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)abbrev domain GTEN GradedTensor
GradedTensor(n:NonNegativeInteger, m:NonNegativeInteger, R:CommutativeRing,dim:NonNegativeInteger): Join(GradedAlgebra(R, NonNegativeInteger), GradedModule(Integer, NonNegativeInteger), Eltable(Integer,R)) with
        coerce: DirectProduct(dim, R) -> GradedTensor(1,0,R,dim)
            ++ coerce(v) views a vector as a (1,0)-tensor.
        coerce: SquareMatrix(dim, R)  -> GradedTensor(0,2,R,dim)
            ++ coerce(m) views a matrix as a (0,2)-tensor.
        coerce: List R -> GradedTensor(0,1,R,dim)
            ++ coerce([r_1,...,r_dim]) allows tensors to be constructed
            ++ using lists.
        coerce: List % -> GradedTensor(n,m+1,R,dim)
            ++ coerce([t_1,...,t_dim]) allows tensors to be constructed
            ++ using lists.
        rank: % -> DirectProduct(2,NonNegativeInteger)
            ++ rank(t) returns the tensorial rank of (n,m)-tensor t
            ++ [n,m] (that is, the number of contravariant and covariant
            ++ indices).
        elt: (%) -> R
            ++ elt(t) gives the component of a rank 0 tensor.
        elt: (%, Integer, Integer) -> R
            ++ elt(t,i,j) gives a component of a rank (2,0) (1,1) or (0,2)-tensor. E.g.
            ++ T(1,1), T(1,-1), T(-1,-1)
        elt: (%, Integer, Integer, Integer) -> R
            ++ elt(t,i,j,k) gives a component of a rank (3,0),(2,1),(1,2) or (0,3)-tensor.
            ++ E.g. T(1,1,1), T(1,1,-1), etc.
        elt: (%, Integer, Integer, Integer, Integer) -> R
            ++ elt(t,i,j,k,l) gives a component of a rank (4,0), (3,1),(2,2),(1,3) or (0,4)-tensor.
            ++ E.g. T(1,1,1,1), T(1,1,1,-1), etc.
        elt: (%, List Integer) -> R
            ++ elt(t,[i1,...,iN]) gives a component of a rank (n,m)-tensor when n+m=N.
            ++ E.g. T[1,1,1,1,1], T[1,1,1,1,-1], etc.
        -- This specializes the documentation from GradedAlgebra.
        product: (%,%) -> %
            ++ product(s,t) is the outer product of the tensors s and t.
            ++ For example, if \spad{r = product(s,t)} for rank 2 tensors s and t,
            ++ then \spad{r} is a rank 4 tensor given by
            ++     \spad{r(i,j,k,l) = s(i,j)t(k,l)}.
        : (%, %) -> %
            ++ st is the inner product of the tensors s and t which contracts
            ++ the last index of s with the first index of t, i.e.
            ++     \spad{ts = contract(t,rank t, s, 1)}
            ++     \spad{t*s = sum(k=1..dim, t[i1,..,iN,-k]*s[k,j1,..,jM])}
            ++ This is compatible with the use of \spad{Mv} to denote
            ++ the matrix-vector inner product.
        contract:  (%, Integer, %, Integer) -> %
            ++ contract(t,i,s,j) is the inner product of tenors s and t
            ++     \spad{r(i1,i2,...,in,j1,j2,...jm) = sum(h=1..dim,s(i1,i2,ii=-h,...,in)t(j1,j2,jj=h,...,jm))}.
        contract:  (%, Integer, Integer)    -> %
            ++ contract(t,i,j) is the contraction of tensor t which
            ++     \spad{r(i1,i2,...,in) = sum(h=1..dim,s(i1,i2,ii=-h,...,ij=h,...,in))}.
        transpose: % -> %
            ++ transpose(t) exchanges the first and last indices of t.
            ++     \spad{r(i,...,l) = t(l,...,i)}.
        transpose: (%, Integer, Integer) -> %
            ++ transpose(t,i,j) exchanges the \spad{i}-th and \spad{j}-th indices of t.
            ++     \spad{r(...,i,...,j,...) = t(...,j,...,i,...)}.
        reindex: (%, List Integer) -> %
            ++ reindex(t,[i1,...,in]) permutes the indices of t.
            ++     \spad{r(j1,j2,...,jn) = t(ji1,ji2,...,jin)}.
        kroneckerDelta:  () -> GradedTensor(1,1,R,dim)
            ++ kroneckerDelta() is the rank (1,1)-tensor defined by
            ++    \spad{kroneckerDelta()(i,j)}
            ++       \spad{= 1  if i = j}
            ++       \spad{= 0 if  i \~= j}
        leviCivitaSymbol: () -> GradedTensor(0,dim,R,dim)
            ++ leviCivitaSymbol() is the rank (0,\spad{dim})-tensor defined by
            ++ \spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1}
            ++ if \spad{i1,...,idim} is an even/is nota /is an odd permutation
            ++ of \spad{minix,...,minix+dim-1}.
        ravel:     % -> List R
            ++ ravel(t) produces a list of components from a tensor such that
            ++   \spad{unravel(ravel(t)) = t}.
        unravel:   List R -> %
            ++ unravel(t) produces a tensor from a list of
            ++ components such that
            ++   \spad{unravel(ravel(t)) = t}.
        sample:    () -> %
            ++ sample() returns an object of type %.
  == add
    Rep == CartesianTensor(1,dim,R)
    -- exports
    rank(t:%):DirectProduct(2,NonNegativeInteger)==directProduct [n,m]

