My life is in teaching. To have a chance to do that with a world audience is just wonderful.

• Computational Science And Engineering Mit
• Computational Science And Engineering Gilbert Strang Pdf Creator Mac
• Computational Science And Engineering Pdf

Computational Science and Engineering Gilbert Strang gs@math.mit.edu Wellesley-Cambridge Press (for ordering information) Book Order Form Related websites: math.mit.edu/18085, math.mit.edu/18086, ocw.mit.edu, math.mit.edu/dela/ CSE Table of Contents MATLAB Codes Problem Solutions FEM Table of Contents. Below are two excerpts from the textbook for this course (Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007.

MIT mathematics Professor Gilbert Strang was among the first MIT faculty members to publish a course on OpenCourseware. He has continued to contribute content through the years. By 2014, he had published five full courses, a video resource, and an online textbook on the OCW website.

He has also shared his knowledge and passion for mathematics in person, traveling extensively around the world. Open thinking has played a major role in his professional career. “A big part of my life is to open mathematics to students everywhere,” says Strang. “I'm very supportive of the whole idea of making these courses available to people around the world. Everyone has the capacity to learn mathematics, and if you can offer a little bit of guidance, the process of discovery is so valuable.”

Computational Science And Engineering Mit

Computational Science And Engineering Gilbert Strang Pdf Creator Mac

Linear Algebra

The concepts in Strang's foundational Linear Algebra course are useful in physics, economics and social sciences, natural sciences, computer sciences, and engineering. Due to its broad range of applications, it has long been one of the most popular courses on OCW. The 18.06 site has received more than 3 million visits since its first publication in 2002. Professor Strang has a website dedicated to his linear algebra teaching.

A new version was released in 2011, in the innovative OCW Scholar format designed for independent learners. The OCW Scholar version of Linear Algebra includes 35 lecture videos and 36 short (and highly-praised) problem-solving help videos by teaching assistants.

Calculus

Professor Strang has also published a collection of other materials on the OCW site including his Calculus textbook. First released in 1991 and still in print from Wellesley-Cambridge Press, the book is a useful resource for educators and independent learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. The book has an online Instructor's Manual and a student Study Guide.

When OCW approached Professor Strang about contributing to Highlights for High School, he offered his support immediately. “I've always wanted to contribute to K-12. I think high school students taking Algebra or Calculus would find some of the study materials useful.”

The result is Highlights of Calculus—a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.

The videos are garnering praise and thanks from viewers around the world. To quote one OCW user, “This series is fabulous! It summarizes the important points of calculus and gives me confidence to learn calculus without being so fearful about it.”

Differential Equations and Linear Algebra Textbook & Videos

Professor Strang has continued to offer new insights into key mathematics subjects. In 2014, he published Differential Equations and Linear Algebra. In 2016, that textbook was developed into a series of 55 short videos supported by The MathWorks® — with parallel videos about numerical solutions by Dr. Cleve Moler, the creator of MATLAB®. The textbook and video lectures helps students in a basic Ordinary Differential Equations (ODE) course. This new series, Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, is also available on The MathWorks website.

Higher Level Mathematics

Computational Science and Engineering

Professor Strang also teaches two graduate-level courses on Computational Science and Engineering, a discipline that deals with the development and application of computational models and simulations. Both courses are on OCW and have full sets of lecture videos:

Wavelets, Filter Banks, and Applications

Another graduate-level course that Professor Strang has published on OCW is Wavelets, Filter Banks, and Applications, a subject with broad applications, including audio and image compression, digital communication, medical imaging and scientific visualization.

OCW Publications by Prof. Gilbert Strang:

All of Professor Strang's books are available through Wellesley-Cambridge Press.

The followingcontentisprovidedunderaCreativeCommonslicense.Your supportwillhelpMITOpenCourseWarecontinuetoofferhigh-qualityeducationalresourcesforfree.Tomake adonation,ortoviewadditionalmaterialsfrom hundredsofMITcourses,visitMITOpenCourseWareatocw.mit.edu.

Computational Science And Engineering Pdf

PROFESSOR STRANG: OK,it'sLaplaceagaintoday.Laplace'sequation.Andtryingtodescribe--That'sabigareathat'salotofpeoplehaveworkedonforcenturies.Andfortheearlycenturies,therewerealwaysanalysismethods.Andthat'swhatwegotstartedonlasttime.Andwe'lldoabitmore.There'snowaywecoulddoeverythingthatpeoplehaveworkedonyearsandyears,tryingtofindideasaboutsolving.Butwecangettheidea.Andthispart,then,isinthesectioncalledLaplace'sequation.AndtheexamWednesdaywouldincludesomeoftheseconstructions.Sothisiswhatwedidlasttime,weidentifieda wholefamilyofsolutionstoLaplace'sequationaspolynomialsinxandy.Ofincreasingdegreen,andthenwhenwewrotetheminpolarformtheywerefantastic.r^n*cos(n*theta)andr^n*sin(n*theta).Somyidea isjust,we'vegotthem,nowlet'susethem.So howtousethesesolutions?Becausewecantakecombinationsofthem,wecancreateseriesofsinesand cosines.SoI'lldothatfirst.Series.AndthenGreen'sfunction,that'sthenameyouremember,we'veseenitbefore.That'sthesolutionwhentherightsideisadeltafunction.WhenwehavePoisson'sequationwithadeltathere.Sothatanimportantone.Andthenwell,abigpartoftwo-dimensionalandthree-dimensionalproblemsisthattheregionitself,notjusttheequationbuttheregionitself,canbeallovertheplace.We'llsolvewhentheregionisnice,likeacircleorasquare,andthenthereisawayto,inprinciple,togetotherregions.Tochangefromacrazyregiontoacircleorasquare,andthensolveitthere.Sothat'scalledconformalmapping.AndIcan'tletthewholecoursegowithoutsayingawordortwoaboutthat.Butsomehowamongnumericalmethods,it'sconformalmapping--Therearepackagesthatdoconformalmapping.Butthey'renotthecentralwaytosolve theseequationsnumerically.Finitedifferences,finiteelementsare.Andthat'swhat'scomingnextweek.Sothisisthefuture;thisisthepresent,rightthere.

