(The Stokes total time derivative in hydrodynamics) Let w = F(x, y, z, t), where x = f(t),...

(The Stokes total time derivative in hydrodynamics) Let w = F(x, y, z, t), where x = f(t), y = g(t), z = h(t), so that w can be expressed in terms of t alone.

Show that

Here both dw/dt and ∂w/∂t = Ft(x, y, z, t) have meaning and are in general unequal. In hydrodynamics, dx/dt, dy/dt, dz/dt are the velocity components of a moving fluid particle, and dw/dt describes the variation of w “following the motion of the fluid.” It is customary, following Stokes, to write Dw/Dt for dw/dt. [See H. Lamb, Hydrodynamics, 6th ed. (Dover: New York, 1945), p. 3.]