\begin{patch}{CartesianTensorXmpPagePatch1}
\begin{paste}{CartesianTensorXmpPageFull1}{CartesianTensorXmpPageEmpty1}
\pastebutton{CartesianTensorXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)\bound{CT i0 }}
\indentrel{3}\begin{verbatim}
   (1)  CartesianTensor(1,2,Integer)
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty1}
\begin{paste}{CartesianTensorXmpPageEmpty1}{CartesianTensorXmpPagePatch1}
\pastebutton{CartesianTensorXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)\bound{CT i0 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch2}
\begin{paste}{CartesianTensorXmpPageFull2}{CartesianTensorXmpPageEmpty2}
\pastebutton{CartesianTensorXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{t0: CT := 8\free{CT }\bound{t0 }}
\indentrel{3}\begin{verbatim}
   (2)  8
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty2}
\begin{paste}{CartesianTensorXmpPageEmpty2}{CartesianTensorXmpPagePatch2}
\pastebutton{CartesianTensorXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{t0: CT := 8\free{CT }\bound{t0 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch3}
\begin{paste}{CartesianTensorXmpPageFull3}{CartesianTensorXmpPageEmpty3}
\pastebutton{CartesianTensorXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{rank t0\free{t0 }}
\indentrel{3}\begin{verbatim}
   (3)  0
                               Type: NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty3}
\begin{paste}{CartesianTensorXmpPageEmpty3}{CartesianTensorXmpPagePatch3}
\pastebutton{CartesianTensorXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{rank t0\free{t0 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch4}
\begin{paste}{CartesianTensorXmpPageFull4}{CartesianTensorXmpPageEmpty4}
\pastebutton{CartesianTensorXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{v: DirectProduct(2, Integer) := directProduct [3,4]\bound{v }}
\indentrel{3}\begin{verbatim}
   (4)  [3, 4]
                         Type: DirectProduct(2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty4}
\begin{paste}{CartesianTensorXmpPageEmpty4}{CartesianTensorXmpPagePatch4}
\pastebutton{CartesianTensorXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{v: DirectProduct(2, Integer) := directProduct [3,4]\bound{v }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch5}
\begin{paste}{CartesianTensorXmpPageFull5}{CartesianTensorXmpPageEmpty5}
\pastebutton{CartesianTensorXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{Tv: CT := v\free{v CT }\bound{Tv }}
\indentrel{3}\begin{verbatim}
   (5)  [3, 4]
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty5}
\begin{paste}{CartesianTensorXmpPageEmpty5}{CartesianTensorXmpPagePatch5}
\pastebutton{CartesianTensorXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{Tv: CT := v\free{v CT }\bound{Tv }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch6}
\begin{paste}{CartesianTensorXmpPageFull6}{CartesianTensorXmpPageEmpty6}
\pastebutton{CartesianTensorXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]]\bound{m }}
\indentrel{3}\begin{verbatim}
        +1  2+
   (6)  |    |
        +4  5+
                          Type: SquareMatrix(2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty6}
\begin{paste}{CartesianTensorXmpPageEmpty6}{CartesianTensorXmpPagePatch6}
\pastebutton{CartesianTensorXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]]\bound{m }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch7}
\begin{paste}{CartesianTensorXmpPageFull7}{CartesianTensorXmpPageEmpty7}
\pastebutton{CartesianTensorXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{Tm: CT := m\bound{Tm }\free{CT m }}
\indentrel{3}\begin{verbatim}
        +1  2+
   (7)  |    |
        +4  5+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty7}
\begin{paste}{CartesianTensorXmpPageEmpty7}{CartesianTensorXmpPagePatch7}
\pastebutton{CartesianTensorXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{Tm: CT := m\bound{Tm }\free{CT m }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch8}
\begin{paste}{CartesianTensorXmpPageFull8}{CartesianTensorXmpPageEmpty8}
