MathTensor

*To*: mathgroup at yoda.physics.unc.edu*Subject*: MathTensor*From*: Steven M. Christensen <mathtensor>*Date*: Thu, 20 Aug 92 03:32:19 EDT

MathGroupers: This is our periodic posting of information on MathTensor which is now in version 2.1. I would appreciate it if you would pass this onto any friends, colleagues, or others who might be interested. Sales of MathTensor indirectly support our other efforts with Mathematica like MathGroup. Thanks. Steve Christensen ------------cut here-------------------------------- MathTensor(tm) Information Sheet Valid after May 15, 1992 Hardware Requirements: MathTensor requires approximately 2 megabytes of disk space. It is recommended that a workstation have at least 8 megabytes of RAM memory. The Macintosh version runs on machines with 8 megabytes or more. 386 versions require at least 5 megabytes of memory, but more is HIGHLY recommended. The Windows 3.1 version works with 8 megabytes of RAM or more. Software Requirements: MathTensor requires Mathematica 1.2, 2.0, or greater. Contact: MathSolutions, Inc. P.O. Box 16175 Chapel Hill, NC 27516 Telephone/Answering Machine/FAX: 919-967-9853 Email: mathtensor at wri.com * MathTensor Product Price List: (Right-to-Use License) Retail Upgrades/yr Media/Shipping Microcomputers: $500 $150 $50 Workstations: $750 $200 $50 Servers: $3000 $1000 $50 (one copy) Extra Manuals: $25 Federal Express Delivery add: $35.00 For definitions of Microcomputers, Workstations and Servers, see the next page. Educational discount available only in the USA. Upgrades and updates are only sent to users with Upgrade Contracts. Volume discounts available. ** North Carolina residents add 6% sales tax. ** Support contracts will be available. ** Purchase orders are accepted. ** Payment is due upon receipt of software shipment. ** Prices, availability and details of functionality of any products or services mentioned on this page are subject to change without notice. ** No credit card orders Availability: MathTensor is available in 1/4" tape (QIC-24) tar format, MS-DOS 3.5" floppies, Macintosh 800K floppies, and Sun 3.5" bar format floppies. Other formats will be made available as needed but may require more time for shipment. Copy Protection: MathTensor, like Mathematica, requires a password on UNIX machines. This password must be obtained from MathSolutions, Inc. Authors: Leonard Parker, Ph.D. and Steven M. Christensen, Ph.D., founders of MathSolutions, Inc. ----------------------- Machine Definitions ----------------------- Microcomputers: Macintosh 386/486 MS-DOS 386/486 Windows 3.0 or greater Workstations: (contact MathSolutions for new machine questions) 386 Unix Data General (AViiON) DEC (VAXstation II, 2000, 3100, 3200, 3520, 3540; MicroVAX II, 2000; VAX11/730; VAX11/750, 11/780, 11/785; MicroVAX 3300, 3400; VAX 4000, 8200, 8250, 8300, 8350; MicroVAX 3500, 3600, 3800, 3900; DECstation 2100, 3100; DECstation 5000) Hewlett-Packard/Apollo (9000 Series 300, 9000 Series 400; 9000 Series 800-835; Domain 2500-4500) IBM RISC System/6000 (Powerstation 320, 520, 530, 730; Powerserver 320, 520, 530) MIPS (RS2030, Magnum 3000, RC2030), NeXT (all models) Silicon Graphics (Personal IRIS 4D/20-35, Personal IRIS 4D/85, 4D/210-220) Sony NEWS (all models) Sun (Sun-3, Sun 386i, SLC, IPC, Sun-4 (all models), SPARCstation 1, 1+ , 2, SPARCserver 2) Servers/Multi-Processor: Convex (C1, C2) Data General (Multi-processor AViiON) DEC (VAX 6210, 6220, 6310, 6320, 64X0, 65X0, 8530, 8550, 8600, 8650, 8700, 8810; VAX 6230, 6240, 6330, 6340, 6350, 8800, 8820; VAX 6360, 8830, 8840, 9000; DECsystem 3100, 5000, 5400, 