Neglecting frictional losses, show that the pressure difference between points 1 and 2 for the flow of a liquid through the convergent tube shown in Figure Q3.1 is given by P1−P2 = (pv²/2)[1-(A2/A1)] = (p/2)(Q²/A²)[1-(A2/A1)]

Step 1: Derive the hydrostatic equation for an incompressible fluid.
The basic hydrostatic vector equation is given as $\nabla P = \rho \vec{g}$.
In a static fluid, the pressure gradient balances the body force due to gravity. Let's consider the z-axis pointing vertically upwards, opposite to the direction of gravity. Thus, the gravitational acceleration vector is $\vec{g} = -g \hat{k}$.
The pressure gradient in the z-direction is $\nabla P = \frac{dP}{dz} \hat{k}$ (since pressure in a static fluid only varies with depth).
Substituting these into the given equation:

$$\frac{dP}{dz} \hat{k} = \rho (-g \hat{k})$$