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
Best Answer
The Poynting vector itself represents the energy flux density of an electromagnetic field. In other words, its magnitude is the energy per unit area per unit time carried by the field at a particular location. Therefore, it has units of W/m^2. If you integrate its divergence over a surface, you obtain the total power radiated through that surface, which of course has units of W.
As for the issue of adding the curl, note that it is a basic vector-calculus identity that the curl of any vector field has zero divergence. Therefore, it must contribute nothing to the integral of the divergence of the Poynting vector. However, we must be very careful in reading your Wikipedia article. The article says that
It does not, however, say that the field that you get when adding the curl is still the energy flux density as computed with $\vec{E}\times\vec{H}$. If you add some weird random field, it will not represent the local energy flux density anymore, but it will still satisfy Poynting's theorem.
In summary, adding a curl of some random field destroys the local meaning of the Poynting vector, but preserves the meaning of the surface integral of its divergence.