Advanced Calculus fi..

I348 Advanced Calculus, Fifth Edition4. a) Write out in detail the steps leading from the Maxwell equations (5.120) to the partialdifferential equations (5.129), (5.130).b) Show from Eqs. (5.120), . . . , (5.130) that each component of E satisfies the wave1equation in each domain free of charge.5. Find the temperature distribution in a solid whose boundaries are two parallel planes, dunits apart, kept at temperatures TI, T2, respectively. (Hint: Take the boundaries to be theplanes x = 0, x = d and note that, by symmetry, T must be independent of y and z.)6. Show that on the basis of the laws of thermodynamics, the line integralis independent of the path in the T V plane. The integrand is minus the differential of thefree energy F.7. Consider a fluid motion in space. A particle occupying position (xo, yo. zo) at time t = 0occupies position (x , y , z) at time t . Thus x , y , z become functions of xo, yo, 20, t :Let the V symbol be used as follows:x = ~ (xo, YO, 20, t), 2y = @(xo, yo* ZO. t), (*)z = x(x0, yo, zo, t).and let J denote the JacobianLet v denote the velocity vector:a(x, Y, Z) 41a(xo7 yo, ZO).ax ay az a4 a@ axv=-i+-j+-k=-i+-j+-k.at at at at at ata) Show that J = V x . V y x Vz.b) Show thataxo vy x vz ay0 v y x- = i. -vz az0=j. - = k. V y x vzax J ' ax J ' ax J 'and obtain similar expressions for(Hint: See Problem 5 following Section 2.12).C) Show thata J--=vv,.vy xvz+vv,.vzxVx+vv,.vxxvy.at 4d) Show that1 aJdivv = --.J at