This document provides an overview of basic calculations and functions in Mathematica. It discusses how Mathematica notebooks allow users to enter input and view output through cells, and how the kernel processes calculations. Functions like N[] are used to obtain numerical approximations. Variables can be defined and lists created to organize objects. Previous results and multiple operations can be referenced. Information about built-in functions is available through ? and ?? commands.

1. BasicCalculations
ME 443Mathematica forEngineers

2. RunningMathematica
Installationand runningstatementscan
be differentfrom one computer system to another,
beaffected byvarious kinds of customization done on Mathematica.
However, the structure of Mathematicacalculations is the same in all cases.
You enter input,
Mathematicaprocesses it,
Mathematicareturnsa result.

3. Notebooks
ThespecialgraphicaluserinterfacesofMathematicaiscalledas“notebook”.
Notebooksareinteractivedocuments,intowhichyoucaninsertMathematicainputaswellasordinarytextandgraphics.
Youcaninteractwithnotebooksby
typingtext,
usingapointingdevicesuchasamouse

4. Kernels
ThefundamentalcomputationalpartofMathematicaiscalledthekernel.
Thefrontendisusuallyaseparateprogramwhichhandlesvariousaspectsofuserinteraction.
Frontendstypicallyallowyoutoprepareinputinvariousways,thensendtheinputtothekernel,andthengetresultsbackfordisplay.
ThenotebookinterfaceforMathematicaisanexampleofaMathematicafrontend.

5. ThenotebookinterfaceiscurrentlyastandardtypeofMathematicafrontend,althoughitisbynomeanstheonlypossibletype.
Notebooks consist of a hierarchyof cells.
Each cell can contain a particularkind of material:
text,
graphics,
Mathematicainputoroutput,
andsoon.
Sequences of cells can be arrangedin groups representingrelated material. A group of cells might, for example, correspond to a section or chapter in your document.

6. An important feature of notebook interfaces is that they allow you to manipulate your documentat several levels.
At the lowest level, you can modify text or other material within a single cell.
At ahigher level, you can do the same kinds of operations on a whole cell at a time.
And beyond that, you can manipulate whole groups of cells.
Notebook interfaces can typically take advantage of the typographical capabilities of your computersystem's graphical user interface.
cells containing text can have a variety of "styles". The styles can involve various fonts, sizes, and so on.
In addition, even within a single cell, you canoften mix several styles, allowing you to produce typographically complex text.

7. By looking at thesebrackets, you canseehow a particular notebookis organized.
Whenagroupofcellscorrespondstoasectionorchapterofyourdocument,thefirstcellinthegrouptypicallygivessomekindofheadingforthesectionorchapter.Notebookinterfacesallowyouto"close"groupsofcellssothatonlytheirfirstcellsarevisible.Ifthefirstcellscontainheadings,youcangetanoutlineofyourdocumentinthisway.
indicatestheextentof cell
indicatestheextentof cellgroups

8. Starting and Quitting
double-click
the Mathematicaicon
text SHIFT+ENTER
choose menu itemQuit
typicalaction for startingMathematica
input for Mathematica
exiting Mathematica

9. AfteryousendinputfromyournotebooktoMathematica, Mathematicawilllabelyourinputsandoutputs.
In[n]:=correspondstonthinputofMathematicasession.
Out[n]:=correspondstonthoutputofMathematicasession.
•type 2 + 2
•pressShift-Return
•Mathematicaprocesses the input
•then it adds the input label In [1] : =
•and gives the outputwithlabelOut[1]:=

10. EnteringInput
PressSHIFT+ENTERtosendinputtoMathematica. Withthiscommand, you
tell Mathematicathat you have finished preparing input for it in a particular cell.
giveall the text in your current cell as input to the Mathematica kernel.
Mathematica requires that the input you give follow a definite syntax.

