**Theorem:**

In an incompressible inviscid fluid let ρ denote the density, **u** the
velocity field, p the pressure and -**F** the body
force (gravity). Then

ρD**u**/Dt = -**∇**p - **F**.

(D/Dt denotes the substantial derivative operator ∂/∂t + **u**.**∇**)

**Proof:**

Consider some fixed surface S enclosing a volume V. The total momentum
of the fluid contained in V is ∫_{V} ρ**u** dV. This will
change with time due to three factors: the force -∫_{S} p
d**S** exerted by the pressure on V, the body force
-∫_{V} **F** dV and the convection
∫_{S} ρ**u**(**u**.d**S**) of momentum across S.
Thus

0 = ∂(∫_{V} ρ**u** dV)/∂t + ∫_{S} p
d**S** + ∫_{V} **F** dV +
∫_{S} ρ**u**(**u**.d**S**)

This equation is the integral form of the Euler momentum equation. By applying the divergence theorem to the surface integrals we obtain

0 = ∫_{V} (ρ∂**u**/∂t + **∇**p +
**F** + ρ(**u**.**∇**)**u**) dV

Since this holds for any volume V the integrand must vanish everywhere, ie.

0 = ρ∂**u**/∂t + ρ(**u**.**∇**)**u** +
**∇**p + **F**

If we drop the assumption that the fluid is inviscid we must add a term
proportional to ∇^{2}**u**, which gives the Navier-Stokes equation (which is rather more difficult to derive).