Prove that the vertical component of the resultant force on submerged curved surface is equal to the weight of the liquid supported by the curved surface.

In fluid mechanics, when analyzing the forces acting on a submerged curved surface, it is essential to determine the vertical component of the resultant force exerted by the static fluid. This force is directly related to the weight of the liquid that is vertically above the curved surface. The proof of this concept is derived using the principles of hydrostatics.

Consider a curved surface submerged in a fluid of density \( \rho \), where the pressure at a depth \( h \) is given by the hydrostatic equation:

\[
P = P_0 + \rho g h
\]

where \( P_0 \) is the atmospheric pressure, \( g \) is the acceleration due to gravity, and \( h \) is the depth of the point under consideration.

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