Advanced Calculus fi..

21 2 Advanced Calculus, Fifth Editionand that(u, u) = lu12.17. a) Prove: If u', vj are contravariant vectors and bijuivj is an invariant, then bij is acovariant tensor.b) hove generally: If ui , vj, . . . , wk, z', . . . are covariant and contravariant vectors andis an invariant, then bi::: is a tensor of the type indicated.18. a) Show that the associated tensor biJ to the curl bij of a covariant vector ui (as inEq. (3.109)) is given byb) Show that for n = 3 in orthogonal coordinate as in Section 3.8, the three nonzerocomponents b23, b31, b12 of the tensor b'J are the same as (3.62). [Note that thecovariant components of p are ap,, pp,, yp, by (3.58').]c) Verify that bij is an alternating tensor.19. Show that in 3-dimensional space the exterior product w,, = u, A v, of two covariantvectors u,, v, can be interpreted in terms of the vector product. In detail, show that, instandard coordinates x, y, z with usual orientation and ul = u,, . . . , vl = v,, . . . , thethree components of the vector product (u,i + . . .) x (v,i + - . .) appear with + or -signs as the nonzero components of w,, .20. Let u;,,,,;, be a covariant tensor in En (r 2 2). Let this tensor be altemating in onecoordinate system. Show that it is alternating in each allowed coordinate system.21. Let u; , v; , w;, be alternating covariant tensors in E ~.a) Show that uiv; fails to be alternating if ulvl # 0.b) Show that u; wjk fails to be alternating if ul w2l # 0.C) In one coordinate system let ui = 1 for i = 1,2,3. Show that if u;v, is alternating,thenv; = Ofor j = l,2,3.22. Show that if u;, ;, is a covariant tensor, then Eq. (3.114) defines a new covariant tensorv;, . im which is altemating.Differential geometry of surfaces in space.In the following problem let S be a surface inspace as in Problems 8-10 following Section 3.8. Thus one has curvilinear coordinates (u, v)on S and a positive definite quadratic differential form ds2 = E du2 + 2F du dv + G dv2,called the frstfundamental form of S. If we write u1 = u, u2 = v, gll = E, g12 = g21 =F, g22 = G. then ds2 = g;, dui duJ and, as in Section 3.10, gij is a covariant second ordertensor. Also g = det (g;;) = EG - F~ > 0.As in Problem 16 following Section 2.13 a unit normal vector for S can be obtained asn = w/ 1 wl, where w = ar/au x ar/a v. The choice of normal n (varying from point to point)will be assumedfxed and unaffected by changes of coordinate on S. (We are here dealingwith an oriented surface as in Section 5.9.)23. Let a path C on S be given by parametric equations u = u(t), v = v(t) in curvilinearcoordinates, so that r = r(u(t), v(t)) is the vector equation of the path. We assume thatt is arc length s on C and that appropriate derivatives exist, so that the Frenet formulas