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+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CartesianTensorXmpTitle}{CartesianTensor}
+\newcommand{\CartesianTensorXmpNumber}{9.7}
+%
+% =====================================================================
+\begin{page}{CartesianTensorXmpPage}{9.7 CartesianTensor}
+% =====================================================================
+\beginscroll
+
+\spadtype{CartesianTensor(i0,dim,R)}
+provides Cartesian tensors with
+components belonging to a commutative ring \spad{R}.
+Tensors can be described as a generalization of vectors and matrices.
+%-% \HDindex{tensor!Cartesian}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+This gives a concise {\it tensor algebra} for multilinear objects
+supported by the \spadtype{CartesianTensor} domain.
+You can form the inner or outer product of any two tensors and you can add
+or subtract tensors with the same number of components.
+Additionally, various forms of traces and transpositions are useful.
+
+The \spadtype{CartesianTensor} constructor allows you to specify the
+minimum index for subscripting.
+In what follows we discuss in detail how to manipulate tensors.
+
+\xtc{
+Here we construct the domain of Cartesian tensors of dimension 2 over the
+integers, with indices starting at 1.
+}{
+\spadpaste{CT := CARTEN(i0 := 1, 2, Integer) \bound{CT i0}}
+}
+
+\subsubsection{Forming tensors}
+
+\labelSpace{2pc}
+\xtc{
+Scalars can be converted to tensors of rank zero.
+}{
+\spadpaste{t0: CT := 8 \free{CT}\bound{t0}}
+}
+\xtc{
+}{
+\spadpaste{rank t0 \free{t0}}
+}
+\xtc{
+Vectors (mathematical direct products, rather than one dimensional array
+structures) can be converted to tensors of rank one.
+}{
+\spadpaste{v: DirectProduct(2, Integer) := directProduct [3,4] \bound{v}}
+}
+\xtc{
+}{
+\spadpaste{Tv: CT := v \free{v CT}\bound{Tv}}
+}
+\xtc{
+Matrices can be converted to tensors of rank two.
+}{
+\spadpaste{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]] \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{Tm: CT := m \bound{Tm}\free{CT m}}
+}
+\xtc{
+}{
+\spadpaste{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]] \bound{n}}
+}
+\xtc{
+}{
+\spadpaste{Tn: CT := n \bound{Tn}\free{CT n}}
+}
+\xtc{
+In general, a tensor of rank \spad{k} can be formed by making a
+list of rank \spad{k-1} tensors or, alternatively, a \spad{k}-deep nested
+list of lists.
+}{
+\spadpaste{t1: CT := [2, 3] \free{CT}\bound{t1}}
+}
+\xtc{
+}{
+\spadpaste{rank t1 \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2: CT := [t1, t1] \free{CT t1}\bound{t2}}
+}
+\xtc{
+}{
+\spadpaste{t3: CT := [t2, t2] \free{CT t2}\bound{t3}}
+}
+\xtc{
+}{
+\spadpaste{tt: CT := [t3, t3]; tt := [tt, tt] \free{CT t3}\bound{tt}}
+}
+\xtc{
+}{
+\spadpaste{rank tt \free{tt}}
+}
+
+\subsubsection{Multiplication}
+%\head{subsection}{Multiplication}{ugxCartenMult}
+
+Given two tensors of rank \spad{k1} and \spad{k2}, the outer
+%-% \HDindex{tensor!outer product}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+\spadfunFrom{product}{CartesianTensor} forms a new tensor of rank
+\spad{k1+k2}.
+\xtc{
+Here
+\texht{$T_{mn}(i,j,k,l) = T_m(i,j) \  T_n(k,l).$}{%
+\spad{Tmn(i,j,k,l) = Tm(i,j)*Tn(k,l).}}
+}{
+\spadpaste{Tmn := product(Tm, Tn)\free{Tm Tn}\bound{Tmn}}
+}
+
+The inner product (\spadfunFrom{contract}{CartesianTensor}) forms a tensor
+%-% \HDindex{tensor!inner product}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+of rank \spad{k1+k2-2}.
+This product generalizes the vector dot product and matrix-vector product
+by summing component products along two indices.
+\xtc{
+Here we sum along the second index of \texht{$T_m$}{\spad{Tm}} and the
+first index of \texht{$T_v$}{\spad{Tv}}.
+Here
+\texht{$T_{mv} = \sum_{j=1}^{\hbox{\tiny\rm dim}} T_m(i,j) \  T_v(j)$}{%
+\spad{Tmv(i) = sum Tm(i,j) * Tv(j) for j in 1..dim.}}
+}{
+\spadpaste{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv}\bound{Tmv}}
+}
+
+The multiplication operator \spadopFrom{*}{CartesianTensor} is scalar
+multiplication or an inner product depending on the ranks of the
+arguments.
+\xtc{
+If either argument is rank zero it is treated as scalar multiplication.
+Otherwise, \spad{a*b} is the inner product summing the last index of
+\spad{a} with the first index of \spad{b}.