5500, 5800) Hewlett-Packard/Apollo (9000 Series 840-850, 9000 Series 855-870/200; Series 10000), IBM RISC System/600 (Powerserver 540, 930) MIPS (RS3240, RC3260, M/2000, RC6280) Silicon Graphics (Power Series 4D/240-280), Sun (SPARCstation 330, 470, SPARCserver 330, 470, 490, all MP machines) Some of the users of MathTensor can be found at: Wake Forest University The University of Maryland Insituto de Fisica Fundamental - Madrid, Spain City College of New York The University of Winnipeg - Canada Stanford University Caltech The University of California - Santa Barbara Texas A&M University The University of North Carolina - Chapel Hill Universitat Konstanz - Germany Schlumberger KK - Japan Sumitomo Corporation - Japan Oakland University Louisiana Tech University Polaroid Lawrence Livermore Labs Los Alamos Labs The University of New Brunswick - Canada United Technologies Research Center Universitat Hannover - Germany Reed College Yale University The University of Bergen - Norway Cotton, Inc. NASA - Langley NASA - Goddard National Center for Supercomputing Applications Wolfram Research The University of Wisconsin - Milwaukee The Hebrew University of Jerusalem - Israel Martin Marietta Cornell University Central Connecticut State University Universite de Liege - Belgium Pune University - India Cal State University - Fullerton Utah State University Southwestern University - Texas University of South Florida Deutsches Klimarenzentrum - Hamburg, Germany University of Oklahoma - Norman Montana State University University of Chicago University of Milan - Italy Wellesley College University of San Francisco University of Cologne - Germany University of California - San Diego University of Berne - Switzerland University d. Bundeswehr - Hamburg, Germany University of Thessaloniki - Greece MathTensor has been mentioned in articles in MacWorld, InfoWorld, the Mathematica Journal, Computers In Physics, Science, and PC Magazine. "The idea of packages and special tools has been taken to a kind of modern limit in the form of MathTensor, a tensor- analysis package built on Mathematica." - Richard Crandall, Howard Vollum Professor of Science, Reed College and Chief Scientist, NeXT Computer, Inc., from Computers in Physics, Nov/Dec 1991. "I would like to take this opportunity to tell you how pleased I have been with MathTensor. I have been using it to study higher-dimensional cosmological solutions of Einstein's Equations. It has been an indispensable tool for me. You have done a great job." - Dwight Vincent, University of Winnipeg - Canada, Feb 1992. "Regarding my comments on MathTensor, you can be assured that they were coming out of my heart, and that they are TRUE. Yes, it is a very nice and useful tool. Many people, like myself, who are getting older and loosing the patience (and in some cases the ability to do long tedious computations) to deal with the "nitty gritty" of things can find here REAL HELP. Also, for those of us who are daring and curious it opens up a door to tackle problems that we otherwise wouldn't." Juan Perez Mercader, Insituto de Fisica Fundamental - Madrid, Spain, Feb 1992. -----------------------General Information-------------------------- MathTensor Tensor analysis is extensively used in applications in physics, mathematics, engineering, and many other areas of scientific research. Problems involving tensors often are extraordinarily large and can be some of the most difficult computations in all of science. Equations with thousands of terms are common and can only be manipulated by computer mathematics systems like Mathematica. MathTensor is