\pastebutton{CartesianTensorXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]]\bound{n }}
\indentrel{3}\begin{verbatim}
        +2  3+
   (8)  |    |
        +0  1+
                          Type: SquareMatrix(2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty8}
\begin{paste}{CartesianTensorXmpPageEmpty8}{CartesianTensorXmpPagePatch8}
\pastebutton{CartesianTensorXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]]\bound{n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch9}
\begin{paste}{CartesianTensorXmpPageFull9}{CartesianTensorXmpPageEmpty9}
\pastebutton{CartesianTensorXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{Tn: CT := n\bound{Tn }\free{CT n }}
\indentrel{3}\begin{verbatim}
        +2  3+
   (9)  |    |
        +0  1+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty9}
\begin{paste}{CartesianTensorXmpPageEmpty9}{CartesianTensorXmpPagePatch9}
\pastebutton{CartesianTensorXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{Tn: CT := n\bound{Tn }\free{CT n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch10}
\begin{paste}{CartesianTensorXmpPageFull10}{CartesianTensorXmpPageEmpty10}
\pastebutton{CartesianTensorXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{t1: CT := [2, 3]\free{CT }\bound{t1 }}
\indentrel{3}\begin{verbatim}
   (10)  [2, 3]
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty10}
\begin{paste}{CartesianTensorXmpPageEmpty10}{CartesianTensorXmpPagePatch10}
\pastebutton{CartesianTensorXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{t1: CT := [2, 3]\free{CT }\bound{t1 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch11}
\begin{paste}{CartesianTensorXmpPageFull11}{CartesianTensorXmpPageEmpty11}
\pastebutton{CartesianTensorXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{rank t1\free{t1 }}
\indentrel{3}\begin{verbatim}
   (11)  1
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty11}
\begin{paste}{CartesianTensorXmpPageEmpty11}{CartesianTensorXmpPagePatch11}
\pastebutton{CartesianTensorXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{rank t1\free{t1 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch12}
\begin{paste}{CartesianTensorXmpPageFull12}{CartesianTensorXmpPageEmpty12}
\pastebutton{CartesianTensorXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{t2: CT := [t1, t1]\free{CT t1 }\bound{t2 }}
\indentrel{3}\begin{verbatim}
         +2  3+
   (12)  |    |
         +2  3+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty12}
\begin{paste}{CartesianTensorXmpPageEmpty12}{CartesianTensorXmpPagePatch12}
\pastebutton{CartesianTensorXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{t2: CT := [t1, t1]\free{CT t1 }\bound{t2 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch13}
\begin{paste}{CartesianTensorXmpPageFull13}{CartesianTensorXmpPageEmpty13}
\pastebutton{CartesianTensorXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{t3: CT := [t2, t2]\free{CT t2 }\bound{t3 }}
\indentrel{3}\begin{verbatim}
          +2  3+  +2  3+
   (13)  [|    |, |    |]
          +2  3+  +2  3+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty13}
\begin{paste}{CartesianTensorXmpPageEmpty13}{CartesianTensorXmpPagePatch13}
\pastebutton{CartesianTensorXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{t3: CT := [t2, t2]\free{CT t2 }\bound{t3 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch14}
\begin{paste}{CartesianTensorXmpPageFull14}{CartesianTensorXmpPageEmpty14}
\pastebutton{CartesianTensorXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]\free{CT t3 }\bound{tt }}
\indentrel{3}\begin{verbatim}
          ++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)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty14}
\begin{paste}{CartesianTensorXmpPageEmpty14}{CartesianTensorXmpPagePatch14}
\pastebutton{CartesianTensorXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]\free{CT t3 }\bound{tt }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch15}
\begin{paste}{CartesianTensorXmpPageFull15}{CartesianTensorXmpPageEmpty15}