11. HistoryofMathematicasessioncanbeobtainonlybyfollowingtheIn[n]:=andOut[n]:=.
SomefeaturesofNotebookinterfaceare;
commandcompletion.
controlof variousparametersusinggraphicaltools.
Notebook interfacestypically allow you tofind and specifycoor- dinatesgraphically.

12. By rotating the box, you can specify a viewpoint, which is then fed to Mathematica in textual form.

13. Getting Information from Mathematica
?Name
??Name
show information on Name
show extra information on Name

14. ?Aaaa*
objects whose names begin with Aaaa
You can ask for information about any object, whether it is built into Mathematica, has been read in from a Mathematica package, or has been introduced by you.

15. Youcanputa*anywhereinthestringyouask?about.Forexample, ?*ExpandwouldgiveyouallobjectswhosenamesendwithExpand. Similarly,?x*0wouldgiveyouobjectswhosenamesstartwithx,andendwith0,andhaveanysequenceofcharactersinbetween.

16. InterruptingCalculations
To terminate or interrupt Mathematica in the middle of a calculation, youcan use
menu item “Quit Kernel” from menu “Evaluation”.
some commands related with kernel control such as
Abort[]
Interrupt[]
Quit[]
Exit[]
On some computer systems, it may take Mathematica some time to respond to your interrupt

17. WarningsandMessages
Mathematicausuallygoesaboutitsworksilently,givingoutputonlywhenithasfinisheddoingthecalculationsyouaskedfor.
However,ifitlooksasifMathematicaisdoingsomethingyoudefinitelydidnotintend,Mathematicawillusuallyprintamessagetowarnyou.ThereisalistofstandardMathematicamessagesavailableasaWolframResearchtechnicalreport,MathematicaWarningMessages.

18. -NumericalCalculations– Arithmetic
You can do arithmetic with Mathematica just as you would on an electronic calculator.

20. ExactAndApproximateResults
Astandardelectroniccalculatordoesallyourcalculationstoaparticularaccuracy,saytendecimaldigits.WithMathematica, however,youcanoftengetexactresults.
Mathemaicagivesexactresultfor2100, eventhoughit has 31 decimaldigits.
YoucantellMathematicatogiveyouanapproximatenumericalresult,justasacalculatorwould,byputtingN[]aroundyourinput. TheNstandsfor"numerical".Itmustbeacapitalletter.

21. When you type in an integer like 7, Mathematica assumes that it is exact. If you type in a number like 4.5, with an explicit decimal point, Mathematica assumes that it is accurate only to a fixed number of decimal places.

23. SomeMathematicalFunctions
Mathematicaincludesaverylargecollectionofmathematicalfunctions.Hereareafewofthecommonones.

24. Theargumentsof allMathematicafunctionsareenclosedin squarebrackets.
Thenamesof built-in Mathematicafunctionsbeginwithcapitalletters.

26. Noticethatthenamesofthesebuilt-inconstantsallbeginwithcapitalletters.

28. Arbitrary-PrecisionCalculations
WhenyouuseNtogetanumericalresult,Mathematicadoeswhatastandardcalculatorwoulddo:itgivesyouaresulttoafixednumberofsignificantfigures.YoucanalsotellMathematicaexactlyhowmanysignificantfigurestokeepinaparticularcalculation.ThisallowsyoutogetnumericalresultsinMathematicatoanydegreeofprecision.

29. Increasingthenumericalprecision,decreasetheroundofferrors.
Thisgivesthenumericalvaluesofπtoafixednumberofsignificantdigits.
This gives πto 40 digits.
Here is square root 7 to 30 digits.

30. GettingUsedtoMathematica
Arguments of functions are given in square brackets.
Names of built-in functions have their first letters capitalized.
Multiplication can be represented by a space.
Powers are denoted by ^.
Numbers in scientific notation are entered, for example, as 2.5 10^- 4.

31. DefiningVariables
Whenyoudolongcalculations,itisoftenconvenienttogivenamestoyourintermediateresults.Justasinstandardmathematics,orinothercomputerlanguages,youcandothisbyintroducingnamedvariables.

