In a book I am studying, the Poynting vector is defined as: $$ \mathcal{P} = \mathbb{E} \times\mathbb{H} $$ and it is described the Poynting's theorem, that states that the flux through a surface wrapping a volume is the energy balance on that volume.
In another part of the book, it says that the only thing that has a physical meaning is the flux of the Poynting vector, while the vector by itself hasn't it. This is because, if you add to the Poynting vector the curl of another vector, it will gave the same flux. $$ \mathcal{P}' = \mathbb{E} \times\mathbb{H} + \operatorname{curl}( \mathbb{F}) $$ $$ \Rightarrow\iint_Σ \mathcal{P}' \cdot n \, dS = \iint_Σ \mathbb{E} \times\mathbb{H} \cdot n\, dS = \iint_Σ \mathcal{P} \cdot n \, dS $$ Looking on the web, I found on the Wikipedia page about Poynting vector, in the section "Adding the curl of a vector field" that (for some reason I cannot fully understand since I have never had a course about special relativity) that the expression for the Poynting vector is unique. To me, this seems to give the Poynting vector a local meaning, not only to its flux through a surface.
Is it true? What is this local meaning (if present)? Where can I look to start comprehend it better and more mathematically?
$\mathbb{E} $: electric field
$\mathbb{H} $: magnetic field
$Σ$: generic closed surface
$\mathcal{P} $: Poynting vector