\pastebutton{CartesianTensorXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{rank tt\free{tt }}
\indentrel{3}\begin{verbatim}
   (15)  5
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty15}
\begin{paste}{CartesianTensorXmpPageEmpty15}{CartesianTensorXmpPagePatch15}
\pastebutton{CartesianTensorXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{rank tt\free{tt }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch16}
\begin{paste}{CartesianTensorXmpPageFull16}{CartesianTensorXmpPageEmpty16}
\pastebutton{CartesianTensorXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{Tmn := product(Tm, Tn)\free{Tm Tn }\bound{Tmn }}
\indentrel{3}\begin{verbatim}
         ++2  3+    +4  6+ +
         ||    |    |    | |
         |+0  1+    +0  2+ |
   (16)  |                 |
         |+8  12+  +10  15+|
         ||     |  |      ||
         ++0  4 +  +0   5 ++
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty16}
\begin{paste}{CartesianTensorXmpPageEmpty16}{CartesianTensorXmpPagePatch16}
\pastebutton{CartesianTensorXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{Tmn := product(Tm, Tn)\free{Tm Tn }\bound{Tmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch17}
\begin{paste}{CartesianTensorXmpPageFull17}{CartesianTensorXmpPageEmpty17}
\pastebutton{CartesianTensorXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv }\bound{Tmv }}
\indentrel{3}\begin{verbatim}
   (17)  [11, 32]
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty17}
\begin{paste}{CartesianTensorXmpPageEmpty17}{CartesianTensorXmpPagePatch17}
\pastebutton{CartesianTensorXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv }\bound{Tmv }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch18}
\begin{paste}{CartesianTensorXmpPageFull18}{CartesianTensorXmpPageEmpty18}
\pastebutton{CartesianTensorXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{Tm*Tv\free{Tm Tv }}
\indentrel{3}\begin{verbatim}
   (18)  [11, 32]
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty18}
\begin{paste}{CartesianTensorXmpPageEmpty18}{CartesianTensorXmpPagePatch18}
\pastebutton{CartesianTensorXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{Tm*Tv\free{Tm Tv }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch19}
\begin{paste}{CartesianTensorXmpPageFull19}{CartesianTensorXmpPageEmpty19}
\pastebutton{CartesianTensorXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{Tmv = m * v\free{Tmv m v }}
\indentrel{3}\begin{verbatim}
   (19)  [11, 32] = [11, 32]
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty19}
\begin{paste}{CartesianTensorXmpPageEmpty19}{CartesianTensorXmpPagePatch19}
\pastebutton{CartesianTensorXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{Tmv = m * v\free{Tmv m v }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch20}
\begin{paste}{CartesianTensorXmpPageFull20}{CartesianTensorXmpPageEmpty20}
\pastebutton{CartesianTensorXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{t0()\free{t0 }}
\indentrel{3}\begin{verbatim}
   (20)  8
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty20}
\begin{paste}{CartesianTensorXmpPageEmpty20}{CartesianTensorXmpPagePatch20}
\pastebutton{CartesianTensorXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{t0()\free{t0 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch21}
\begin{paste}{CartesianTensorXmpPageFull21}{CartesianTensorXmpPageEmpty21}
\pastebutton{CartesianTensorXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{t1(1+1)\free{t1 }}
\indentrel{3}\begin{verbatim}
   (21)  3
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty21}
\begin{paste}{CartesianTensorXmpPageEmpty21}{CartesianTensorXmpPagePatch21}
\pastebutton{CartesianTensorXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{t1(1+1)\free{t1 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch22}
\begin{paste}{CartesianTensorXmpPageFull22}{CartesianTensorXmpPageEmpty22}
\pastebutton{CartesianTensorXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{t2(2,1)\free{t2 }}
\indentrel{3}\begin{verbatim}
   (22)  2
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty22}
