Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 21 16. Prove (3.87) from (3.84) and (3.86).7. Prove that g = det (glj) is positive. [Hint: Show from (3.84) that (gij) is the product oftwo matrices and then take determinants.]8. a) Show that, the tensors vj, and wfi being given in a neighborhood D, Eq. (3.90) defines tul, as components of a tensor.b) Generalize the result of part (a) to addition of two tensors of arbitrary type.9. Show that multiplication of all tensor components by the same invariant, as in (3.91),defines a new tensor.10. Let u,, v,, wk, zl, pmh be tensors. Show that each of the following tensor products is atensor:a) u1vj = si, b) uiwk = t,! C) wkzl = qkl d) wkp,h = d:,,11. a) Let ui be a tensor of the type indicated. Show that each of the following is a tensor:w: =u;;/I,v; =u;.b) Prove generally that contraction of a tensor yields a tensor.12. Show from (3.93) that the rfil can be expressed as in (3.94). [Hint: Show first thatr;[glS = (a2(a/ax~ax1)(a(a/axs). Next, with the aid of (3.84), compute the quantityin parentheses in (3.94) and show that it equals 2rilglS. Now multiply both sides bygSt.l13. Derive the formula (3.104). [Hint: First show thatdiv u = (aua/axa) + rtlul.Next put i = j = a! in (3.94), multiply out, and show that the last two terns cancel,so that rtl = ~gas(agaS/ax1). Now obtain ag/axl, from the determinant g, as a sum ofn determinants, the ath of which is obtained from the given determinant by differentiatingeach element in the ath row with respect to x'. Expand each of these determinants byminors of the ath row to obtainwhere A,, is the cofactor of g,, in the determinant g. Now ga" gsa = A,,/g (seeProblem 7 following Section 1.9). Thus conclude that rtl = (2g)-'(ag/axl) and showthat this permits one to write div u in the form asserted.]14. Derive (3.109) from (3.108), using (3.95).15. We define the norm of a vector u to be its norm or length in standard coordinates. Thusfor components U1 or Ui the length isShow that Jul can be written in the forms:lul = (u,u')f, ("1lul = (u'u')~ = (u'u')~.= (gtju1uj)4, lul = (g1ju1uj)~,each of which shows that the norm is an invariant.16. The inner product of two vectors u, v is defined to be the invariant(u, v) = u;vl.Show that this can be written in the forms: