Advanced Calculus fi..

166 Advanced Calculus, Fifth Edition ~ns-In the original coordinates r], = r],(ul, ..., u,) and the functional dependencies areas in (2.170), with FI(UI, ..., u,) = v,.+I, ..., F~(uI, ..., Un) = qn(ul, . .., u,). Ifr = n, there are no functional dependencies.For proofs of the results just stated one is referred to pp. 86-87 of DifferentialGeometry, Lie Groups, and Symmetric Spaces, by S. Helgason (American MathematicalSociety, Providence, R.I., 2001).The range of the mapping u = f(x) just considered is often a smooth curve,surface or hypersurface in En. As the expression of the mapping in local coordinatesshows, S has dimension r. One calls the mapping a submersion of D onto S.-9EXAMPLE 2 Let fl(x1, x2, x3) = f2(x1, XZ, x3) = sin (XI + xz + x + 3),f3fx1, x2, x3) = cos (XI + x2 + ~3). Then A = (af,/ax,) has rank 1 in E~ and thereare two dependencies FJ( fl , f2, f3) E 0; one choice with (a Fj/auk) of rank 2 is(See Problem 12 below.)PROBLEMS1. A function f (x, y), defined for all (x, y), satisfies the conditionsf(x,O)=sinx, --af = 0.Evaluate f (~r/2,2), f (n, 3), f (x, 1).2. Two functions f (x, y) and g(x, y) are such thatin a domain D. Show thatVf = Vgf =g+cfor some constant c.3. Determine all functions f (x, y) whose second partial derivatives are identically 0.4. A function f (x, y), defined for all (x, y), is such that = 0. Show that there is a functiong(x) such thata~f (x, Y ) = g(x).5. Determine all functions f (x, y) such that && = 0 for all (x, y). [Hint: Cf. Problem 4.16. Show that the following sets of functions are functionally dependent:a) f =:, g=x+y 5 2 ;b) f = x2 + 2xy + y2 + 2x + 2y, g = exeY;c) f = X ~ ~ - X ~ ~ + X g=xy+x-y+z,~ Z ,~ = X ~ + ~ ~ + Z ~ - ~ ~ Zd) f =u+v-x, g=x-y+u, h=u-2~+5~-3y.7. Find an identity relating each of the sets of functions of Problem 6.8. Plot the level curves of the functions f and g of Problem 6(a) and (b).d4