$\vec F$ = force vector,
$\vec A$ = area vector,
$P$ = pressure
Mathematically $\vec F = P \vec A$. By product rule we get,
$$ {\rm d}\vec F = P {\rm d}\vec A + \vec A {\rm d}P $$
Why do we often compute Force over a surface as $\vec F = \int P {\rm d}\vec A$ whilst ignoring the term $\int \vec A {\rm d}P$ ?
If the area under consideration is a more complicated surface than a flat plane, and, if the pressure varies with position spatially on the surface, then the force per unit area at a given location on the surface is $p\mathbf{n}$, where $\mathbf{n}$ is the unit normal to the surface, and the differential element of area over which the force is applied is dA, so the vectorial force on the surface is $\mathbf{F}=\int{p\mathbf{n}dA}$.
The definition $\vec F = P \vec A$ is only valid if P is a constant, and in such case your second term is zero. The definition that makes physical sense when P varies with position is $d\vec F = P d\vec A$.
