Poynting vector and differential forms It is well known that electromagnetic field is a 2-form and Maxwell's equation can be reformulated in language of differential forms.
What is the Poynting vector in this language?
 A: Poynting vector is a 3d vector which can be thought of as a part of 4d stress-energy tensor $T$. In invariant language, you fix your reference frame by choosing a time-like 4-velocity $u$. Consider $T(u)$: it gives you a flow of energy-momentum. $T(u,u)$ is the flow of energy, and the vector $P=T(u)+T(u,u)u$ is the flow of momentum. By construction, $P(u)=0$ so you can think of it as 3d space-like vector.
So far, this applies to any stress-energy $T$. For Maxwell $T$, express it in terms of $F$: $T = F.F - 1/4 g (F,F)$, where $g$ is metric and $F.F$ is a partial scalar product of $F$ with itself, with respect to the first index: this $T$ is traceless. You can use this to express $T(u)$ and $T(u,u)$ and $P$ in terms of $F(u)$ (aka the electric field) and $(^*\!F)(u)$ (aka the magnetic field). In the end you should find something like $P=u\wedge F(u)\wedge (^*\!F)(u)$.
A: Too long for a comment.
It is worth mentioning the controversy that arose about the expression of the Poynting vector $P$, whether the Abraham form $E\times H$ or the Minkowski form $D\times B$ is valid. The notations are that the electro-magnetic field, a closed $2$-form in Minkowski space, has coordinates $(E,B)$. The vectors $(D,H)$ are given by
$$D=\frac{\partial L}{\partial E},\qquad H=-\frac{\partial L}{\partial B}$$
where $L(E,B)$ is the density of the Lagrangian from which the Maxwell's equations derive. For instance, the standard equations follow from the choice $L=\frac12(|E|^2-|B|^2)$ (here the light speed is set to $1$). Other choices are possible, like that of Born-Infeld, which yield non-linear models.
The amazing resolution of the controversy is that both expressions equal each other if, and only if $L$ is Lorentz invariant. In mathematical terms, this means that $L$ is a function of the quantities
$$\frac12(|E|^2-|B|^2),\qquad E\cdot B$$
only.
Going further, the Maxwell's equations imply that the following tensor is (row-wise) divergence free
$$T=\begin{pmatrix}
W & E\times H \\
D\times B &
-W_B\otimes B-W_D\otimes D+\sigma I_3 
\end{pmatrix}$$
where
$$\sigma:=B\cdot W_B+D\cdot W_D-W.$$
Hereabove $W:=D\cdot E-L(E,B)$ is the energy density (a partial Legendre transform of $L$), and subscripts are differentials. That ${\rm Div}_{t,x}T=0$ expresses the conservation laws of the energy and of the Poynting vector. The important point is that $T$ is a symmetric tensor, and this is ensured by the Lorentz invariant.