+}{
+\spadpaste{Tm*Tv \free{Tm Tv}}
+}
+\xtc{
+This definition is consistent with the inner product on matrices
+and vectors.
+}{
+\spadpaste{Tmv = m * v \free{Tmv m v}}
+}
+
+\subsubsection{Selecting Components}
+%\head{subsection}{Selecting Components}{ugxCartenSelect}
+
+\labelSpace{2pc}
+\xtc{
+For tensors of low rank (that is, four or less), components can be selected
+by applying the tensor to its indices.
+}{
+\spadpaste{t0()      \free{t0}}
+}
+\xtc{
+}{
+\spadpaste{t1(1+1)   \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2(2,1)   \free{t2}}
+}
+\xtc{
+}{
+\spadpaste{t3(2,1,2) \free{t3}}
+}
+\xtc{
+}{
+\spadpaste{Tmn(2,1,2,1) \free{Tmn}}
+}
+\xtc{
+A general indexing mechanism is provided for a list of indices.
+}{
+\spadpaste{t0[]        \free{t0}}
+}
+\xtc{
+}{
+\spadpaste{t1[2]        \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2[2,1]      \free{t2}}
+}
+\xtc{
+The general mechanism works for tensors of arbitrary rank, but is
+somewhat less efficient since the intermediate index list must be created.
+}{
+\spadpaste{t3[2,1,2]    \free{t3}}
+}
+\xtc{
+}{
+\spadpaste{Tmn[2,1,2,1] \free{Tmn}}
+}
+
+\subsubsection{Contraction}
+%\head{subsection}{Contraction}{ugxCartenContract}
+
+A ``contraction'' between two tensors is an inner product, as we have
+%-% \HDindex{tensor!contraction}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+seen above.
+You can also contract a pair of indices of a single tensor.
+This corresponds to a ``trace'' in linear algebra.
+The expression \spad{contract(t,k1,k2)} forms a new tensor by summing the
+diagonal given by indices in position \spad{k1} and \spad{k2}.
+\xtc{
+This is the tensor given by
+\texht{$xT_{mn} = \sum_{k=1}^{\hbox{\tiny\rm dim}} T_{mn}(k,k,i,j).$}{%
+\spad{cTmn(i,j) = sum Tmn(k,k,i,j) for k in 1..dim.}}
+}{
+\spadpaste{cTmn := contract(Tmn,1,2) \free{Tmn}\bound{cTmn}}
+}
+\xtc{
+Since \spad{Tmn} is the outer product of matrix \spad{m} and matrix \spad{n},
+the above is equivalent to this.
+}{
+\spadpaste{trace(m) * n \free{m n}}
+}
+\xtc{
+In this and the next few examples,
+we show all possible contractions of \spad{Tmn} and their matrix
+algebra equivalents.
+}{
+\spadpaste{contract(Tmn,1,2) = trace(m) * n      \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,1,3) = transpose(m) * n  \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,1,4) = transpose(m) * transpose(n) \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,2,3) = m * n             \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,2,4) = m * transpose(n)  \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,3,4) = trace(n) * m      \free{Tmn m n}}
+}
+
+\subsubsection{Transpositions}
+%\head{subsection}{Transpositions}{ugxCartenTranspos}
+
+You can exchange any desired pair of indices using the
+\spadfunFrom{transpose}{CartesianTensor} operation.
+
+\labelSpace{1pc}
+\xtc{
+Here the indices in positions one and three are exchanged, that is,
+\texht{$tT_{mn}(i,j,k,l) = T_{mn}(k,j,i,l).$}{%
+\spad{tTmn(i,j,k,l) = Tmn(k,j,i,l).}}
+}{
+\spadpaste{tTmn := transpose(Tmn,1,3) \free{tTmn}}
+}
+\xtc{
+If no indices are specified, the first and last index are exchanged.
+}{
+\spadpaste{transpose Tmn \free{Tmn}}
+}
+\xtc{
+This is consistent with the matrix transpose.
+}{
+\spadpaste{transpose Tm = transpose m \free{Tm m}}
+}
+
+If a more complicated reordering of the indices is required, then the
+\spadfunFrom{reindex}{CartesianTensor} operation can be used.
+This operation allows the indices to be arbitrarily permuted.
+\xtc{
+This defines
+\texht{$rT_{mn}(i,j,k,l) = \allowbreak T_{mn}(i,l,j,k).$}{%
+\spad{rTmn(i,j,k,l) = Tmn(i,l,j,k).}}
+}{
+\spadpaste{rTmn := reindex(Tmn, [1,4,2,3]) \free{Tmn}\bound{rTmn}}
+}
+
+\subsubsection{Arithmetic}
+%\head{subsection}{Arithmetic}{ugxCartenArith}
+
+\labelSpace{2pc}
+\xtc{
+Tensors of equal rank can be added or subtracted so arithmetic
+expressions can be used to produce new tensors.
