Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 1174. Given thata) find ciu and dv in tcrms of tix, tiy, and rlz at the point x = I, y = 1, ; = 2, u = 3,ri = 2;b) find (au/a.u),,; and (av/ay),., at this point;C) lind approximately the values of u and u for x = 1.1, y = 1.2, z = 1.8.5. If xy + 2xu + 3xv + uv - 1 = 0, 2xjl + 3jju - 2x11 + 2uv + 2 = 0, lind the Jacobianmatrix ['flI,,$:I.6. Let equations Fl (.rl. x?, xj, x4) = 0, F7(xI, x?. xj, x4) = 0 be given. At a certain pointwhere the equations arc satisfied, it is known thata) Evaluatc (ijxl lilxj),, and (axl/i)x4),, at the point.b) Evaluatc (a.ul /a.u3),, and (a.r4/ax3),, at the point.c) Evaluate ij(xl. .ur)/a(x3, x4) and a(x3, x4)/a(xl, x?) at the point.7. Prove: If F(x, y, z) = 0, then (g),,(g)7($)x = -I.8. In therrnociynclrnic.~ the variables p (pressure), T (temperature), U (internal energy), andV (volume) occur. For each substance these are related by two equations, so that any twoof the four variables can be chosen as independent, the other two then being dependent.In addition, the second law of thermodynamics implies the relationiiU i)pa) m-Tn+p=O,when V and T are independent. Show that this relation can be written in each of thefollowing forms:i) 7' i)p iJ 7'b) m+Tm-pm=O (U,Vindep.),T I iJ(T,u);,(