\begin{spad}
)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]
\end{spad}
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l.203 \end{spad}
\newpage
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[]\T1/cmr/m/n/12 )abbrev do-main GTEN Grad-edTen-sor Grad-edTen-sor(n:NonNegati
veInteger, m:NonNegativeInteger, R:CommutativeRing,dim:NonNegativeInteger):
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\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 +
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\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 ^^@
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\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 ^^@
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\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 :
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\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 :
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\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 +
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\OT1/cmr/m/n/12 +[] + +[] + +\OML/cmm/m/it/12 Thisiscompatiblewiththeuseof[]tod
enote \OT1/cmr/m/n/12 +
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\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 :
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\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 +
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\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 :
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\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 + +[] +
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\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 +
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\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)
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