We consider the space of solenoidal vector fields in an unbounded domain $\Omega\subset \mathbb{R}^n$ whose boundary is given as a Lipschitz graph. It is shown that the space of solenoidal vector fields is isomorphic to the $n-1$ product space of the space of scalar functions in some natural topology such as $L^2(\Omega)$. As an application, we introduce a systematic reduction of the equations describing the motion of incompressible flows. This gives a new perspective of the derivation of Ukai's solution formula for the Stokes equations in the half space and provides a key step for the generalization of Ukai's approach to the Stokes semigroup in the case of the curved boundary.