Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 349IFigure 5.39 Problem 7.[Hint: By the chain rule,Use the result of (b) to show thatIand obtain similar expressions for av,/ay, av,/az. Add the results and use the resultof (~1.1Remark The Jacobian J can be interpreted as the ratio of the volume occupied by a smallpiece of the fluid at time t to the volume occupied by this piece when t = 0, as in Fig. 5.39.Hence by (d) the divergence of the velocity vector can be interpreted as measuring thepercentage of change in this ratio per unit time or simply as the rate of change of volume perunit volume of the moving piece of fluid.8. Consider a piece of the fluid of Problem 7 (not necessarily a "small" piece) occupying aregion R = R(t) at time t and a region Ro = R(0) when t = 0. Let F(x, y, z, t) be afunction differentiable throughout the part of space concerned.a) Show thatIJJJ~(x,y.Z,t)dxd~d~=R(0 RoFW(XO, YO, ZO. t). ...I J ~XO YO ~ Z O .[Hint: Use Eq. (5.109), noting that the degree 6 must be 1 here.]b) Show thatd /// F(x, y, z, t)dx dy dz =dtR(0 R(t)I/I/ [$ + div(Fv) dx dy dz.[Hint: Use (a) and apply Leibnitz's Rule of Section 4.9 to differentiate the right-handside. Use the result of part (d) of Problem 7 to simplify the result. Then return to theoriginal variables by (a) again.]9. Let p = p(x, y, z, t) be the density of the fluid motion of Problems 7 and 8. The integral