Sun Jun 15 16:46:04 MDT 1997 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ----------[ R e d u c e ]---------- % ---------- Initialization ---------- on time; Time: 0 ms % ---------- Tensor Analysis ---------- % 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. %kdelta({i, k}, {j, h}); % Levi-Civita symbol: [epsilon(2,1,3), epsilon(1,3,1)] => [-1, 0] vector i1, i2, i3; Time: 20 ms {eps(i2, i1, i3), eps(i1, i3, i1)}; { - eps(i1,i2,i3),0} Time: 0 ms % 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]] load_package(linalg)$ Time: 110 ms a:= mat((1, -2), (3, 4))$ Time: 0 ms b:= mat((5, 6), (-7, 8))$ Time: 0 ms kronecker_product(a, b); [ 5 6 -10 -12] [ ] [-7 8 14 -16] [ ] [15 18 20 24 ] [ ] [-21 24 -28 32 ] Time: 10 ms clear a, b; Time: 0 ms % 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 % 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 operator T; Time: 0 ms T({i}, {j}); t({i},{j}) Time: 10 ms % 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 % ---------- Quit ---------- quit; Quitting real 7.10 user 0.27 sys 0.89