the largest Mathematica package yet developed outside of Wolfram Research. It adds over 250 new functions and objects to Mathematica to give the user both elementary and advanced tensor analysis functionality. MathTensor is a general tool for handling both indicial and concrete tensor indices. Standard objects like Riemann tensor, Ricci tensor, metric and others are built into the system along with common functions like the covariant derivative, index commutation, raising and lowering of indices, and various differential forms operations. MathTensor has been under development by Leonard Parker and Steven M. Christensen since the first alpha test release of Mathematica. It contains over 25,000 lines of Mathematica code contained in nearly 100 files totalling approximately 2.0 Megabytes of disk space. MathTensor will run on any machine that runs Mathematica and has sufficient RAM memory (generally 8 Megabytes or more) and disk space for file storage and swap. MathTensor runs under versions 1.2 and 2.0 or greater of Mathematica. MathSolutions is in constant contact with Wolfram Research regarding Mathematica changes and versions. Included in this mailing are a price list, machine list, a list of the current functions and objects, and a set of example MathTensor computations. For other information, contact us at the address above. MathSolutions, Inc. was formed by Leonard Parker and Steven M. Christensen. MathSolutions has its office in Chapel Hill, North Carolina. Both Parker and Christensen are theoretical physicists, specializing in research in Einstein's Special and General Theory of Relativity, quantum field theory, black hole theory, and cosmology. Parker obtained his Ph.D. in Physics from Harvard University and Christensen his Ph.D. in Physics from the University of Texas at Austin. MathTensor is a trademark of MathSolutions, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. ----------------------MathTensor Examples ---------------------------- MathTensor provides commands for simplifying and manipulating tensor expressions, as well as a knowledge base of transformation rules and definitions required for dealing with some of the more important tensors. MathTensor is designed to work along with the functions of Mathematica to provide users with the functions and objects they need to devise their own custom tensor analysis programs. MathTensor provides most of the basic structures needed for doing tensor computations and for programming new functions. As MathTensor is used and special applications are developed, they will be added to future versions of MathTensor. The following pages give a few examples of how MathTensor works. (* First, after starting Mathematica 2.0, we load a file which in turn loads many other files containing the MathTensor function and object definitions. *) In[1]:= <<MathTensor.m ==================================================== MathTensor (TM) 2.1 (UNIX (R)) (April 20, 1992) by Leonard Parker and Steven M. Christensen Copyright (c) 1991, MathSolutions, Inc. Runs with Mathematica (R) Versions 1.2, 2.0 or 2.1. Licensed to machine sunny. ==================================================== No unit system is chosen. If you want one, you must edit the file called Conventions.m, or enter a command to interactively set units. Units: {} Sign conventions: Rmsign = 1 Rcsign = 1 MetricgSign = 1 DetgSign = -1 TensorForm turned on, ShowTime turned off, MetricgFlag = True. ========================================= (* MathTensor has a number of tensors already defined. Lower indices are entered with an "l" in front, while upper indices