32. Onceyouhaveassignedavaluetoaparticularvariable,thevaluewillbekeptuntil
youexplicitlyremoveitor
youstartawholenewMathematicasession.
ForgettingaboutdefinitionsyoumadeearlieristhesinglemostcommoncauseofmistakeswhenusingMathematica.Ifyousetx=5, Mathematicaassumesthatyoualwayswantxtohavethevalue5,untilorunlessyouexplicitlytellitotherwise.Toavoidmistakes,youshouldremovevaluesyouhavedefinedassoonasyouhavefinishedusingthem.

33. The variables you define can have almost any names.
There is no limit on the length of their names. One constraint, however, is that variable names can never start with numbers. For example, x2 could be a variable, but 2x means 2*x.
Mathematica uses both upper-and lower-case letters.
There is a convention that built-in Mathematica objects always have names starting with upper-case (capital) letters.
To avoid confusion, you should always choose names for your own variables that start with lower-case letters.

34. YoucantypeformulasinvolvingvariablesinMathematicaalmostexactlyasyouwouldinmathematics.Thereareafewimportantpointstowatch,however.
xymeansxtimesy.
xywith no space is the variablewith name xy.
5xmeans5timesx.
x^2ymeans(x^2)y,notx^(2y)

35. UsingPreviousResults
In doing calculations, you will often need to use previous results that you have got. In Mathematica, % always stands for your last result.

37. Sequences of Operations
In doing a calculation with Mathematica, you usually go through a sequence of steps. If you want to, you can do each step on a separate line. Often, however, you will find it convenient to put several steps on the same line. You can do this simply by separating the pieces of input you want to give with semicolons.

38. Ifyouendyourinputwithasemicolon,itisasifyouaregivingasequenceofoperations,withan"empty"oneattheend.ThishastheeffectofmakingMathematicaperformtheoperationsyouspecify,butdisplaynooutput.

39. MakingListsof Objects
Indoingcalculations,itisoftenconvenienttocollecttogetherseveralobjects,andtreatthemasasingleentity.ListsgiveyouawaytomakecollectionsofobjectsinMathematica.Asyouwillseelater,listsareveryimportantandgeneralstructuresinMathematica.
Alistsuchas{3,5,1}isacollectionofthreeobjects.Butinmanyways,youcantreatthewholelistasasingleobject.Youcan,forexample,doarithmeticonthewholelistatonce,orassignthewholelisttobethevalueofavariable.

41. ManipulatingElementsof Lists
ManyofthemostpowerfullistmanipulationoperationsinMathematicatreatwholelistsassingleobjects.Sometimes,however, youneedtopickoutorsetindividualelementsinalist.
YoucanrefertoanelementofaMathematicalistbygivingits"index".Theelementsarenumberedinorder,startingat1.

42. Byassigningavariabletobealist,youcanuseMathematicalistsmuchlike"arrays"inothercomputerlanguages.

44. The Four Kinds of Bracketing in Mathematica
Eachkindofbracketinghasaverydifferentmeaning.
(term)parenthesesforgrouping
f[x]squarebracketsforfunctions
{a,b,c}curlybracesforlists
v[[i]]doublebracketsforindexing

45. Typesof Numbers
Four types of numbers are built into Mathematica.
Integerarbitrary-length exact integer
Rationalinteger/integerin lowest terms
Realapproximate real number, with any specified precision
Complexcomplexnumber of the form number+ numberi

47. NumericalFunctions

49. ElementaryTranscendentalFunctions
MathematicalfunctionsinMathematicaaregivennamesaccordingtodefiniterules.
AswithmostMathematicafunctions,thenamesareusuallycompleteEnglishwords,fullyspelledout.
Forafewverycommonfunctions,Mathematicausesthetraditionalabbreviations.Thusthetangentfunction,forexample,isTan,notTangent.

52. ComplexNumbers
You can enter complex numbers in Mathematica just by including the constant I, equal to squareroot-1. Make sure that you type a capital I.