\begin{paste}{CartesianTensorXmpPageEmpty22}{CartesianTensorXmpPagePatch22}
\pastebutton{CartesianTensorXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{t2(2,1)\free{t2 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch23}
\begin{paste}{CartesianTensorXmpPageFull23}{CartesianTensorXmpPageEmpty23}
\pastebutton{CartesianTensorXmpPageFull23}{\hidepaste}
\tab{5}\spadcommand{t3(2,1,2)\free{t3 }}
\indentrel{3}\begin{verbatim}
   (23)  3
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty23}
\begin{paste}{CartesianTensorXmpPageEmpty23}{CartesianTensorXmpPagePatch23}
\pastebutton{CartesianTensorXmpPageEmpty23}{\showpaste}
\tab{5}\spadcommand{t3(2,1,2)\free{t3 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch24}
\begin{paste}{CartesianTensorXmpPageFull24}{CartesianTensorXmpPageEmpty24}
\pastebutton{CartesianTensorXmpPageFull24}{\hidepaste}
\tab{5}\spadcommand{Tmn(2,1,2,1)\free{Tmn }}
\indentrel{3}\begin{verbatim}
   (24)  0
                               Type: NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty24}
\begin{paste}{CartesianTensorXmpPageEmpty24}{CartesianTensorXmpPagePatch24}
\pastebutton{CartesianTensorXmpPageEmpty24}{\showpaste}
\tab{5}\spadcommand{Tmn(2,1,2,1)\free{Tmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch25}
\begin{paste}{CartesianTensorXmpPageFull25}{CartesianTensorXmpPageEmpty25}
\pastebutton{CartesianTensorXmpPageFull25}{\hidepaste}
\tab{5}\spadcommand{t0[]\free{t0 }}
\indentrel{3}\begin{verbatim}
   (25)  8
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty25}
\begin{paste}{CartesianTensorXmpPageEmpty25}{CartesianTensorXmpPagePatch25}
\pastebutton{CartesianTensorXmpPageEmpty25}{\showpaste}
\tab{5}\spadcommand{t0[]\free{t0 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch26}
\begin{paste}{CartesianTensorXmpPageFull26}{CartesianTensorXmpPageEmpty26}
\pastebutton{CartesianTensorXmpPageFull26}{\hidepaste}
\tab{5}\spadcommand{t1[2]\free{t1 }}
\indentrel{3}\begin{verbatim}
   (26)  3
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty26}
\begin{paste}{CartesianTensorXmpPageEmpty26}{CartesianTensorXmpPagePatch26}
\pastebutton{CartesianTensorXmpPageEmpty26}{\showpaste}
\tab{5}\spadcommand{t1[2]\free{t1 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch27}
\begin{paste}{CartesianTensorXmpPageFull27}{CartesianTensorXmpPageEmpty27}
\pastebutton{CartesianTensorXmpPageFull27}{\hidepaste}
\tab{5}\spadcommand{t2[2,1]\free{t2 }}
\indentrel{3}\begin{verbatim}
   (27)  2
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty27}
\begin{paste}{CartesianTensorXmpPageEmpty27}{CartesianTensorXmpPagePatch27}
\pastebutton{CartesianTensorXmpPageEmpty27}{\showpaste}
\tab{5}\spadcommand{t2[2,1]\free{t2 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch28}
\begin{paste}{CartesianTensorXmpPageFull28}{CartesianTensorXmpPageEmpty28}
\pastebutton{CartesianTensorXmpPageFull28}{\hidepaste}
\tab{5}\spadcommand{t3[2,1,2]\free{t3 }}
\indentrel{3}\begin{verbatim}
   (28)  3
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty28}
\begin{paste}{CartesianTensorXmpPageEmpty28}{CartesianTensorXmpPagePatch28}
\pastebutton{CartesianTensorXmpPageEmpty28}{\showpaste}
\tab{5}\spadcommand{t3[2,1,2]\free{t3 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch29}
\begin{paste}{CartesianTensorXmpPageFull29}{CartesianTensorXmpPageEmpty29}
\pastebutton{CartesianTensorXmpPageFull29}{\hidepaste}
\tab{5}\spadcommand{Tmn[2,1,2,1]\free{Tmn }}
\indentrel{3}\begin{verbatim}
   (29)  0
                               Type: NonNegativeInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty29}
\begin{paste}{CartesianTensorXmpPageEmpty29}{CartesianTensorXmpPagePatch29}
\pastebutton{CartesianTensorXmpPageEmpty29}{\showpaste}
\tab{5}\spadcommand{Tmn[2,1,2,1]\free{Tmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch30}
\begin{paste}{CartesianTensorXmpPageFull30}{CartesianTensorXmpPageEmpty30}
\pastebutton{CartesianTensorXmpPageFull30}{\hidepaste}
\tab{5}\spadcommand{cTmn := contract(Tmn,1,2)\free{Tmn }\bound{cTmn }}
\indentrel{3}\begin{verbatim}
         +12  18+
   (30)  |      |
