# On the defining properties of hydrostatic center of pressure

I'm having some issues understanding the defining properties of the center of pressure (Sadly, I have an awful teacher which follows a believe-me-its-true strategy and uses a book which I find too superficial on this topic, its Çengel's Fluid Mechanics - Fundamentals and Applications). Section 3-6 is devoted to hydrostatic pressure on curved surfaces.

To make my point, let's consider the following scenario: A wall with a curved boundary on hydrostatic conditions (taken from Cengel).

It is clear to me that to calculate the total force that water makes on the wall, we just integrate the pressure throughout the surface,

$$F = \int p \ dA$$

where the usual formula applies $$\Delta p = \rho g \Delta z$$

These definitions are just fine to me, I believe it applies to all surfaces that I can think of. Now, Çengel (and other textbooks) introduce the concept of center of pressure. In its most fundamental form, I see the center of pressure to be a vector that represents an hypothetical force which mimics the total force that acts on the wall, similar to the definition of center of gravity, which makes calculations easier but no actual physical force acts only on a single point, but rather on every particle present on the system.

It is even intuitive how to calculate the components of this vector. By Newton's second law, just balance horizontal and vertical forces acting on the system (figure 3-32 above).

My problem is with the place where the center of pressure acts. For example, Çengel mentions that, for the plane plate (See figure below), the location of the center of pressure it's chosen in such a way that the torque around a chosen axis it's the same whether we calculate it in the integral form (whole system) or just as a force acting on a point (center of pressure). I agree that this unequivocally determines where the center of pressure is, and I see how this definition complies with the objective of chosing a point in such a way that mimics the whole system, since when calculating torques or balancing forces, it does not matter whether I use the whole system or just a point-like distribution of forces.

Here are my questions:

1. Does this location where the center of pressure acts is always well defined? Which definition is the most appropiate to consider for the location of the center of pressure?, and, more importantly;
2. Is this point always located on the curved surface? Çengel mentions that the horizontal forces $$F_H$$ and $$F_x$$ have the same line of action for the curved surface case (see image below), and so a practical way of determining the position of the center of pressure is by intersecting the extension of the line of action of $$F_x$$ with the wall's surface. By this practical construction, it is clear that the center of pressure it's always located on the surface. However, I can't see why the torque property mentioned for plane plates it's satisfied.

I would also appreciate another references (books) on this topic, as I find Çengel's textbook too hollow at moments.