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# Tensor

## Introduction to Tensor

- Indicial Tensor Manipulation package. It may be loaded by LOADFILE("itensr"); A manual for the Tensor packages is available in share/tensor.descr. A demo is available by DEMO("itenso.dem1"); (and additional demos are in ("itenso.dem2"), ("itenso.dem3") and following).

- There are two tensor packages in MACSYMA, CTENSR and ITENSR. CTENSR is Component Tensor Manipulation, and may be accessed with LOAD(CTENSR); . ITENSR is Indicial Tensor Manipulation, and is loaded by doing LOAD(ITENSR); A manual for CTENSR AND ITENSR is available from the LCS Publications Office. Request MIT/LCS/TM-167. In addition, demos exist on the TENSOR; directory under the filenames CTENSO DEMO1, DEMO2, etc. and ITENSO DEMO1, DEMO2, etc. Do DEMO("ctenso.dem1"); or DEMO("itenso.dem2"); Send bugs or comments to RP or TENSOR.

## Definitions for Tensor

Function: CANFORM (exp)
  [Tensor Package] Simplifies exp by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If ALLSYM is TRUE then all indices are assumed symmetric, otherwise symmetry information provided by DECSYM declarations will be used. The dummy indices are renamed in the same manner as in the RENAME function. When CANFORM is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling RENAME on the expression first. Also see the example under DECSYM. Note: CANFORM may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result.

Function: CANTEN (exp)
  [Tensor Package] Simplifies exp by renaming (see RENAME) and permuting dummy indices. CANTEN is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if CANFORM is not capable of carrying out the required simplification.

Function: CARG (exp)
  returns the argument (phase angle) of exp. Due to the conventions and restrictions, principal value cannot be guaranteed unless exp is numeric.

Variable: COUNTER
  default: [1] determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option DUMMYX[#].

Function: DEFCON (tensor1,<tensor2,tensor3>)
  gives tensor1 the property that the contraction of a product of tensor1 and tensor2 results in tensor3 with the appropriate indices. If only one argument, tensor1, is given, then the contraction of the product of tensor1 with any indexed object having the appropriate indices (say tensor) will yield an indexed object with that name, i.e.tensor, and with a new set of indices reflecting the contractions performed. For example, if METRIC: G, then DEFCON(G) will implement the raising and lowering of indices through contraction with the metric tensor. More than one DEFCON can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. CONTRACTIONS is a list of those indexed objects which have been given contraction properties with DEFCON.

Function: FLUSH (exp,tensor1,tensor2,...)
  Tensor Package - will set to zero, in exp, all occurrences of the tensori that have no derivative indices.

Function: FLUSHD (exp,tensor1,tensor2,...)
  Tensor Package - will set to zero, in exp, all occurrences of the tensori that have derivative indices.

Function: FLUSHND (exp,tensor,n)
  Tensor Package - will set to zero, in exp, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.

      (%i1) SHOW(A([I],[J,R],K,R)+A([I],[J,R,S],K,R,S));
                                     J R S      J R
      (%o1)                          A        + A
                                     I,K R S    I,K R
      (%i2) SHOW(FLUSHND(%o1,A,3));
                                           J R
      (%o2)                                A
                                           I,K R

Function: KDELTA (L1,L2)
  is the generalized Kronecker delta function defined in the Tensor package with L1 the list of covariant indices and L2 the list of contravariant indices. KDELTA([i],[j]) returns the ordinary Kronecker delta. The command EV(EXP,KDELTA) causes the evaluation of an expression containing KDELTA([],[]) to the dimension of the manifold.

Function: LC (L)
  is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated.

Function: LORENTZ (exp)
  imposes the Lorentz condition by substituting 0 for all indexed objects in exp that have a derivative index identical to a contravariant index.

Function: MAKEBOX (exp)
  will display exp in the same manner as SHOW; however, any tensor d'Alembertian occurring in exp will be indicated using the symbol []. For example, []P([M],[N]) represents G([],[I,J])*P([M],[N],I,J).

Function: METRIC (G)
  specifies the metric by assigning the variable METRIC:G; in addition, the contraction properties of the metric G are set up by executing the commands DEFCON(G), DEFCON(G,G,KDELTA). The variable METRIC, default: [], is bound to the metric, assigned by the METRIC(g) command.

Function: NTERMSG ()
  gives the user a quick picture of the "size" of the Einstein tensor. It returns a list of pairs whose second elements give the number of terms in the components specified by the first elements.

Function: NTERMSRCI ()
  returns a list of pairs, whose second elements give the number of terms in the RICCI component specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.

Function: NZETA (Z)
  returns the complex value of the Plasma Dispersion Function for complex Z.

      NZETAR(Z) ==> REALPART(NZETA(Z))

  NZETAI(Z) returns IMAGPART(NZETA(Z)). This function is related to the complex error function by

      NZETA(Z) = %I*SQRT(%PI)*EXP(-Z^2)*(1-ERF(-%I*Z)).

Function: RAISERIEMANN (dis)
  returns the contravariant components of the Riemann curvature tensor as array elements UR[I,J,K,L]. These are displayed if dis is TRUE.

Variable: RATEINSTEIN
  default: [] - if TRUE rational simplification will be performed on the non-zero components of Einstein tensors; if FACRAT:TRUE then the components will also be factored.

Variable: RATRIEMAN
  - This switch has been renamed RATRIEMANN.

Variable: RATRIEMANN
  default: [] - one of the switches which controls simplification of Riemann tensors; if TRUE, then rational simplification will be done; if FACRAT:TRUE then each of the components will also be factored.

Function: REMCON (tensor1,tensor2,...)
  removes all the contraction properties from the tensori. REMCON(ALL) removes all contraction properties from all indexed objects.

Function: RICCICOM (dis)
  Tensor package) This function first computes the covariant components LR[i,j] of the Ricci tensor (LR is a mnemonic for "lower Ricci"). Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument to RICCICOM is TRUE, then these mixed components, RICCI[i,j] (the index i is covariant and the index j is contravariant), will be displayed directly. Otherwise, RICCICOM(FALSE) will simply compute the entries of the array RICCI[i,j] without displaying the results.

Function: RINVARIANT ()
  Tensor package) forms the invariant obtained by contracting the tensors

      R[i,j,k,l]*UR[i,j,k,l].

  This object is not

  automatically simplified since it can be very large.

Function: SCURVATURE ()
  returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.

Function: SETUP ()
  this has been renamed to TSETUP(); Sets up a metric for Tensor calculations.

Function: WEYL (dis)
  computes the Weyl conformal tensor. If the argument dis is TRUE, the non-zero components W[I,J,K,L] will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch RATWEYL is set to TRUE, then the components will be rationally simplified; if FACRAT is TRUE then the results will be factored as well.

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