         +0   6 +
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty30}
\begin{paste}{CartesianTensorXmpPageEmpty30}{CartesianTensorXmpPagePatch30}
\pastebutton{CartesianTensorXmpPageEmpty30}{\showpaste}
\tab{5}\spadcommand{cTmn := contract(Tmn,1,2)\free{Tmn }\bound{cTmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch31}
\begin{paste}{CartesianTensorXmpPageFull31}{CartesianTensorXmpPageEmpty31}
\pastebutton{CartesianTensorXmpPageFull31}{\hidepaste}
\tab{5}\spadcommand{trace(m) * n\free{m n }}
\indentrel{3}\begin{verbatim}
         +12  18+
   (31)  |      |
         +0   6 +
                          Type: SquareMatrix(2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty31}
\begin{paste}{CartesianTensorXmpPageEmpty31}{CartesianTensorXmpPagePatch31}
\pastebutton{CartesianTensorXmpPageEmpty31}{\showpaste}
\tab{5}\spadcommand{trace(m) * n\free{m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch32}
\begin{paste}{CartesianTensorXmpPageFull32}{CartesianTensorXmpPageEmpty32}
\pastebutton{CartesianTensorXmpPageFull32}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,1,2) = trace(m) * n\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +12  18+   +12  18+
   (32)  |      | = |      |
         +0   6 +   +0   6 +
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty32}
\begin{paste}{CartesianTensorXmpPageEmpty32}{CartesianTensorXmpPagePatch32}
\pastebutton{CartesianTensorXmpPageEmpty32}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,1,2) = trace(m) * n\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch33}
\begin{paste}{CartesianTensorXmpPageFull33}{CartesianTensorXmpPageEmpty33}
\pastebutton{CartesianTensorXmpPageFull33}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,1,3) = transpose(m) * n\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +2  7 +   +2  7 +
   (33)  |     | = |     |
         +4  11+   +4  11+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty33}
\begin{paste}{CartesianTensorXmpPageEmpty33}{CartesianTensorXmpPagePatch33}
\pastebutton{CartesianTensorXmpPageEmpty33}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,1,3) = transpose(m) * n\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch34}
\begin{paste}{CartesianTensorXmpPageFull34}{CartesianTensorXmpPageEmpty34}
\pastebutton{CartesianTensorXmpPageFull34}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +14  4+   +14  4+
   (34)  |     | = |     |
         +19  5+   +19  5+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty34}
\begin{paste}{CartesianTensorXmpPageEmpty34}{CartesianTensorXmpPagePatch34}
\pastebutton{CartesianTensorXmpPageEmpty34}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch35}
\begin{paste}{CartesianTensorXmpPageFull35}{CartesianTensorXmpPageEmpty35}
\pastebutton{CartesianTensorXmpPageFull35}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,2,3) = m * n\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +2  5 +   +2  5 +
   (35)  |     | = |     |
         +8  17+   +8  17+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty35}
\begin{paste}{CartesianTensorXmpPageEmpty35}{CartesianTensorXmpPagePatch35}
\pastebutton{CartesianTensorXmpPageEmpty35}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,2,3) = m * n\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch36}
\begin{paste}{CartesianTensorXmpPageFull36}{CartesianTensorXmpPageEmpty36}
\pastebutton{CartesianTensorXmpPageFull36}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,2,4) = m * transpose(n)\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +8   2+   +8   2+
   (36)  |     | = |     |
         +23  5+   +23  5+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty36}
\begin{paste}{CartesianTensorXmpPageEmpty36}{CartesianTensorXmpPagePatch36}
\pastebutton{CartesianTensorXmpPageEmpty36}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,2,4) = m * transpose(n)\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch37}
\begin{paste}{CartesianTensorXmpPageFull37}{CartesianTensorXmpPageEmpty37}