55. DefiningMathematica Functions
Mathematicacontainshundredsofbuilt-inmathematicalfunctionslikeRound,Sqrt,andSin,andthesecanbecombinedtoformanendlessnumberofpossibleexpressions.Theseexpressionscanbesimplelike1/xormorecomplicatedlikeAbs[Abs[x]-1].
ThefirststepinMathematicaprogrammingistodefineafunctiontoabbreviatesuchanexpression.

58. Numerical Precision
Mathematicacanhandleapproximaterealnumberswithanynumberofdigits.
Ingeneral,theprecisionofanapproximaterealnumberisthenumberofdecimaldigitsinitwhicharetreatedassignificantforcomputations.
Theaccuracyofanapproximaterealnumberisthenumberofdecimaldigitswhichappeartotherightofthedecimalpoint. Precisionisthusameasureoftherelativeerrorinanumber,whileaccuracyisameasureofabsoluteerror.

60. WhenyouuseN[expr,n],Mathematicaevaluatestheexpressionexprstartingwithnumbersthathavendigitsofprecision.However,thefactthatsuchnumbersareuseddoesnotnecessarilymeanthattheresultsyougetwillhavendigitsofprecision.Inmostcases,yourresultswillhaveatleastslightlyfewerdigitsofprecision.

61. Indoingnumericalcomputations,itisinevitablethatyouwillsometimesendupwithresultsthatarelessprecisethanyouwant. Particularlywhenyougetnumericalresultsthatareveryclosetozero,youmaywellwanttoassumethattheresultsshouldbeexactlyzero.ThefunctionChopallowsyoutoreplaceapproximaterealnumbersthatareclosetozerobytheexactinteger0.

66. Algebra
OneofthemostimportantfeaturesofMathematicaisthatitcandosymbolic,aswellasnumericalcalculations.Thismeansthatitcanhandlealgebraicformulasaswellasnumbers.

68. Youcanusespacestodenotemultiplication.Becarefulnottoforgetthespaceinxy.Ifyoutypeinxywithnospace,Mathematicawillinterpretthisasasinglesymbol,withthenamexy,notasaproductofthetwosymbolsxandy.

69. Whenyoutypeinmorecomplicatedexpressions,itisimportantthatyouputparenthesesintherightplaces.Thus,forexample,youhavetogivetheexpressionx4yintheformofx^(4y).Ifyouleaveouttheparentheses,yougetx4yinstead.

70. Whenyoutypeinanexpression,Mathematicaautomaticallyappliesitslargerepertoireofrulesfortransformingexpressions.Theserulesincludethestandardrulesofalgebra,suchasx-x=0,togetherwithmuchmoresophisticatedrulesinvolvinghighermathematicalfunctions.

71. Thenotionoftransformationrulesisaverygeneralone.Infact,youcanthinkofthewholeofMathematicaassimplyasystemforapplyingacollectionoftransformationrulestomanydifferentkindsofexpressions.
ThegeneralprinciplethatMathematicafollowsissimpletostate.Ittakesanyexpressionyouinput,andgetsresultsbyapplyingasuccessionoftransformationrules,stoppingwhenitknowsnomoretransformationrulesthatcanbeapplied.

72. UsingSymbolsforUnits
TherearemanywaystousesymbolsinMathematica.Sofar,wehaveconcentratedonusingsymbolstostorevaluesandtorepresentmathematicalvariables.ThissubsectiondescribesanotherwaytousesymbolsinMathematica.
Theideaistousesymbolsas“tags”fordifferenttypesofobjects.
Workingwithphysicalunitsgivesonesimpleexample.Whenyouspecifythelengthofanobject,youwanttogivenotonlyanumber,butalsotheunitsinwhichthelengthismeasured.Instandardnotation,youmightwritealengthas12meters.
YoucanimitatethisnotationalmostdirectlyinMathematica.Youcanforexamplesimplyuseasymbolmeterstoindicatetheunitsofourmeasurement.

