The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations, ∂ ∂ u ∂ ∂ t = Π Π S ( − − ( u ⋅ ⋅ ∇ ∇ ) u + ν ν ∇ ∇ 2 u ) + f S ρ ρ − − 1 ∇ ∇ p = Π Π I ( − − ( u ⋅ ⋅ ∇ ∇ ) u + ν ν ∇ ∇ 2 u ) + f I where Π Π S and Π Π I are solenoidal and irrotational projection operators satisfying Π Π S + Π Π I = 1 , and f S and f I are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation.