Is this application of product rule for the definition of pressure correct?

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 $pmathbf{n}$, 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 $mathbf{F}=int{pmathbf{n}dA}$.

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 $dvec F = P dvec A$.