Maths - Tensor Programs

There are a number of open source programs that can work with tensors. I have used Axiom, how to install Axiom here.

I have put user input in red: 

(1) -> CT := CARTEN(i0 := 1, 2,Integer)
           (1)  CartesianTensor(1,2,Integer)
           Type: Domain

(2) -> t0: CT := 8
           (2)  8
           Type: CartesianTensor(1,2,Integer)

(3) -> rank t0
 (3)  0
             Type: NonNegativeInteger

(4) -> v: DirectProduct(2, Integer) := directProduct [3,4]
 (4)  [3,4]
             Type: DirectProduct(2,Integer)

(5) -> Tv: CT := v
             +1  2+
        (5)  |    |
             +4  5+
        Type: SquareMatrix(2,Integer)

(6) -> m: SquareMatrix(2, Integer) := matrix[[1,2],[4,5]]
             +1  2+
        (6)  |    |
             +4  5+
        Type: SquareMatrix(2,Integer)

(7) -> Tm: CT := m
             +1  2+
        (7)  |    |
             +4  5+
        Type: CartesianTensor(1,2,Integer)

(8) -> n: SquareMatrix(2, Integer) := matrix[[2,3],[0,1]]
             +2  3+
        (8)  |    |
             +0  1+
        Type: SquareMatrix(2,Integer)

(9) -> Tn: CT := n
             +2  3+
        (9)  |    |
             +0  1+
        Type: CartesianTensor(1,2,Integer)
 
(10) -> t1: CT := [2,3]
             (10)  [2,3]
        Type: CartesianTensor(1,2,Integer)

(11) -> rank t1
             (11)  1
        Type: PositiveInteger

(12) -> t2: CT := [t1,t1]
             +2  3+
       (12)  |    |
             +2  3+
       Type: CartesianTensor(1,2,Integer)

(13) -> t3: CT := [t2,t2]
             +2  3+ +2  3+
      (13)  [|    |,|    |]
             +2  3+ +2  3+
       Type: CartesianTensor(1,2,Integer)

(14) -> tt: CT := [t3,t3]; tt := [tt,tt]
             ++2  3+  +2  3++ ++2  3+  +2  3++
             ||    |  |    || ||    |  |    ||
             |+2  3+  +2  3+| |+2  3+  +2  3+|
      (14)  [|              |,|              |]
             |+2  3+  +2  3+| |+2  3+  +2  3+|
             ||    |  |    || ||    |  |    ||
             ++2  3+  +2  3++ ++2  3+  +2  3++
       Type: CartesianTensor(1,2,Integer)
 
(15) -> rank tt
 (15)  5
         Type: PositiveInteger

(16) -> Tmn := product(Tm,Tn)
             ++2  3+    +4  6+ +
             ||    |    |    | |
             |+0  1+    +0  2+ |
       (16)  |                 |
             |+8  12+  +10  15+|
             ||     |  |      ||
             ++0  4 +  +0   5 ++
         Type: CartesianTensor(1,2,Integer)

(17) -> Tmv := contract(Tm,2,Tv,1)
 >> Error detected within library code:
             Improper index for contraction
(17) -> Tm*Tv
             +Tv   2Tv+
       (17)  |        |
             +4Tv  5Tv+
       Type: CartesianTensor(1,2,Polynomial Integer)

(18) -> Tmv = m * v
 (18)  [Tmv,Tmv]= [11,32]
             Type: Equation DirectProduct(2,Polynomial Integer)

(19) -> t0()
 (19)  8
             Type: PositiveInteger

(20) -> t1(1+1)
 (20)  3
             Type: PositiveInteger

(21) -> t2(2,1)
 (21)  2
             Type: PositiveInteger

(22) -> t3(2,1,2)
 (22)  3
             Type: PositiveInteger

(23) -> Tmn(2,1,2,1)
 (23)  0
             Type: NonNegativeInteger

(24) -> t0[]
 (24)  8
             Type: PositiveInteger

(25) -> t1[2]
 (25)  3
             Type: PositiveInteger

(26) -> t2[2,1]
 (26)  2
             Type: PositiveInteger

(27) -> t3[2,1,2]
 (27)  3
             Type: PositiveInteger

(28) -> Tmn[2,1,2,1]
 (28)  0
             Type: NonNegativeInteger

(29) -> cTmn := contract(Tmn,1,2)
             +12  18+
       (29)  |      |
             +0   6 +
        Type: CartesianTensor(1,2,Integer)

(30) -> trace(m) * n
             +12  18+
       (30)  |      |
             +0   6 +
       Type: SquareMatrix(2,Integer)