+}{
+\spadpaste{tt := transpose(Tm)*Tn - Tn*transpose(Tm) \free{Tm Tn}\bound{tt2}}
+}
+\xtc{
+}{
+\spadpaste{Tv*(tt+Tn) \free{tt2 Tv Tn}}
+}
+\xtc{
+}{
+\spadpaste{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm) \free{Tm Tn}}
+}
+
+\subsubsection{Specific Tensors}
+%\head{subsection}{Specific Tensors}{ugxCartenSpecific}
+
+Two specific tensors have properties which depend only on the
+dimension.
+
+\labelSpace{1pc}
+\xtc{
+The Kronecker delta satisfies
+%-% \HDindex{Kronecker delta}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+\texht{}{%
+\begin{verbatim}
+             +-
+             |   1  if i  = j
+delta(i,j) = |
+             |   0  if i ^= j
+             +-
+\end{verbatim}
+}%
+}{
+\spadpaste{delta:  CT := kroneckerDelta()   \free{CT}\bound{delta}}
+}
+\xtc{
+This can be used to reindex via contraction.
+}{
+\spadpaste{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2]) \free{Tmn delta}}
+}
+
+\xtc{
+The Levi Civita symbol determines the sign of a permutation of indices.
+}{
+\spadpaste{epsilon:CT := leviCivitaSymbol() \free{CT}\bound{epsilon}}
+}
+Here we have:
+\texht{}{%
+\begin{verbatim}
+epsilon(i1,...,idim)
+     = +1  if i1,...,idim is an even permutation of i0,...,i0+dim-1
+     = -1  if i1,...,idim is an  odd permutation of i0,...,i0+dim-1
+     =  0  if i1,...,idim is not   a permutation of i0,...,i0+dim-1
+\end{verbatim}
+}
+\xtc{
+This property can be used to form determinants.
+}{
+\spadpaste{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m \free{epsilon Tm m}}
+}
+
+\subsubsection{Properties of the CartesianTensor domain}
+%\head{subsection}{Properties of the CartesianTensor domain}{ugxCartenProp}
+
+\spadtype{GradedModule(R,E)} denotes ``\spad{E}-graded \spad{R}-module'',
+that is, a collection of \spad{R}-modules indexed by an abelian monoid
+\spad{E.}
+%-% \HDexptypeindex{GradedModule}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+An element \spad{g} of \spad{G[s]} for some specific \spad{s}
+in \spad{E} is said to be an element of \spad{G} with
+\spadfunFrom{degree}{GradedModule} \spad{s}.
+Sums are defined in each module \spad{G[s]} so two elements of
+\spad{G} can be added if they have the same degree.
+Morphisms can be defined and composed by degree to give the
+mathematical category of graded modules.
+
+\spadtype{GradedAlgebra(R,E)} denotes ``\spad{E}-graded \spad{R}-algebra.''
+%-% \HDexptypeindex{GradedAlgebra}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+A graded algebra is a graded module together with a degree preserving
+\spad{R}-bilinear map, called the \spadfunFrom{product}{GradedAlgebra}.
+
+\begin{verbatim}
+degree(product(a,b))= degree(a) + degree(b)
+
+product(r*a,b) = product(a,r*b) = r*product(a,b)
+product(a1+a2,b) = product(a1,b) + product(a2,b)
+product(a,b1+b2) = product(a,b1) + product(a,b2)
+product(a,product(b,c)) = product(product(a,b),c)
+\end{verbatim}
+
+The domain \spadtype{CartesianTensor(i0, dim, R)} belongs
+to the category \spadtype{GradedAlgebra(R, NonNegativeInteger)}.
+The non-negative integer \spadfunFrom{degree}{GradedAlgebra}
+is the tensor rank
+and the graded algebra \spadfunFrom{product}{GradedAlgebra}
+is the tensor outer product.
+The graded module addition captures the notion that
+only tensors of equal rank can be added.
+
+If \spad{V} is a vector space of dimension \spad{dim} over \spad{R},
+then the tensor module \spad{T[k](V)} is defined as
+\begin{verbatim}
+T[0](V) = R
+T[k](V) = T[k-1](V) * V
+\end{verbatim}
+where \spadop{*} denotes the \spad{R}-module tensor
+\spadfunFrom{product}{GradedAlgebra}.
+\spadtype{CartesianTensor(i0,dim,R)} is the graded algebra in which
+the degree \spad{k} module is \spad{T[k](V)}.
+
+\subsubsection{Tensor Calculus}
+%\head{subsection}{Tensor Calculus}{ugxCartenCalculus}
+
+It should be noted here that often tensors are used in the context of
+tensor-valued manifold maps.
+This leads to the notion of covariant and contravariant bases with tensor
+component functions transforming in specific ways under a change of
+coordinates on the manifold.
+This is no more directly supported by the \spadtype{CartesianTensor}
+domain than it is by the \spadtype{Vector} domain.
+However, it is possible to have the components implicitly represent
+component maps by choosing a polynomial or expression type for the
+components.
+In this case, it is up to the user to satisfy any constraints which arise
+on the basis of this interpretation.
+\endscroll
+\autobuttons
+\end{page}
+%