are entered with a "u" in front. In output lines the indices are placed where they should be on the tensor. *) In[2]:= RiemannR[la,lb,lc,ld] Out[2]= R abcd In[3]:= RiemannR[la,ub,lc,ud] b d Out[3]= R a c (* Operations like covariant differentiation are done simply with the CD function, with the derivative index placed after the tensor. *) In[4]:= CD[RicciR[la,lb],uc] c Out[4]= R ab; (* The standard summation convention is recognized. Here indices are summed using the standard Mathematica substitution operation which in the case below simply renames the upper index. *) In[5]:= % /. uc->ub b Out[5]= R ab; (* The DefineTensor function permits you to define your own tensors. The input name of the object defined below is "tensor", and its print name, which appears in output lines, is "t". The last argument indicates that it will have two indices, which upon interchange result in multiplication of the object by a weight factor of 1 -- that is, it is a symmetric tensor. *) In[6]:= DefineTensor[tensor,"t",{{2,1},1}] PermWeight::sym: Symmetries of t assigned PermWeight::def: Object t defined In[7]:= tensor[la,lb] Out[7]= t ab (* MathTensor now automatically reorders symmetric indices into lexical order. *) In[8]:= tensor[lb,la] Out[8]= t ab (* The standard symmetries of the Riemann tensor are built into its definition. *) In[9]:= RiemannR[lb,la,lc,ld] Out[9]= -R abcd In[10]:= RiemannR[lc,ld,la,lb] Out[10]= R abcd (* MathTensor knows that the appropriate sum of indices on the Riemann tensor gives the Ricci tensor. *) In[11]:= RiemannR[la,lb,ua,lc] Out[11]= R bc (* The sum of the indices on the Ricci tensor gives the Riemann Scalar. *) In[12]:= RicciR[la,ua] Out[12]= R (* Now we define a new tensor with four indices and no symmetries. *) In[13]:= DefineTensor[T,"T",{{1,2,3,4},1}] PermWeight::sym: Symmetries of T assigned PermWeight::def: Object T defined (* We produce a complicated product of seven of these tensor with multiple summations of indices and then add it to another similar object. *) In[14]:= SevenTensorTest := T[la,lb,uc,ud] T[lc,ld,ue,uf] T[le,lf,ug,uh] T[lg,li,ui,uj] * T[lh,lj,uk,ul] T[lk,ll,um,un] T[lm,ln,ua,ub] - T[la,lb,uc,ud] T[lc,le,ue,uf] * T[ld,lf,ug,uh] T[lg,lh,ui,uj] T[li,lj,uk,ul] T[lk,ll,um,un] T[lm,ln,ua,ub] (* Trying to find some simplification of SevenTensorTest is not easy by hand. *) In[15]:= SevenTensorTest cd ef gh ij kl mn ab Out[15]= T T T T T T T - ab cd ef gi hj kl mn cd ef gh ij kl mn ab > T T T T T T T ab ce df gh ij kl mn (* But the MathTensor command Tsimplify rapidly finds that the two terms are equal. *) In[16]:= Tsimplify[%] Out[16]= 0 (* Some terms differ only by the renaming of summation indices. *) In[17]:= T[la,lb,lc,ld] RiemannR[ua,ub,uc,ud] + T[le,lf,lg,lh] RiemannR[ue,uf,ug,uh] abcd efgh Out[17]= R T + R T abcd efgh (* MathTensor's canonicalization functions can rename indices and combine terms. *) In[18]:= Canonicalize[%] pqrs Out[18]= 2 R T pqrs (* MathTensor can symmetrize or antisymmetrize pairs of indices. *) In[19]:= Symmetrize[T[la,lb,lc,ld], {la,lb}] T + T abcd bacd Out[19]= ------------- 2 In[20]:= Expand[%] T T abcd bacd Out[20]= ----- + ----- 2 2 In[21]:= Antisymmetrize[%,{lc,ld}] T T T T abcd abdc bacd badc ----- - ----- + ----- - ----- 2 2 2 2 Out[21]= ----------------------------- 2 In[22]:= Expand[%] T T T T abcd abdc bacd badc Out[22]= ----- - ----- + ----- - ----- 4 4 4 4 (* MathTensor understands how to convert covariant to ordinary partial derivatives with affine connection terms added. *) In[23]:= CD[RicciR[la,lb],lc] Out[23]= R ab;c In[24]:= CDtoOD[%] p p Out[24]= R - G R - G R ab,c bc