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Overfull \hbox (46.44412pt too wide) in paragraph at lines 122--203 []\T1/cmr/m/n/12 )abbrev do-main GTEN Grad-edTen-sor Grad-edTen-sor(n:NonNegati veInteger, m:NonNegativeInteger, R:CommutativeRing,dim:NonNegativeInteger):

Overfull \hbox (100.47577pt too wide) in paragraph at lines 122--203 \T1/cmr/m/n/12 sor(0,2,R,dim) ++ co-erce(m) views a ma-trix as a (0,2)-tensor. co-erce: List R -> Grad-edTen-sor(0,1,R,dim) ++ coerce([r$[]\OML/cmm/m/it/12 ; :::; r[]im\OT1/cmr/m/n/12 ])\OML/cmm/m/it/12 allowstensorstobeconstructed \OT1/ cmr/m/n/12 +

Overfull \hbox (27.92737pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +\OML/cmm/m/it/12 usinglists:coerce \OT1/cmr/m/n/12 : \OML/cmm/ m/it/12 List \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 coerce\OT1/cmr/m/n/12 ([\OML/c mm/m/it/12 t[]; :::; t[]im\OT1/cmr/m/n/12 ])\OML/cmm/m/it/12 allowstensorstobec onstructed \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 usinglists:rank \OT1/cmr/m/n/12 : + + \OML/cmm/m/it/12 rank\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t\OT1/cmr/m/n/12 ) \OML/cmm/m/it/12 returnsthetensorialrankof\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 n; m\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^@

Overfull \hbox (149.74619pt too wide) in paragraph at lines 122--203 \OML/cmm/m/it/12 elt\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; i; j\OT1/cmr/m/n/12 )\ OML/cmm/m/it/12 givesacomponentofarank\OT1/cmr/m/n/12 (2\OML/cmm/m/it/12 ; \OT1 /cmr/m/n/12 0)(1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 or\OT1/cm r/m/n/12 (0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2) \OMS/cmsy/m/n/12 ^^@ \OML/cmm/ m/it/12 tensor:E:g: \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 T\OT1/cmr/m/n/12 (1\OML /cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; T\OT1/cmr/m/n/12 (1\OML/cmm /m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; T\OT1/cmr/m /n/12 (\OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 elt \OT1/cmr/m/n/12 : (+ + \OML/cmm/m/it /12 elt\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; i; j; k\OT1/cmr/m/n/12 )\OML/cmm/m/ it/12 givesacomponentofarank\OT1/cmr/m/n/12 (3\OML/cmm/m/it/12 ; \OT1/cmr/m/n/1 2 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (2\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\ OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/c mm/m/it/12 or\OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 3) \OMS/cmsy/ m/n/12 ^^@

Overfull \hbox (145.59494pt too wide) in paragraph at lines 122--203 \OML/cmm/m/it/12 tensor: \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 E:g:T\OT1/cmr/m/n/ 12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\ OML/cmm/m/it/12 ; T\OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/c mm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; etc:elt \ OT1/cmr/m/n/12 : (+ + \OML/cmm/m/it/12 elt\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; i; j; k; l\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 givesacomponentofarank\OT1/cmr/m/n/ 12 (4\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 0)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (3 \OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (2\OML/ cmm/m/it/12 ; \OT1/cmr/m/n/12 2)\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 (1\OML/cmm/m /it/12 ; \OT1/cmr/m/n/12 3)\OML/cmm/m/it/12 or\OT1/cmr/m/n/12 (0\OML/cmm/m/it/1 2 ; \OT1/cmr/m/n/12 4) \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 tensor: \OT1/cmr/m /n/12 + +\OML/cmm/m/it/12 E:g:T\OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/ n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1) \OML/cmm/m/it/12 ; T\OT1/cmr/m/n/12 (1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/ cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cmr/ m/n/12 1)\OML/cmm/m/it/12 ; etc:elt \OT1/cmr/m/n/12 :

