tensor(deprecated)/contract - Help

tensor

 contract
 contract a tensor over one or more pairs of indices

 Calling Sequence contract(A, [i1,i2], ...)

Parameters

 A - tensor_type object of rank > 1 [i1, i2] - pair of indices of opposite index character to be contracted over given as a list. There must be at least one pair in the call, and the total number of pairs cannot exceed rank(A)/2.

Description

Important: The tensor package has been deprecated. Use the superseding command DifferentialGeometry[Tensor][ContractIndices].

 • The function contract(A, [i1,i2], [i3,i4], ...) computes the contraction of the tensor A over the pairs of indices i1 and i2, i3 and i4, etc.
 • There must not be any duplicates in the given indices (it is impossible to contract over a single index more than once) and the indices of each pair must be of opposite covariant-contravariant character.
 • The return value is the resultant tensor_type object of rank equal to rank(A) - 2 * (# of pairs in the call).
 • Simplification:  This routine uses the tensor/prod/simp routine for simplification purposes.  The simplification routine is applied to each component of the result after it is computed.  By default, tensor/prod/simp is initialized to the tensor/simp routine. It is recommended that the tensor/prod/simp routine be customized to suit the needs of the particular problem.
 • This function is part of the tensor package, and so can be used in the form contract(..) only after performing the command with(tensor), or with(tensor,contract).  This function can always be accessed in the long form tensor[contract](..).

Examples

Important: The tensor package has been deprecated. Use the superseding command DifferentialGeometry[Tensor][ContractIndices].

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

Create a rank-2 tensor and contract it to form a scalar (the trace)

 > $T≔\mathrm{create}\left(\left[1,-1\right],\mathrm{array}\left(\left[\left[w,x,0\right],\left[y,z,0\right],\left[0,{y}^{2},xyw\right]\right]\right)\right)$
 ${T}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{ccc}{w}& {x}& {0}\\ {y}& {z}& {0}\\ {0}& {{y}}^{{2}}& {x}{}{y}{}{w}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{-1}\right]\right]\right)$ (1)
 > $\mathrm{contract}\left(T,\left[1,2\right]\right)$
 ${table}{}\left(\left[{\mathrm{compts}}{=}{x}{}{y}{}{w}{+}{w}{+}{z}{,}{\mathrm{index_char}}{=}\left[\right]\right]\right)$ (2)