(31) -> contract(Tmn,1,2) = trace(m) * n
             +12  18+  +12  18+
       (31)  |      |= |      |
             +0   6 +  +0   6 +
      Type: Equation CartesianTensor(1,2,Integer)

(32) -> contract(Tmn,1,3) = transpose(m) * n
             +2  7 +  +2  7 +
       (32)  |     |= |     |
             +4  11+  +4  11+
        Type: Equation CartesianTensor(1,2,Integer)

(33) -> contract(Tmn,1,4) = transpose(m) * transpose(n)
             +14  4+  +14  4+
       (33)  |     |= |     |
             +19  5+  +19  5+
       Type: Equation CartesianTensor(1,2,Integer)

(34) -> contract(Tmn,2,3) = m * n
             +2  5 +  +2  5 +
       (34)  |     |= |     |
             +8  17+  +8  17+
       Type: Equation CartesianTensor(1,2,Integer)

(35) -> contract(Tmn,2,4) = m * transpose(n)
             +8   2+  +8   2+
       (35)  |     |= |     |
             +23  5+  +23  5+
       Type: Equation CartesianTensor(1,2,Integer)

(36) -> contract(Tmn,3,4) = trace(n) * m
             +3   6 +  +3   6 +
       (36)  |      |= |      |
             +12  15+  +12  15+
       Type: Equation CartesianTensor(1,2,Integer)

(37) -> tTmn := transpose(Tmn,1,3)
             ++2  3 +  +4   6 ++
             ||     |  |      ||
             |+8  12+  +10  15+|
       (37)  |                 |
             |+0  1+    +0  2+ |
             ||    |    |    | |
             ++0  4+    +0  5+ +
       Type: CartesianTensor(1,2,Integer)

(38) -> transpose Tmn
             ++2  8+   +4  10++
             ||    |   |     ||
             |+0  0+   +0  0 +|
       (38)  |                |
             |+3  12+  +6  15+|
             ||     |  |     ||
             ++1  4 +  +2  5 ++
       Type: CartesianTensor(1,2,Integer)

(39) -> transpose Tm=transpose m
             +1  4+  +1  4+
       (39)  |    |= |    |
             +2  5+  +2  5+
       Type: Equation CartesianTensor(1,2,Integer)

(40) -> rTmn := reindex(Tmn,[1,4,2,3])
             ++2  0+   +3  1+ +
             ||    |   |    | |
             |+4  0+   +6  2+ |
       (40)  |                |
             |+8   0+  +12  4+|
             ||     |  |     ||
             ++10  0+  +15  5++
       Type: CartesianTensor(1,2,Integer)

(41) -> tt:=transpose(Tm)*Tn - Tn*transpose(Tm)
             +- 6  - 16+
       (41)  |         |
             + 2    6  +
       Type: CartesianTensor(1,2,Integer)

(42) -> Tv*(tt+Tn)
             +- 4Tv  - 13Tv+
       (42)  |             |
             + 2Tv    7Tv  +
        Type: CartesianTensor(1,2,Polynomial Integer)

(43) -> reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)
             ++46   84 +  +57   114++
             ||        |  |        ||
             |+174  212+  +228  285+|
       (43)  |                      |
             | +18  24+    +17  30+ |
             | |      |    |      | |
             + +57  63+    +63  76+ +
        Type: CartesianTensor(1,2,Integer)

(44) -> delta: CT := kroneckerDelta()
             +1  0+
       (44)  |    |
             +0  1+
        Type: CartesianTensor(1,2,Integer)

(45) -> contract(Tmn, 2, delta, 1) =reindex(Tmn,[1,3,4,2])
             + +2  4+   +0  0++  + +2  4+   +0  0++
             | |    |   |    ||  | |    |   |    ||
             | +3  6+   +1  2+|  | +3  6+   +1  2+|
       (45)  |                |= |                |
             |+8   10+  +0  0+|  |+8   10+  +0  0+|
             ||      |  |    ||  ||      |  |    ||
             ++12  15+  +4  5++  ++12  15+  +4  5++
        Type: Equation CartesianTensor(1,2,Integer)

(46) -> epsilon:CT := leviCivitaSymbol()
             + 0   1+
       (46)  |      |
             +- 1  0+
       Type: CartesianTensor(1,2,Integer)

(47) -> contract(epsilon*Tm*epsilon,1,2) = 2 * determinant m
 (47)  - 6= - 6
             Type: Equation CartesianTensor(1,2,Integer)
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