \text{for}\; m \leq n \\ \frac{1}{(m-n)!} \underline{e}^{\gamma_{n+1}} \wedge \cdots \wedge \underline{e}^{\gamma_m} \delta^{\alpha_1 \cdots \alpha_m}_{\gamma_{n+1}\cdots \gamma_m \beta_1 \cdots \beta_n}& \;\;\;\; \text{for}\; m \geq n \end{cases}$$
Using these relations to calculate traditional magnetic flux density $\vec{B}$,
$$\vec{B}=\star(\underline{\nabla} \wedge \underline{A}) = (\underline{\nabla} \wedge \underline{A}) \cdot \vec{\vec{\vec{\epsilon}}}{}^{-1} = \left( \partial_{\alpha} A_{\beta} \underline{e}^{\alpha} \wedge \underline{e}^{\beta} \right) \left( \frac{1}{3! \epsilon_{\alpha'\beta'\gamma}} \vec{e}_{\alpha'} \wedge \vec{e}_{\beta'} \wedge \vec{e}_{\gamma}\right)$$
Since $\star^{-1}=\star$ in 3D, we can calculate potential form of traditional magnetic flux density by
$$B^{\alpha}=\epsilon^{\alpha \beta \gamma} (\partial_{\beta} A_{\gamma} - \partial_{\gamma} A_{\beta})$$
where volume form (levi-civita) tensor are related to the geometry of our space since $\epsilon_{1\cdots N}=\sqrt{|g|}$.
(4) Electric field $\underline{E}$ is naturally a 1-form. Since it is the exterior derivative of some scalar field.
$$\underline{E}=-\underline{\nabla}\phi - \dot{\underline{A}}$$
Our traditional vector representation $\vec{E}$ is metric dual($\diamond$) of 1-form $\underline{E}$.
$$\vec{E} = \diamond \underline{E} \;\;\;\; \Leftrightarrow \;\;\;\; E^{\alpha}=g^{\alpha\beta} (\partial_{\beta} \phi - \dot{A}_{\beta})$$
(5) Duality, duality, duality, ... : In 3D, volume dual ($\star$) , \metric dual($\diamond$) and Hodge dual($*$) can be used to map from one anti-symmetric tensor to the other. See the below.
Fig 1. Duality map (Image from profstewart.org)
(6) Electric displacement field (Electric density field) $\underline{\underline{D}}$ is naturally a 2-form. It is connected to the electric field $\underline{E}$ by
$$\underline{\underline{D}}=\underline{\underline{\vec{\varepsilon}_0}} \cdot \underline{E} + \underline{\underline{P}}=\varepsilon_0 *\underline{E} + \underline{\underline{P}} $$
where $*=\star \diamond$ is Hodge dual, which gives a mapping between $n$-forms and $(3-n)$-forms.
$$\varepsilon_0 {}^{\mathbf{c}}_{\mathbf{ab}}=\varepsilon_0 \epsilon_{[\mathbf{abd}]} g^{\mathbf{dc}}=\varepsilon_0 *$$
(7) Divergence of n-vector $v$ and n-form $w$ is defined by
$$ \nabla \cdot v = (-1)^{n-1} \star^{-1}\nabla \wedge \star v$$
$$ \nabla \cdot w = \diamond \nabla \cdot \diamond w = (-1)^{n-1} *^{-1} \nabla \wedge * w$$
For example, divergence of the traditional magnetic flux density can be written as
$$\underline{\nabla} \cdot \vec{B} = \underline{\nabla} \cdot \underline{\underline{B}} \cdot \vec{\vec{\vec{\epsilon}}}{}^{-1} = \star \left( \underline{\nabla} \wedge \underline{\underline{B}} \right)$$
Traditional 'nabla' vector notation can make a confuse, simply $\vec{\nabla}$ is wrong. $\nabla$ operation maps n-forms to (n+1)-form. So the proper notation is $\underline{\nabla}$.
