Mon Jun 16 00:48:27 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- 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. delta : CartesianTensor(1, 4, Integer) := kroneckerDelta() +1 0 0 0+ | | |0 1 0 0| (1) | | |0 0 1 0| | | +0 0 0 1+ Type: CartesianTensor(1,4,Integer) Time: 0.02 (EV) + 0.03 (OT) + 0.02 (GC) = 0.07 sec delta([i, k], [j, h]) There are no exposed library operations named delta but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op delta to learn more about the available operation. Cannot find a definition or applicable library operation named delta with argument type(s) List OrderedVariableList [i,k] List OrderedVariableList [j,h] )clear properties delta -- Levi-Civita symbol: [epsilon(2,1,3), epsilon(1,3,1)] => [-1, 0] epsilon : CartesianTensor(1, 3, Integer) := leviCivitaSymbol() +0 0 0+ +0 0 - 1+ + 0 1 0+ | | | | | | (2) [|0 0 1|,|0 0 0 |,|- 1 0 0|] | | | | | | +0 - 1 0+ +1 0 0 + + 0 0 0+ Type: CartesianTensor(1,3,Integer) Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec [epsilon(2, 1, 3), epsilon(1, 3, 1)] (3) [- 1,0] Type: List Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (GC) = 0.05 sec )clear properties epsilon -- 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]]); Type: Matrix Integer Time: 0.02 (IN) + 0.02 (EV) + 0.02 (OT) + 0.03 (GC) = 0.09 sec b:= matrix([[5, 6], [-7, 8]]); Type: Matrix Integer Time: 0.02 (IN) = 0.02 sec product(a :: CartesianTensor(1, 2, Integer), _ b :: CartesianTensor(1, 2, Integer)) + + 5 6+ +- 10 - 12++ | | | | || | +- 7 8+ + 14 - 16+| (6) | | |+ 15 18+ + 20 24+ | || | | | | ++- 21 24+ +- 28 32+ + Type: CartesianTensor(1,2,Integer) Time: 0.15 (IN) + 0.08 (OT) = 0.23 sec )clear properties a b -- 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 T:= operator('T); Type: BasicOperator Time: 0.08 (IN) + 0.02 (EV) + 0.02 (GC) = 0.12 sec T([i], [j]) (8) T([i],[j]) Type: Expression List Symbol Time: 0.65 (IN) + 0.02 (EV) + 0.12 (OT) + 0.05 (GC) = 0.83 sec -- 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 real 4.6 user 2.7 sys 0.2