\pastebutton{CartesianTensorXmpPageFull37}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn,3,4) = trace(n) * m\free{Tmn m n }}
\indentrel{3}\begin{verbatim}
         +3   6 +   +3   6 +
   (37)  |      | = |      |
         +12  15+   +12  15+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty37}
\begin{paste}{CartesianTensorXmpPageEmpty37}{CartesianTensorXmpPagePatch37}
\pastebutton{CartesianTensorXmpPageEmpty37}{\showpaste}
\tab{5}\spadcommand{contract(Tmn,3,4) = trace(n) * m\free{Tmn m n }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch38}
\begin{paste}{CartesianTensorXmpPageFull38}{CartesianTensorXmpPageEmpty38}
\pastebutton{CartesianTensorXmpPageFull38}{\hidepaste}
\tab{5}\spadcommand{tTmn := transpose(Tmn,1,3)\free{tTmn }}
\indentrel{3}\begin{verbatim}
         ++2  3 +  +4   6 ++
         ||     |  |      ||
         |+8  12+  +10  15+|
   (38)  |                 |
         |+0  1+    +0  2+ |
         ||    |    |    | |
         ++0  4+    +0  5+ +
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty38}
\begin{paste}{CartesianTensorXmpPageEmpty38}{CartesianTensorXmpPagePatch38}
\pastebutton{CartesianTensorXmpPageEmpty38}{\showpaste}
\tab{5}\spadcommand{tTmn := transpose(Tmn,1,3)\free{tTmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch39}
\begin{paste}{CartesianTensorXmpPageFull39}{CartesianTensorXmpPageEmpty39}
\pastebutton{CartesianTensorXmpPageFull39}{\hidepaste}
\tab{5}\spadcommand{transpose Tmn\free{Tmn }}
\indentrel{3}\begin{verbatim}
         ++2  8+   +4  10++
         ||    |   |     ||
         |+0  0+   +0  0 +|
   (39)  |                |
         |+3  12+  +6  15+|
         ||     |  |     ||
         ++1  4 +  +2  5 ++
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty39}
\begin{paste}{CartesianTensorXmpPageEmpty39}{CartesianTensorXmpPagePatch39}
\pastebutton{CartesianTensorXmpPageEmpty39}{\showpaste}
\tab{5}\spadcommand{transpose Tmn\free{Tmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch40}
\begin{paste}{CartesianTensorXmpPageFull40}{CartesianTensorXmpPageEmpty40}
\pastebutton{CartesianTensorXmpPageFull40}{\hidepaste}
\tab{5}\spadcommand{transpose Tm = transpose m\free{Tm m }}
\indentrel{3}\begin{verbatim}
         +1  4+   +1  4+
   (40)  |    | = |    |
         +2  5+   +2  5+
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty40}
\begin{paste}{CartesianTensorXmpPageEmpty40}{CartesianTensorXmpPagePatch40}
\pastebutton{CartesianTensorXmpPageEmpty40}{\showpaste}
\tab{5}\spadcommand{transpose Tm = transpose m\free{Tm m }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch41}
\begin{paste}{CartesianTensorXmpPageFull41}{CartesianTensorXmpPageEmpty41}
\pastebutton{CartesianTensorXmpPageFull41}{\hidepaste}
\tab{5}\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])\free{Tmn }\bound{rTmn }}
\indentrel{3}\begin{verbatim}
         ++2  0+   +3  1+ +
         ||    |   |    | |
         |+4  0+   +6  2+ |
   (41)  |                |
         |+8   0+  +12  4+|
         ||     |  |     ||
         ++10  0+  +15  5++
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty41}
\begin{paste}{CartesianTensorXmpPageEmpty41}{CartesianTensorXmpPagePatch41}
\pastebutton{CartesianTensorXmpPageEmpty41}{\showpaste}
\tab{5}\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])\free{Tmn }\bound{rTmn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch42}
\begin{paste}{CartesianTensorXmpPageFull42}{CartesianTensorXmpPageEmpty42}
\pastebutton{CartesianTensorXmpPageFull42}{\hidepaste}
\tab{5}\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)\free{Tm Tn }\bound{tt2 }}
\indentrel{3}\begin{verbatim}
         +- 6  - 16+
   (42)  |         |
         + 2    6  +
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty42}
\begin{paste}{CartesianTensorXmpPageEmpty42}{CartesianTensorXmpPagePatch42}
\pastebutton{CartesianTensorXmpPageEmpty42}{\showpaste}
\tab{5}\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)\free{Tm Tn }\bound{tt2 }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch43}
\begin{paste}{CartesianTensorXmpPageFull43}{CartesianTensorXmpPageEmpty43}
