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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)}.
*: (%, %) -> %
++ s*t 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{t*s = 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{M*v} 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]</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1176878541039946027-25px001.spad
using old system compiler.
GTEN abbreviates domain GradedTensor
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