I've read 4 different books and yet nobody explains why forces $F_1$ ($=p_1A_1$) and $F_2$ ($=p_2A_2$) point in different directions. Shouldn't $F_2$ point in the same direction as $v_2$?
Since we're assuming that parts of fluid between $a$ and $b$ have the same kinetic and potential energies (same holds for $c$ and $d$), why do all proofs state that the change in work: $W_2 - W_1$ is equal to the change in energy $E_2 - E_1$? Work is equal to the change in kinetic energy, so $W_2 = W_1 = 0$ (because we assumed that fluid between each pair of points has the same energy).
Then there's the problem of signs, how do we determine which sign to choose and how do potential energies come into the equation?