74. ThereisinfactastandardMathematicapackagethatallowsyoutoworkwithunits.Thepackagedefinesmanysymbolsthatrepresentstandardtypesofunits.

75. ValuesforSymbols
WhenMathematicatransformsanexpressionsuchasx+xinto2x,itistreatingthevariablexinapurelysymbolicorformalfashion.Insuchcases, xisasymbolwhichcanstandforanyexpression.
Often,however,youneedtoreplaceasymbollikexwithadefinite“value”. Sometimesthisvaluewillbeanumber;oftenitwillbeanotherexpression.
Totakeanexpressionsuchas1+2xandreplacethesymbolxthatappearsinitwithadefinitevalue,youcancreateaMathematicatransformationrule, andthenapplythisruletotheexpression.Toreplacexwiththevalue3, youwouldcreatethetransformationrulex→3.Youmusttype“-”+”>”asapaircharacters,withnospaceinbetween.Youcanthinkofx→3asbeingaruleinwhich“xgoesto3”.
ToapplyatransformationruletoaparticularMathematicaexpression,youtypeexpr/.rule.The“replacementoperator”/.istypedasapairofcharacters,withnospaceinbetween.

77. Thereplacementoperator/.allowsyoutoapplytransformationrulestoaparticularexpression.Sometimes,however,youwillwanttodefinetransformationrulethatshouldalwaysbeapplied.Forexample,youmightwanttoreplacexwith3wheneverxoccurs.

78. You can define the value of a symbol to be any expression, not just a number. You should realize that once you have given such a definition, the definition will continue to be used whenever the symbol appears, until you explicitly change or remove the definition. For most people, forgetting to remove values you have assigned to symbols is the single most common source of mistakes in using Mathematica.

82. TransformingAlgebraicExpressions
Thereareoftenmanydifferentwaystowritethesamealgebraicexpression.Asoneexample,theexpression(1+x)2canbewrittenas1+2x+x2.Mathematicaprovidesalargecollectionoffunctionsforconvertingbetweendifferentformsofalgebraicexpressions.

85. SimplifyingAlgebraicExpressions
Therearemanysituationswhereyouwanttowriteaparticularalgebraicexpressioninthesimplestpossibleform.Althoughitisdifficulttoknowexactlywhatonemeansinallcasesbythe"simplestform",aworthwhilepracticalprocedureistolookatmanydifferentformsofanexpression,andpickouttheonethatinvolvesthesmallestnumberofparts.

86. For many simple algebraic calculations, you may find it convenient to use Simplify routinely on your results.

87. Inmorecomplicatedcalculations,however,youoftenneedtoexercisemorecontrolovertheexactformofanswerthatyouget.Inaddition,whenyourexpressionsarecomplicated,Simplifymayspendalongtimetestingalargenumberofpossibleformsinitsattempttofindthesimplestone.
Mathematicadoesnotautomaticallyexpandoutexpressionsoftheform(ab)^cexceptwhencisaninteger.Ingeneralitisonlycorrecttodothisexpansionifaandbarepositivereals.Nevertheless,thefunctionPowerExpanddoestheexpansion,effectivelyassumingthataandbareindeedpositivereals.

89. SumsandProducts

92. AdvancedTopic: TransformingTrigonometricExpressions
Aswehaveseen,evenwhenyourestrictyourselftopolynomialsandrationalexpressions,therearemanydifferentwaystowriteanyparticularexpression.Ifyouconsidermorecomplicatedexpressions, involving,forexample,highermathematicalfunctions,thevarietyofpossibleformsbecomesstillgreater.Asaresult,itistotallyinfeasibletohaveaspecificfunctionbuiltintoMathematicatoproduceeachpossibleform.Rather,Mathematicaallowsyoutoconstructarbitrarysetsoftransformationrulesforconvertingbetweendifferentforms.

