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$P + \frac{1}{2}\rho v^2 + \rho g y = \rm{const.}$

$F_1 = P_1 A_1 , F_2 = P_2 A_2$

$\Delta m = \rho \Delta V = \rho A_1 \Delta x_1 = \rho A_2 \Delta x_2$

$A_1 \frac{\Delta x_1}{\Delta t} = A_2 \frac{\Delta x_2}{\Delta t}$

$A_1 v_1 = A_2 v_2 = Av = \rm{const}$

$W_1 = F_1 \Delta x_1 = P_1 A_1 \Delta x_1 = P_1 \Delta V$

$W_2 = F_2 \Delta x_2 = P_2 A_2 \Delta x_2 = P_2 \Delta V$

$W= W_1 - W_2 = (P_1 - P_2 ) \Delta V = \Delta E_k + \Delta E_p$

$E_k = \frac{1}{2}(\Delta m) v_2^2 - \frac{1}{2}(\Delta m) v_1^2$

$E_p = (\Delta m) g y_2 - (\Delta m) g y_1$

$(P_1 - P_2 ) \Delta V = \frac{1}{2}(\Delta m) v_2^2 - \frac{1}{2}(\Delta m) v_1^2 + (\Delta m) g y_2 - (\Delta m) g y_1$

$\begin{equation*}\begin{aligned}P_1 - P_2 &= \frac{\Delta m}{\Delta V}\frac{1}{2}(v_2^2 - v_1^2) + \frac{\Delta m}{\Delat V} g(y_2 - y_1) \\&= \frac{1}{2}\rho (v_2^2 - v_1^2 ) + \rho g (y_2 -y_1)\end{aligned}\end{equation*}$

$\therefore P_1 + \frac{1}{2} \rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g y_2$

$\therefore P + \frac{1}{2}\rho v^2 + \rho g y = \rm{const.}$

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