(8) Natural form of flux density and density field in 3D : n-from field naturally contracts with n-dimensional area to make a scalar. Flux density has the dimension of $L^{-2}$. For example, current density combined with some surface elements makes scalar current.
$$I=\int_S \underline{\underline{J}} \cdot d\vec{\vec{S}}$$
And density field has the dimension of $L^{-3}$, so it contracts with volume element to make a scalar. For example, electric charge density contracts with to give a scalar charge.
$$Q=\int_V \underline{\underline{\underline{\rho}}} \cdot d\vec{\vec{\vec{V}}}$$
The exterior derivative of polarization density yields the bound charge density (3-form). Thus natural form of polarization density is 2-form.
$$\underline{\nabla} \wedge \underline{\underline{P}}= -\underline{\underline{\underline{\rho}}}{}_{\text{bound}}$$
The exterior derivative of of magnetization density yields the bound current density (2-form). Thus the natural form of magnetization density is 1-form.
$$\underline{\nabla} \wedge \underline{M} = \underline{\underline{J}}{}_{\text{bound}}$$
(9) Then our traditional Maxwell equations are naturally expressed by
$$\begin{align} \vec{\nabla} \cdot \vec{B}=0 &\;\;\;\;\; \rightarrow \;\;\;\;\;& \underline{\nabla} \wedge \underline{\underline{B}} = 0 \\ \vec{\nabla} \times \vec{E} + \dot{\vec{B}}=0 &\;\;\;\;\; \rightarrow \;\;\;\;\;& \underline{\nabla} \wedge \underline{E} + \dot{\underline{\underline{B}}}=0 \\ \vec{\nabla} \cdot \vec{D} = \rho_{\text{free}} &\;\;\;\;\; \rightarrow \;\;\;\;\;& \underline{\nabla} \wedge \underline{\underline{D}} = \underline{\underline{\underline{\rho}}}{}_{\text{free}} \\ \vec{\nabla} \times \vec{H} - \dot{\vec{D}}= \vec{J}_{\text{free}} &\;\;\;\;\; \rightarrow \;\;\;\;\; &\underline{\nabla} \wedge \underline{H} - \dot{\underline{\underline{D}}}=\underline{\underline{J}}{}_{\text{free}} \end{align}$$
(10) Other quantities are naturally expressed like below.
Energy density :
$$u=\frac{1}{2} (\vec{E} \cdot \vec{D} + \vec{B} \cdot \vec{H}) \;\;\;\;\; \rightarrow \;\;\;\;\; \underline{\underline{\underline{u}}}=\frac{1}{2} \left( \underline{E} \wedge \underline{\underline{D}} + \underline{\underline{B}} \wedge \underline{H} \right)$$
Poynting vector :
$$\vec{S}=\vec{E} \times \vec{H} \;\;\;\;\; \rightarrow \;\;\;\;\; \underline{\underline{S}}=\underline{E} \wedge \underline{H}$$
Potential form :
$$\vec{E} = -\vec{\nabla} \phi - \dot{\vec{A}} \;\;\;\;\; \rightarrow \;\;\;\;\; \underline{E}=-\underline{\nabla} \phi - \dot{\underline{A}}$$
$$\vec{B} = \vec{\nabla} \times \vec{A} \;\;\;\;\; \rightarrow \;\;\;\;\; \underline{\underline{B}}=\underline{\nabla} \wedge \underline{A}$$
Also it has gauge invariance :
$$\phi \rightarrow \phi + \xi, \;\;\;\;\; \underline{A} \rightarrow \underline{A}-\underline{\nabla}\xi$$
Continuity equation :
$$\dot{\rho} + \vec{\nabla} \cdot \vec{J}=0 \;\;\;\;\; \rightarrow \;\;\;\;\; \dot{\underline{\underline{\underline{\rho}}}} + \underline{\nabla} \wedge \underline{\underline{J}} = 0$$
