Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 373In the discussion at the end of Section 5.21 differential forms are related toalternating covariant tensors. One can in fact define a differential form of order mas a multilinear function @ of tangent vectors L1, . . . , L, (depending differentiablyon P), which is alternating: that is, @ = @(P, L1, . . . , L,), whereand, in general, @ reverses sign when two of the Li are interchanged. (For m = 0or 1, the condition is satisfied automatically.) In a coordinate system, the tangentvectors can be represented in terms of components dxl, . . . , dxn and, as in (5.216)(Section 5.21),For a full discussion, one is referred to the book of Boothby cited at the end ofthe chapter.PROBLEMS1. 1n E~ let CY = x1x4 (0-form), = x4dx1 - x1 dx4, = x1 dx2dx3 + x2dx3dx1 +-x3 dxl dx2. Calculate and simplify:a) b) B2-Y c) BY +yB d) y2e) da f) d2B g) dy i. ,A2. Illustrate by examples of 1-forms and 2-forms in E~ that ar = arl and B = jJ1 implya + B = CYI + PI and CYB = a1B1.3. Prove the rule CYB = (- l)m"'QBa of (5.207). [Hint: By the distributive laws (5.208) itsuffices to prove it fora = Adxi1 . . . dxl"'l, jl = Bdxj' . . . dxh,. ;3If ik = je for some k, l, both sides are 0. Otherwise, as in Example 2 of Section 5.19, onehastoverifythatonecangofrom(iI, ..., i,,, jl, ..., jrn,)to(jl, ..., jm,,il, ..., i,,)by m 1 m2 interchanges.]4. a) Prove the rule (5.211).b) Prove therule(5.212)forml = l,m2 = l,n =4.c) Prove the rule (5.213) for m = 1, n = 4.5. For y as in Problem 1, apply the given change of variables to obtain a form jj in fl, ..., fand simplify:a) x1 = f1 + f2, ~2 = 22 + 2~3, x3 = f3 + 3~4, x4 = 4 ~1 + ~4b) x' = (~112, ~2 = ~ 1 ~ x3 2 = , 5123, x4 = ili4.6. Under the (invertible) change of coordinates in E ~: x1 = f I , x2 = z1 + f2, z3 =fl + f3,a smooth path C: x' = xl(t), a i t 5 b and a smooth surface S: x1 = xi(u, v), i =1,2,3, (u, v) in R,,, become and 3 respectively. Let XI denote xl(xl, x2, x3) withx', x2, x3 expressed in terms of fl, f2, f3.a) Express the line integral JcXl dxl + X2 dx2 + X3 dx3 as an integral over z.b) Express the surface integral JJsXl dx2 dx3 + X2 dx3 dxl + X3 dxl dx2 as a surfaceintegral over 3.