Determination of radiation pressure Consider  the incidence of an electromagnetic wave on the plane $x=0$.

We have that:
$$f_x=\frac{dF}{dV}$$
$f_x$ is the volumic force density on the medium.
My doubt is purely mathematical. I want to the determinate the radiation pressure:
$$P_{rad}=\frac{F}{S}$$
I could say that:
$$\int_0^{\infty} f(x) dx=\int_0^{\infty} \frac{dF}{dx.dy.dz}dx=$$
$$=\int_a^b \frac{dF}{dS}=$$
$$=\int_{P(0)}^{P(\infty)}dP_{rad} \hspace{15pt}?$$
As you can see on the second row I didn't chose the integral limits, because we have a integral upon two elementar variables.
So, is it correct to write $\frac{dF}{dS}=d\left(\frac{F}{S}\right)$?

Note: I have this question because I saw this solution written on a book and I know, for example, that $d(xy)=dx.y+dy.x\neq dx.dy$
 A: 1) The relation $\frac{dF}{dS}=d\left(\frac{F}{S}\right)$ is certainly incorrect as @Floris has mentioned in the comment. As the simplest counter-example, consider a linear function, $F(S) = \alpha \, S$, with $\alpha \neq 0$ as a proportionality constant. Then, one could easily see that
$$ \frac{dF}{dS}= \alpha \neq 0 = d\left(\frac{F}{S}\right) = d\left(\frac{\alpha S}{S}\right) = d \left( \alpha \right)~. $$
Notice that we have abused the notation above: on the left-hand-side, there is a derivative while on the right-hand-side, there is a differential; but for the moment, we neglect this abuse.
2) Furthermore, I cannot see what “volumic force density” is and why such a force, $f_x$, is relevant to this problem.
As far as I can understand, one should find the (normal) force per surface area (= pressure) exerted by the radiation on the surface $S$ on the $y-z$-plane (as in the posted figure). In the simplest case of a constant normal force $F$ the radiation pressure will be
$$ P = \frac{F}{S} ~.$$
The concise mathematical expression for a slightly more general case of a spatially varying force normal on a surface $A$ will be
$$ P \equiv \frac{d F}{d A} ~, $$
where $\mathcal{d}{A}$ is the area differential, and $\frac{d F}{d A}$ represents the surface force density, or the force exerted on an infinitesimal area $dA$. The infinitesimal area can be given in any coordinate system; for instance, in the case given above, using Cartesian coordinates, $d A  = d S = dy \, dz$.
For a more detailed discussion and more rigorous formulation, see < https://en.wikipedia.org/wiki/Pressure#Formula >.
