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