pa ac pb (* Using positive and negative index values, MathTensor can deal with concrete contravariant or covariant indices. *) In[25]:= RiemannR[1,2,3,4] 1234 Out[25]= R In[26]:= RiemannR[-1,2,-3,4] 4 2 Out[26]= R 3 1 (* We can set the dimension of the spacetime to some value, like 4. *) In[27]:= Dimension = 4 Out[27]= 4 (* Then using the MakeSum function, we can explicitly write out sums in terms of concrete indices. *) In[28]:= MakeSum[RicciR[la,lb] RicciR[lc,ub]] 1 2 3 4 Out[28]= R R + R R + R R + R R 1a c 2a c 3a c 4a c (* Now we define tensor T with two indices that are symmetric. *) In[29]:= DefineTensor[T,"T",{{2,1},1}] PermWeight::sym: Symmetries of T assigned PermWeight::def: Object T defined (* Then the lower components of T can be defined in terms of the components of other tensors like the Ricci tensor. *) In[30]:= SetComponents[T[la,lb],RicciR[la,lc] RicciR[lb,uc]] Components assigned to T (* We can ask for a specific covariant component of T. *) In[31]:= T[-1,-1] 1 2 3 4 Out[31]= R R + R R + R R + R R 11 1 21 1 31 1 41 1 (* One example application built into MathTensor does variations with respect to the metric tensor of structures that are functions of the metric tensor, Metricg. The variation of the square root of the determinant of the metric times the Riemann scalar gives terms in the variation of the metric, called h. *) In[32]:= Sqrt[Detg] ScalarR Out[32]= Sqrt[g] R In[33]:= Variation[%,Metricg] MetricgFlag::off: MetricgFlag is turned off by this operation pq p q Out33]= Sqrt[g] h - Sqrt[g] h + pq; p ;q pq Sqrt[g] R g h pq pq ----------------- - Sqrt[g] R h 2 pq (* A more complicated variation is just as easily found. *) In[34]:= Sqrt[Detg] RicciR[la,lb] RicciR[ua,ub] ab Out[34]= Sqrt[g] R R ab In[35]:= Variation[%,Metricg] pq r -(Sqrt[g] h R ) ;r pq p rq Out[35]= --------------------- + Sqrt[g] h R - 2 q; pr p qr r pq Sqrt[g] h R Sqrt[g] h R p ; qr pq;r ------------------ - ------------------ + 2 2 p qr Sqrt[g] h R q pr p ;qr Sqrt[g] h R - ------------------ + pq;r 2 pq rs Sqrt[g] g R R h rs pq qr p ----------------------- - Sqrt[g] R R h - 2 pq r pr q Sqrt[g] R R h pq r (* MathTensor's ApplyRules function permits the user to define large sets of rules that can be applied to expressions to simplify them. One set of rules that is provided as an example are the RiemannRules. Familiar rules are applied in the next two examples. MathTensor includes several functions DefUnique and RuleUnique that help the user devise their own rules and save them for later use. *) In[36]:= CD[RicciR[la,lb],ub] b Out[36]= R ab; In[37]:= ApplyRules[%,RiemannRules] R ;a Out[37]= --- 2 In[38]:= RiemannR[la,lb,lc,ld] RiemannR[ua,uc,ub,ud] acbd Out[38]= R R abcd In[39]:= ApplyRules[%,RiemannRules] pqrs R R pqrs Out[39]= ----------- 2 (* Suppose we define a tensor with four indices. *) In[40]:= DefineTensor[tensor,"t",{{1,2,3,4},1}] PermWeight::sym: Symmetries of t assigned PermWeight::def: Object t defined (* MathTensor's multiple index facility permits the user to add indices with one, two or three primes to be used as extra non-spacetime indices. *) In[41]:= AddIndexTypes In[42]:= tensor[ala,blb,clc,ld] Out[42]= t a'b''c'''d (* Tools for building rules involving these extra indices are provided. Future releases of MathTensor will extend this functionality. *) (* Shipped with MathTensor is the file Components.m which may be run separately from MathTensor. Components.m takes a file like CompInSchw.m, listed next, and computes the components of the affine connection, the Riemann tensor, Ricci tensor, Riemann scalar, Weyl tensor, Einstein tensor and several