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If we consider an infinitesimal surface element, the normal force acting on it divided by its area is defined to be the pressure in the point in which the surface element lies.

Force acts on masses, but a surface doesn't possess any mass. So it doesn't make sense to say the force on the surface element.

What is the exact, unambiguous, formal definition of pressure of a point?

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    $\begingroup$ Consider water. The tiny surface element is made up of water molecules. These molecules aren't massless, are they? $\endgroup$
    – Yashas
    Feb 27, 2017 at 16:38
  • $\begingroup$ I agree with Yashas. The surface element should still exist, so it should have mass. Are you confused because they use infinitesimally small masses to represent a infinitesimal surface element? If that's the case, you have to realize that the forces and dimensions are all infinitesimally small when looking at infinitesimal elements. $\endgroup$
    – JMac
    Feb 27, 2017 at 16:45

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A more correct definition is to say that pressure equals normal momentum flux across a surface. With this general definition you can define pressure at a point inside the fluid by imagining an infinitesimal surface there. In the case of an impenetrable surface (such as container wall) this momentum flux simply becomes momentum exchange with the surface, and this equals force exerted on the surface.

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