In a straight uniform current of fluid of density [rho], flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle [theta] with the velocity, is [rho]Aq cos [theta], the product of the density [rho], the area A, and q cos [theta] the component velocity normal to the plane. Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and [theta] the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (1) = flux across the surface into the interior _ _ / / = - | | [rho]q cos [theta] dS, _/_/ the integral equation of continuity. In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time. To transfer the integral equation into the differential equation of continuity, Green's transformation is required again, namely, _ _ _ _ _ / / / /d[xi] d[eta] d[zeta] \ / / | | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, (2) _/_/_/ \ dx dy dz / _/_/ or individually _ _ _ _ _ / / / d[xi] / / | | | ----- dx dy dz = | | l[xi] dS,..., (3) _/_/_/ dx _/_/ where the integrations extend throughout the volume and over the surface of a closed space S; l, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and [xi], [eta], [zeta] any continuous functions of x, y, z. The integral equation of continuity (1) may now be written _ _ _ _ _ / / / d[rho] / / | | | ----- dx dy dz = | | (l[rho]u + m[rho]v + n[rho]w) dS = 0, (4) _/_/_/ dt _/_/ which becomes by Green's transformation _ _ _ / / / /d[rho] d([rho]u) d([rho]v) d([rho]w)\ | | | ( ------ + --------- + --------- + -------- ) dx dy dz = 0, (5) _/_/_/ \ dt dx dy dz / leading to the differential equation of continuity when the integration is removed. 22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S.