other objects. CompInSchw.m is the input file containing information about the famous Schwarzschild metric. This metric represents the curved space of a gravitating spherically symmetric object with mass M in otherwise empty space. *) ---------------------- CompInSchw.m file listing -------------------- (* CompInSchw.m , sample input file for Components. This contains the Schwarzschild exterior solution of the vacuum Einstein equations. The convention is that negative integers from -Dimension to -1 denote covariant index values, and positive integers from 1 to Dimension denote contravariant index values. The value 0 is not used for an index value because its sign can not be used to distinguish between a covariant and contravariant index. *) (* Give the value of Dimension, in this case the value is 4 *) Dimension = 4 (* Give the names of the coordinates x[1],..,x[Dimension] *) x/: x[1] = r x/: x[2] = theta x/: x[3] = phi x/: x[4] = t (* Give the covariant components of the metric tensor. Negative integers denote covariant components. Since Metricg is symmetric, only the components with ordered indices should be specified. Note that -3, for example, is lexically before -1, so Metricg[-3,-1] is specified and not Metricg[-1,-3]. All components having the first index less than or equal to the second index, up to Metricg[-Dimension, -Dimension], should be specified. *) Metricg/: Metricg[-1, -1] = (1 - (2*G*M)/r)^(-1) Metricg/: Metricg[-2, -1] = 0 Metricg/: Metricg[-3, -1] = 0 Metricg/: Metricg[-4, -1] = 0 Metricg/: Metricg[-2, -2] = r^2 Metricg/: Metricg[-3, -2] = 0 Metricg/: Metricg[-4, -2] = 0 Metricg/: Metricg[-3, -3] = r^2*Sin[theta]^2 Metricg/: Metricg[-4, -3] = 0 Metricg/: Metricg[-4, -4] = -(1 - (2*G*M)/r) (* Next give the flag settings for various options explained below. *) Rmsign = 1 Rcsign = 1 CalcEinstein = 1 (* Calculate the covariant Einstein tensor components EinsteinG[la,lb]. The value 0 tells Components not to calculate these components. *) CalcRiemann = 1 (* Calculate the covariant Riemann tensor components RiemannR[la,lb]. The value 0 tells Components not to calculate these components. *) CalcWeyl = 1 (* Calculate the covariant Weyl tensor components WeylC[la,lb]. The value 0 tells Components not to calculate these components. *) Perturb[,] (* End of file CompInSchw.m *) --------------------------------------------------------------------- (* If we load Components.m into Mathematica we can run the following command to produce from CompInSchw.m two new files, CompOutSchw.m, which contains results that can be used immediately in MathTensor, and CompOutSchw.out which can be printed. The computation below takes from just a few seconds to less than one minute on typical workstations. More complex metrics can take a bit longer. *) In[1]:= <<Components.m ==================================================== MathTensor (TM) 2.1 (UNIX (R)) (April 20, 1992) Components Package by Leonard Parker and Steven M. Christensen Copyright (c) 1991, MathSolutions, Inc. Runs with Mathematica (R) Versions 1.2, 2.0 and 2.1. Licensed to machine sunny. ==================================================== In[2]:= Components["CompInSchw.m","CompOutSchw.m", "CompOutSchw.out"] The following tensors have been calculated and stored in the file CompOutSchw.m in InputForm, and in the file CompOutSchw.out in OutputForm: Metricg MatrixMetricgLower MatrixMetricgUpper Detg AffineG[ua,lb,lc] RicciR[la,lb] ScalarR EinsteinG[la,lb,lc,ld] RiemannR[la,lb,lc,ld] WeylC[la,lb,lc,ld] You can edit CompOutSchw.out to print a record of the results. (* The listing of the output file is too