Tue Feb 23 22:34:29 MST 1999 aquarius% macsyma Starting Macsyma math engine with no window system... /opt/local/macsyma_422/macsyma.422 local This is Macsyma 422.0 for Sparc (Solaris 2.x) computers. Copyright (c) 1982 - 1998 Macsyma Inc. All rights reserved. Portions copyright (c) 1982 Massachusetts Institute of Technology. All rights reserved. Type "DESCRIBE(TRADE_SECRET);" to see important legal notices. Type "HELP();" for more information. /aquarius/data2/opt/local/macsyma_422/system/init.lsp being loaded. /aquarius/home/wester/macsyma-init.lsp being loaded. (c1) (c2) /* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ symbol_display_case: lower_case$ Time= 0 msecs (c3) showtime: all$ Time= 0 msecs (c4) prederror: false$ Time= 0 msecs (c5) /* ---------- Tensor Analysis ---------- */ init_itensor()$ /aquarius/data2/opt/local/macsyma_422/tensor/itensor.so being loaded. /aquarius/data2/opt/local/macsyma_422/tensor/isymtry.so being loaded. Time= 600 msecs (c6) /* Generalized Kronecker delta: delta([j, h], [i, k]) = delta(j, i) delta(h, k) - delta(j, k) delta(h, i). See David Lovelock and Hanno Rund, _Tensors, Differential Forms, & Variational Principles_, John Wiley & Sons, Inc., 1975, p. 109. */ ishow(kdelta([i, k, @j, @h])); h j j h (d6) kdelta kdelta - kdelta kdelta i k i k Time= 10 msecs (c7) /* Levi-Civita symbol: [epsilon(2,1,3), epsilon(1,3,1)] => [-1, 0] */ [levi_civita([2, 1, 3]), levi_civita([1, 3, 1])]; Time= 0 msecs (d7) [- 1, 0] (c8) /* Tensor outer product: [[ 5 6] [-10 -12]] [1 -2] [ 5 6] [[ -7 8] [ 14 -16]] ij ij [3 4] X [-7 8] = [ ] c = a b [[ 15 18] [ 20 24]] kl kl [[-21 24] [-28 32]] */ a: matrix([1, -2], [3, 4])$ Time= 0 msecs (c9) b: matrix([5, 6], [-7, 8])$ Time= 0 msecs (c10) outermap("*", a, b); Time= 0 msecs [ [ 5 6 ] [ - 10 - 12 ] ] [ [ ] [ ] ] [ [ - 7 8 ] [ 14 - 16 ] ] (d10) [ ] [ [ 15 18 ] [ 20 24 ] ] [ [ ] [ ] ] [ [ - 21 24 ] [ - 28 32 ] ] (c11) remvalue(a, b)$ Time= 0 msecs (c12) /* Definition of the Christoffel symbol of the first kind (a is the metric tensor) [Lovelock and Rund, p. 81] d a d a d a 1 kh hl lk Chr1 = - (----- + ----- - -----) lhk 2 l k h d x d x d x */ imetric: a$ Time= 0 msecs (c13) ishow(ichr1([l, h, k])); a + a - a l k,h k h,l h l,k (d13) ------------------------ 2 Time= 10 msecs (c14) /* Partial covariant derivative of a type (1, 1) tensor field (Chr2 is the Christoffel symbol of the second kind) [Lovelock and Rund, p. 77] i d i i m m i T = ---- T + Chr2 T - Chr2 T j|k k j m k j j k m d x */ ishow(T([@i, j])); i (d14) t j Time= 10 msecs (c15) ishow(covdiff(%, k)); i %1 i i %1 (d15) - t ichr2 + t + ichr2 t %1 k j j,k k %1 j Time= 10 msecs (c16) /* Verify the Bianchi identity for a symmetric connection (K is