Advanced Calculus fi..

196 Advanced Calculus. Fifth Editionc) the components of a vector p are given byp, = p,sin~cos8+pysin~sin8+p,cos~,p@ = pxcos~cos8+pycos~sin8 - p,sin$,pe = -px sin8 + p, cos8;au ru 1d) grad U has components v, a, ,au;1 ae) div p = - F - - [ S ~ ~ ~ ~ ( P ~ P ~ )P sin@f) curl p has components:+ P&(P$ sin41 + p$@];g) V~U is given by (3.67).8. (Curvilinear coordinates on a surface) EquationsI x = f(u, v), Y = g(u, v), z = h(u, V)can be interpreted as parametric equations of a surface S in space. They can be consideredas a special case of (3.42), in which w is restricted to a constant value, while (u, v) variesover a domain D of the uv plane; the surface S then corresponds to a surface w = constfor (3.42). We consider u, v as curvilinear coordinates on S. The two sets of curves u =const and v = const on S form families like the parallels to the axes in the xy-plane.Graph the surface and the lines u = const, v = const, for the following cases:a) sphere: x = sin u cos v, y = sin u sin v, z = cos u;b) cylinder: x = cos u, y = sin u, z = v;c) cone: x = sinh u sin v, y = sinh u cos v, z = sinh u.9. For the surface of Problem 8, assume that f, g, h have continuous first partial derivativesin D and that the Jacobian matrixhas rank 2 in D (Section 1.17). Show that each point (uo, vo) of D has a neighborhoodDo in which the correspondence between points (u, v) and points of the surface is oneto-one.[Hint: If, for example, a( f, g)/a(u, v) # 0 at (uo, vo), then apply the ImplicitFunction Theorem of Section 2.10, as in Section 2.12, to obtain inverse functions u =4(x, y), v = $(x, y) of x = f (u, v), y = g(u, v) in Do. Show that each point (x, y, z)of the surface, for (u, v) in Do, comes from a unique (u, v) in Do.]10. Let a surface S be given as in Problems 8 and 9 and let r = xi + yj + zk.a) Show that arlau and arlav are vectors tangent to the lines v = const, u = const onthe surface.b) Show that the curves v = const, u = const intersect at right angles, so that thecoordinates are orthogonal, if and only if