Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 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, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity.
The integral equation of continuity (I) may now be written l f fdxdydz+ff (lpu+mpv+npdso, (4) which becomes by Green's transformation (dt +d dz dy dx (p u) + d (p v) + d (p w) l I dxdydz - o, dp leading to the differential equation of continuity when the integration is removed.
To transfer the integral equation into the differential equation of continuity, Green's transformation is required again, namely, (++) dxdydz= ff (l +mr t } ndS, (2) or individually dxdydz = f flldS, ..., (3) where the integrations extend throughout the volume and over the surface of a closed space S; 1, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and, 77, any continuous functions of x, y, z.
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