the Riemann curvature tensor) [Lovelock and Rund, p. 94] h h h K + K + K = 0 i jk|l i kl|j i lj|k */ ishow(covdiff(icurvature([@h, i, j, k]), l) + covdiff(icurvature([@h, i, k, l]), j) + covdiff(icurvature([@h, i, l, j]), k)); %2 h %8 h %8 (d16) - ichr2 (- ichr2 ichr2 - ichr2 ichr2 j i %8 %2 l k k %8 l %2 h h %8 h %5 + ichr2 + ichr2 ichr2 ) + ichr2 ichr2 k %2 ,l l %8 k %2 j i ,%5 l k h %3 h - ichr2 ichr2 + ichr2 %3 i ,j l k j %2 %2 %10 %2 %10 %2 (- ichr2 ichr2 - ichr2 ichr2 + ichr2 %10 i l k k %10 l i k i ,l %2 %10 %2 + ichr2 ichr2 ) + ichr2 l %10 k i k i h %9 h %9 h (- ichr2 ichr2 - ichr2 ichr2 + ichr2 %9 %2 l j j %9 l %2 j %2 ,l h %9 h + ichr2 ichr2 ) - ichr2 l %9 j %2 k %2 %2 %7 %2 %7 %2 %7 (- ichr2 ichr2 - ichr2 ichr2 + ichr2 ichr2 %7 i l j j %7 l i l %7 j i %2 h %5 h %3 + ichr2 ) + ichr2 ichr2 - ichr2 ichr2 j i ,l %5 i ,k l j k i ,%3 l j %28 h %12 h + ichr2 ichr2 - ichr2 ichr2 k j l i ,%28 j k l i ,%12 h %5 h %3 + ichr2 ichr2 - ichr2 ichr2 j %5 ,k l i k %3 ,j l i h %28 h - ichr2 ichr2 - ichr2 k %28 l i ,j j %20 %20 %26 %20 %26 %20 (ichr2 ichr2 + ichr2 - ichr2 ichr2 k %26 l i l i ,k k i l %26 %20 %26 h %25 - ichr2 ichr2 ) - (- ichr2 ichr2 %26 i k l %25 %20 k j h %25 h h %25 - ichr2 ichr2 + ichr2 + ichr2 ichr2 ) j %25 k %20 j %20 ,k k %25 j %20 %20 h %11 %19 %11 ichr2 + ichr2 (ichr2 ichr2 + ichr2 l i k %11 j %19 l i l i ,j %19 %11 %11 %19 - ichr2 ichr2 - ichr2 ichr2 ) j i l %19 %19 i j l h %12 %18 h h + ichr2 ichr2 + (- ichr2 ichr2 + ichr2 j %12 l i ,k j %11 k %18 k %11 ,j h %18 h %18 %11 + ichr2 ichr2 - ichr2 ichr2 ) ichr2 j %18 k %11 %18 %11 j k l i %5 h %28 h - ichr2 ichr2 + ichr2 ichr2 j i ,k l %5 k i l %28 ,j %20 %23 h h + ichr2 (- ichr2 ichr2 + ichr2 j i k %20 l %23 l %20 ,k h %23 h %23 + ichr2 ichr2 - ichr2 ichr2 ) k %23 l %20 %23 %20 k l %20 %22 %20 %22 + (- ichr2 ichr2 - ichr2 ichr2 %22 i k j j %22 k i %20 %22 %20 h + ichr2 ichr2 + ichr2 ) ichr2 k %22 j i j i ,k l %20 %11 %17 h h - ichr2 (- ichr2 ichr2 + ichr2 k i j %11 l %17 l %11 ,j h %17 h %17 + ichr2 ichr2 - ichr2 ichr2 ) j %17 l %11 %17 %11 j l %12 h %11 %16 %11 - ichr2 ichr2 - (ichr2 ichr2 + ichr2 j i l %12 ,k j %16 k i k i ,j %16 %11 %11 %16 h - ichr2 ichr2 - ichr2 ichr2 ) ichr2 j i k %16 %16 i j k l %11 h %28 h %21 + ichr2 ichr2 - ichr2 ichr2 %28 i ,j k l j i ,%21 k l h %21 %14 h - ichr2 ichr2 + ichr2 ichr2 %21 i ,l k j j l k i ,%14 h %3 h %21 + ichr2 ichr2 - ichr2 ichr2 l %3 k i ,j j %21 ,l k i h %14 %14 h - ichr2 ichr2 + ichr2 ichr2 j %14 k i ,l j i k %14 ,l h %12 h %14 - ichr2 ichr2 + ichr2 ichr2 %12 i ,k j l %14 i ,l j k h %21 + ichr2 ichr2 k %21 j i ,l Time= 590 msecs (c17) canform(%); Time= 550 msecs (d17) 0 (c18) /* ---------- Quit ---------- */ quit(); Bye. real 5.03 user 2.09 sys 1.00