93. There are nevertheless a few additional built-in Mathematicafunctions for transforming expressions.

95. SolvingEquations
Theassignmentssuchasx=ywhichsetxequaltoy.Thissectiondiscussesequations,whichtestequality.Theequationx==ytestswhetherxisequaltoy.
Itisveryimportantthatyoudonotconfusex=ywithx==y. Whilex=yisanimperativestatementthatactuallycausesanassignmenttobedone,x==ymerelytestewhetherxandyareequal,andcausesnoexplicitaction.

97. Thetestswehaveusedsofarinvolveonlynumbers,andalwaysgiveadefiniteanswer,eitherTrueorFalse.Youcanalsodotestsonsymbolicexpressions.

98. Evenwhenyoudotestsonsymbolicexpressions,therearesomecaseswhereyoucangetdefiniteresults.Animportantoneiswhenyoutesttheequalityoftwoexpressionsthatareidentical.Whateverthenumericalvaluesofthevariablesintheseexpressionsmaybe, Mathematicaknowsthattheexpressionsmustalwaysbeequal.

99. Expressionslikex==4representequationsinMathematica.TherearemanyfunctionsinMathematicaformanipulatingandsolvingequations.

100. SolvingAlgebraicEquations
Anexpressionlikex2+2x-7==0representsanequationinMathematica.Youwilloftenneedtosolveequationslikethis,tofindoutforwhatvaluesofxtheyaretrue.

101. Solvealwaystriestogiveyouexplicitformulasforthesolutionstoequations.However,itisabasicmathematicalresultthat,forsufficientlycomplicatedequations,explicitalgebraicformulascannotbegiven.Ifyouhaveanalgebraicequationinonevariable,andthehighestpowerofthevariableisatmostfour,thenMathematicacanalwaysgiveyouformulasforthesolutions.However,ifthehighestpowerisfiveormore,itmaybemathematicallyimpossibletogiveexplicitalgebraicformulasforallthesolutions.

104. SolvingSimultaneousEquations
YoucanalsouseMathematicatosolvesetsofsimultaneousequations. Yousimplygivethelistofequations,andspecifythelistofvariablestosolvefor.

105. Mathematicacan solve any set of simultaneous linear equations. Mathematicacan also solve a largeclass of simultaneous polynomial equations. Even when it does not manage to solve the equationsexplicitly, Mathematicawill still usually reduce them to a much simpler form.

106. NumericalSolutionof PolynomialEquations

107. NSolvewill always give you the complete set of numerical solutions to any polynomial equation inone variable.

108. SolvingNon-AlgebraicEquations
ThemainequationsthatSolveandrelatedMathematicafunctionsdealwitharepolynomialequations.Inadditiontobeingabletosolvepurelyalgebraicequations,Mathematicacanalsosolvesomeequationsinvolvingotherfunctions.
Itisimportanttorealizethatanequationsuchassin(x)=aactuallyhasaninfinitenumberofpossiblesolutions,inthiscasedifferingbymultiplesof2tt.However,Solvebydefaultreturnsjustonesolution, butprintsamessagetellingyouthatothersolutionsmayexist.

110. Solvecanalsohandleequationsinvolvingsymbolicfunctions.Insuchcases,itagainprintsawarning,thengivesresultsintermsofformalinversefunctions.

111. EliminatingVariables
Whenyouareworkingwithsetsofequationsinseveralvariables,itisoftenconvenienttoreorganizetheequationsbyeliminatingsomevariablesbetweenthem.
WhenyouwritedownasetofsimultaneousequationsinMathematica,youarespecifyingacollectionofconstraintsbetweenvariables.WhenyouuseSolve,youarefindingvaluesforsomeofthevariablesintermsofothers,subjecttotheconstraintsrepresentedbytheequations.

113. Insomecases,youmaywanttoconstructexplicitlyequationsinwhichvariableshavebeeneliminated.YoucandothisusingEliminate.

114. As a more sophisticated example of Eliminate, consider the problem of writing x5 + y5 in terms ofthe "symmetric polynomials" x+yand xy.