\pastebutton{CartesianTensorXmpPageFull43}{\hidepaste}
\tab{5}\spadcommand{Tv*(tt+Tn)\free{tt2 Tv Tn }}
\indentrel{3}\begin{verbatim}
   (43)  [- 4, - 11]
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty43}
\begin{paste}{CartesianTensorXmpPageEmpty43}{CartesianTensorXmpPagePatch43}
\pastebutton{CartesianTensorXmpPageEmpty43}{\showpaste}
\tab{5}\spadcommand{Tv*(tt+Tn)\free{tt2 Tv Tn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch44}
\begin{paste}{CartesianTensorXmpPageFull44}{CartesianTensorXmpPageEmpty44}
\pastebutton{CartesianTensorXmpPageFull44}{\hidepaste}
\tab{5}\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)\free{Tm Tn }}
\indentrel{3}\begin{verbatim}
         ++46   84 +  +57   114++
         ||        |  |        ||
         |+174  212+  +228  285+|
   (44)  |                      |
         | +18  24+    +17  30+ |
         | |      |    |      | |
         + +57  63+    +63  76+ +
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty44}
\begin{paste}{CartesianTensorXmpPageEmpty44}{CartesianTensorXmpPagePatch44}
\pastebutton{CartesianTensorXmpPageEmpty44}{\showpaste}
\tab{5}\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)\free{Tm Tn }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch45}
\begin{paste}{CartesianTensorXmpPageFull45}{CartesianTensorXmpPageEmpty45}
\pastebutton{CartesianTensorXmpPageFull45}{\hidepaste}
\tab{5}\spadcommand{delta: CT := kroneckerDelta()\free{CT }\bound{delta }}
\indentrel{3}\begin{verbatim}
         +1  0+
   (45)  |    |
         +0  1+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty45}
\begin{paste}{CartesianTensorXmpPageEmpty45}{CartesianTensorXmpPagePatch45}
\pastebutton{CartesianTensorXmpPageEmpty45}{\showpaste}
\tab{5}\spadcommand{delta: CT := kroneckerDelta()\free{CT }\bound{delta }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch46}
\begin{paste}{CartesianTensorXmpPageFull46}{CartesianTensorXmpPageEmpty46}
\pastebutton{CartesianTensorXmpPageFull46}{\hidepaste}
\tab{5}\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])\free{Tmn delta }}
\indentrel{3}\begin{verbatim}
         + +2  4+   +0  0++   + +2  4+   +0  0++
         | |    |   |    ||   | |    |   |    ||
         | +3  6+   +1  2+|   | +3  6+   +1  2+|
   (46)  |                | = |                |
         |+8   10+  +0  0+|   |+8   10+  +0  0+|
         ||      |  |    ||   ||      |  |    ||
         ++12  15+  +4  5++   ++12  15+  +4  5++
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty46}
\begin{paste}{CartesianTensorXmpPageEmpty46}{CartesianTensorXmpPagePatch46}
\pastebutton{CartesianTensorXmpPageEmpty46}{\showpaste}
\tab{5}\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])\free{Tmn delta }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch47}
\begin{paste}{CartesianTensorXmpPageFull47}{CartesianTensorXmpPageEmpty47}
\pastebutton{CartesianTensorXmpPageFull47}{\hidepaste}
\tab{5}\spadcommand{epsilon:CT := leviCivitaSymbol()\free{CT }\bound{epsilon }}
\indentrel{3}\begin{verbatim}
         + 0   1+
   (47)  |      |
         +- 1  0+
                     Type: CartesianTensor(1,2,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty47}
\begin{paste}{CartesianTensorXmpPageEmpty47}{CartesianTensorXmpPagePatch47}
\pastebutton{CartesianTensorXmpPageEmpty47}{\showpaste}
\tab{5}\spadcommand{epsilon:CT := leviCivitaSymbol()\free{CT }\bound{epsilon }}
\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPagePatch48}
\begin{paste}{CartesianTensorXmpPageFull48}{CartesianTensorXmpPageEmpty48}
\pastebutton{CartesianTensorXmpPageFull48}{\hidepaste}
\tab{5}\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m\free{epsilon Tm m }}
\indentrel{3}\begin{verbatim}
   (48)  - 6 = - 6
           Type: Equation(CartesianTensor(1,2,Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{CartesianTensorXmpPageEmpty48}
\begin{paste}{CartesianTensorXmpPageEmpty48}{CartesianTensorXmpPagePatch48}
\pastebutton{CartesianTensorXmpPageEmpty48}{\showpaste}
\tab{5}\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m\free{epsilon Tm m }}
\end{paste}\end{patch}