Overfull \hbox (236.17693pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 (+ + \OML/cmm/m/it/12 elt\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; \ OT1/cmr/m/n/12 [\OML/cmm/m/it/12 i\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; :::; iN\O T1/cmr/m/n/12 ])\OML/cmm/m/it/12 givesacomponentofarank\OT1/cmr/m/n/12 (\OML/cm m/m/it/12 n; m\OT1/cmr/m/n/12 ) \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 tensorwhe nn \OT1/cmr/m/n/12 + \OML/cmm/m/it/12 m \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 N: \ OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 E:g:T\OT1/cmr/m/n/12 [1\OML/cmm/m/it/12 ; \O T1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/ m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1]\OML/cmm/m/it/12 ; T\OT1/cmr/m/n/1 2 [1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OM L/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; \OMS/cmsy/m/n/12 ^^@\OT1/cm r/m/n/12 1]\OML/cmm/m/it/12 ; etc: \OMS/cmsy/m/n/12 ^^@ ^^@\OML/cmm/m/it/12 Thi sspecializesthedocumentationfromGradedAlgebra:product \OT1/cmr/m/n/12 :

Overfull \hbox (40.319pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 (+ + \OML/cmm/m/it/12 product\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 s; t\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 istheouterproductofthetensorssandt: \OT1/ cmr/m/n/12 + +\OML/cmm/m/it/12 Forexample; if[]forrank\OT1/cmr/m/n/12 2\OML/cmm /m/it/12 tensorssandt; \OT1/cmr/m/n/12 + + \OML/cmm/m/it/12 thenrisarank\OT1/cm r/m/n/12 4\OML/cmm/m/it/12 tensorgivenby \OT1/cmr/m/n/12 +

Overfull \hbox (0.12308pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +[] + +[] + +\OML/cmm/m/it/12 Thisiscompatiblewiththeuseof[]tod enote \OT1/cmr/m/n/12 +

Overfull \hbox (344.44069pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +\OML/cmm/m/it/12 thematrix \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/ 12 vectorinnerproduct:contract \OT1/cmr/m/n/12 : (+ + \OML/cmm/m/it/12 contract \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; i; s; j\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 i stheinnerproductoftenorssandt \OT1/cmr/m/n/12 + +[]\OML/cmm/m/it/12 :contract \ OT1/cmr/m/n/12 :

Overfull \hbox (30.7633pt too wide) in paragraph at lines 122--203 \OML/cmm/m/it/12 transpose\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t\OT1/cmr/m/n/12 )\ OML/cmm/m/it/12 exchangesthefirstandlastindicesoft: \OT1/cmr/m/n/12 + +[]\OML/c mm/m/it/12 :transpose \OT1/cmr/m/n/12 : (+ + \OML/cmm/m/it/12 transpose\OT1/cmr /m/n/12 (\OML/cmm/m/it/12 t; i; j\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 exchangesthe i \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 thandj \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/ it/12 thindicesoft: \OT1/cmr/m/n/12 +

Overfull \hbox (43.71811pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 :reindex \OT1/cmr/m/n/12 : (+ + \OML/cmm/m/ it/12 reindex\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t; \OT1/cmr/m/n/12 [\OML/cmm/m/i t/12 i\OT1/cmr/m/n/12 1\OML/cmm/m/it/12 ; :::; in\OT1/cmr/m/n/12 ])\OML/cmm/m/i t/12 permutestheindicesoft: \OT1/cmr/m/n/12 + +[]\OML/cmm/m/it/12 :kroneckerDel ta \OT1/cmr/m/n/12 :

Overfull \hbox (190.21642pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 leviCivitaSymbol \OT1/cmr/m/n/12 : ()\OMS/c msy/m/n/12 ^^@ \OML/cmm/m/it/12 > GradedTensor\OT1/cmr/m/n/12 (0\OML/cmm/m/it/1 2 ; dim; R; dim\OT1/cmr/m/n/12 ) + +\OML/cmm/m/it/12 leviCivitaSymbol\OT1/cmr/m /n/12 ()\OML/cmm/m/it/12 istherank\OT1/cmr/m/n/12 (0\OML/cmm/m/it/12 ; []\OT1/c mr/m/n/12 ) \OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 tensordefinedby \OT1/cmr/m/n/ 12 + +[] +

Overfull \hbox (85.94875pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +\OML/cmm/m/it/12 if[]isaneven=isnota=isanoddpermutation \OT1/c mr/m/n/12 + +\OML/cmm/m/it/12 of[]:ravel \OT1/cmr/m/n/12 : + + \OML/cmm/m/it/12 ravel\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 prod ucesalistofcomponentsfromatensorsuchthat \OT1/cmr/m/n/12 +

Overfull \hbox (31.33415pt too wide) in paragraph at lines 122--203 \OT1/cmr/m/n/12 +[]\OML/cmm/m/it/12 :unravel \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 ListR\OMS/cmsy/m/n/12 ^^@ \OML/cmm/m/it/12 > \OT1/cmr/m/n/12 + + \OML/cmm/m/it /12 unravel\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 producesatensorfromalistof \OT1/cmr/m/n/12 + +\OML/cmm/m/it/12 componentssucht hat \OT1/cmr/m/n/12 + +[]\OML/cmm/m/it/12 :sample \OT1/cmr/m/n/12 : [1] [2] (./4903831854760167283-16.0px.aux) ) (see the transcript file for additional information) Output written on 4903831854760167283-16.0px.dvi (2 pages, 5992 bytes). Transcript written on 4903831854760167283-16.0px.log.

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