long to give here, but it gives the calculated values of the components of the Riemann and related tensors in the Schwarzschild spacetime. *) The names of the various commands and other objects available in MathTensor is given below: A List of MathTensor Functions and Objects (12/15/91) Absorb Epsilon InvertFast Absorbg EpsilonProductTensor Kdelta AbsorbKdelta EpsilonProductTensorRule KdeltaRule AbsorbRule EpsilonSign Kill AddIndexTypes EpsilonToEpsDownRule Lap AffineG EpsilonToEpsUpRule LieD AffineToMetric EpsUp LieDtoCD AffineToMetricRule EpsUpToEpsDownRule LieDtoCDrule AllSymmetries EpsUpToEpsilonRule Lightc Antisymmetrize EqApart LorentzGaugeRule ApplyRules EqApply Lower ApplyRulesRepeated EqCancel Lowera Arglist EqCollect LowerAllPairs ArglistAllTypes EqDivide LowerAllTypes AskSignsFlag EqExpand Lowerb AskSignsProcedure EqExpandAll Lowerc BianchiFirstPairRule EqFactor LowerIndexAllTypesQ BianchiSecondPairRule EqFactorTerms LowerIndexaQ CanAll EqMinus LowerIndexbQ CanApplyRules EqPlus LowerIndexcQ CanApplyRulesFast EqPower LowerIndexQ CanApplyRulesRepeated EqReverse MakeAllSymmetries CanDum EqSimplify MakePermWeightGroup Cannn EqSolve MakeSum CannnDum EqSubtract Matchlist CanNonInvert EqTimes Matchlista Canonicalize EqTogether MatchlistAllTypes CanSame EqTwoDivide Matchlistb CanSuperApplyRules EqTwoPlus Matchlistc CD EqTwoSubtract MatchlistOrd CDtoOD EqTwoTimes $MathTensorVersionNumber ClearComponents esuUnits MaxwellA ClearUnits EvaluateODFlag MaxwellB CommuteCD Evenlist MaxwellCyclicEquation Components Explode MaxwellCyclicRule CountNewDums FirstCubicRiemannRule MaxwellDivergenceEquation CoXD FirstQuadraticRiemannRule MaxwellDivergenceRule DefineForm FreeList MaxwellE DefineTensor FtoC MaxwellF DefUnique FtoCrule MaxwellJ Detg GaussianUnits Maxwellk1 DetgSign GenLap Maxwellk3 Dimension GravitationalUnits Maxwellrho Downdummylist hbar MaxwellT Downlist HeavisideLorentzUnits MaxwellTexpression Downuserlist HodgeStar MaxwellTtoFrule DualStar Implode MaxwellVectorPotentialRule Dum IndexAllTypesQ Metricg DumAllTypes IndexaQ MetricgFlag EinsteinG IndexbQ MetricgFlagOff EinsteinToRicciRule IndexcQ MetricgFlagOn emuUnits IndexQ MetricgSign Eps0 IndexTypes Mu0 EpsDown IndicesAndNotOrderedQ NaturalUnits EpsDownToEpsilonRule InListQ NegIntegerQ EpsDownToEpsUpRule Invert NewtonG NonTensorPart Symmetries OD SymmetriesOfSymbol Oddlist Symmetrize OrderedArgsQ SyntaxCheck Pair SyntaxCheckOff PairAllTypes SyntaxCheckOn PairAllTypesQ TensorForm PairaQ TensorPart PairbQ TensorPartSameQ PaircQ TensorQ Pairdum TensorSimp PairQ TensorSimpAfter PermWeight TraceFreeRicciR PIntegrate TraceFreeRicciToRicciRule PosIntegerQ Tsimplify PrettyOff TsimplifyAfter PrettyOn Units Raise Unlist Raisea Updowndummylist RaiseAllTypes Updummylist Raiseb Uplist Raisec UpLo RankForm UpLoa RationalizedGaussianUnit UpLob RationalizedMKSUnits UpLoc Rcsign UpperIndexAllTypesQ RicciR UpperIndexaQ RicciSquared UpperIndexbQ RicciToAffine UpperIndexcQ RicciToAffineRule UpperIndexQ RicciToTraceFreeRicciRule Upuserlist RiemannCyclicFirstThreeRule Var RiemannCyclicSecondThreeRule Varg RiemannR Variation RiemannRules VariationalDerivative RiemannSquared VectorA RiemannToAffine VectorAFlag RiemannToAffineRule WeylC RiemannToWeylRule WeylToRiemannRule Rmsign XD Rulelists XDtoCDflag RuleUnique XP RuleUniqueAllTypes ZeroFormQ ScalarR ScalarRtoAffine ScalarRtoAffineRule SecondCubicRiemannRule SecondQuadraticRiemannRule SetAntisymmetric SetComponents SetSymmetric ShowNumbers ShowTime SIUnits SuperApplyRules SwapDum

**minor revision of histogram.m**

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