So,canIjuststartwithanexampleortwo?Like,howwouldyousolveLaplace'sequationinacircle?So,inacircle.Thisistheideahere.IhaveLaplace'sequation.OK,soI'vegotawholelotofsolutions.AndI'veevengotchalktowritethemdown.OK,sohere'smycircle,mightaswellmakeittheunitcircle.Radiusone.AndinsidehereisLaplace.u_xx+u_yy=0.Nosourcesinside.Sowehavetohavesourcesfromsomewhere,andtheywillcomefromtheboundary.Soontheboundary,wekeep--Letmethinkofuastemperature.SoIsetthetemperatureontheboundary.uequalssomeu_0,someknownfunction.Thisisgiven.This istheboundarycondition.Thisisthegivenboundarycondition.Andit'safunctionof--I'mgoing tousepolarcoordinates.Polarcoordinatesare naturalforacircle.Sothisisatr=1,somaybeIshouldsayontheboundary,whichis r=1,andgoingaroundthe angletheta,isgiven.Thisisu_0(theta).That'smygivenboundarycondition.ThisproblemisnamedafterDirichlet,becauseit'slikegivingfixedconditionsandnotNeumannconditions.OK,soI'mjustlookingforacombination.ForafunctionthatsolvesLaplace'sequationinsidethecircle,andtakesonsomevaluesaroundtheboundary.Andofcoursetheboundaryvaluesmightbeplusoneonthetopandminusoneonthebottom.Ortheboundaryconditionmightvaryaround,itmight,variablethen comearound.Butnoticethatthisisaperiodicfunction.Thisis2pi-periodic,becausetheproblem'sthesame.IfIincreasethetaby2piI'vecomebacktothesamepoint.Soit'sgottohavethatvalue.OK,sowellletmegiveyouacoupleofexamplesfirst.

Supposeu_0,supposethisfunction--Example1,easy.Supposeu_0issin(3theta).SothatmeansI'vegotaregionhere,I'mprescribingitstemperatureontheboundary.AndIwanttosaywhatdoesitlooklikeinside?AndI'mprescribingrightnowthesin(3theta),sotherethetaiszero,soit'szero,theboundarycondition'szerothere,climbstoone,backtozero,downtominusone.backtozero.Threetimes,andagaincomesbacktozeroagainthere.So,I'mlookingforasolutiontoLaplace'sequation --andI'vegotaprettygoodlist --thatwillmatchu_0whenrisone.Sothat'stheboundary,risone.Soyoucantellmewhatitis.Soyoucansolvethisproblem,rightaway.Theanswerisuofrandthetais,whatfunctionwill--?RememberI'vegotmyeyeonthatlist.Youtoo,right?I'mjusttryingtogetonethatwhenrisoneitwillmatchsin(3theta).What'sthegoodguy?It'llbeonthatlist.Whichofthose,byitself,hereIdon'tneedaseriesbecauseI'vegotsuchaneatu_0function.I'llgetitrightwithoneanswer,andwhatisthatanswer?Ilookthere.IsaywhatdoIdo,sothatatr=1,I'llmatchsin(3theta).I'llusercubedsin(3theta).So thegoodwinnerwillbercubedsin(3theta).ThatsolvesLaplace'sequation.Wecheckeditout,it'stheimaginarypartofx+iycubed.Wecouldwriteitinxandycoordinatesifwewantedbutwedon'twantto.Anditmatcheswhenrisone,itgivesussin(3theta).That'sit.AndofcourseIcouldtakeanyone.

NowsupposeI'mtryingtomatchsomethingthat'snotassimpleassin(3theta).Inthatcase,Imayhavetouseallof them.Imean,it'svery,veryflukythatonetermisgoing todoit.Usually,somymainexampleswouldbeI'llhavetomatchallofthem.SowhatdoIdo?Atr=1,somygeneralsolutionisacombinationoftheseguysIworkedsohardtoget.Thesolutionisofthisform.It'ssomea_0,theconstant.Andthena_1*r*cos(theta),andb_1*r*sin(theta).Anda_2rsquaredcos(2theta).Andsoon.I'mjusttakinganycombinationof,I'musingthea'sasthecoefficientsforthecosineguys.Andtheb's,b_1,b_2, b_3wouldbethecoefficientsforthesineone.OK,that'smygeneralsolution.ThatsolvesLaplace'sequation.Everytermdid,soeverycombinationwill.Now,setr=1.Tomatchthatr=1--andmatchtheboundary.Andmatchu_0(theta),therequiredtemperaturearoundthe--ontheboundary.The boundarybeing whererisone.Sothisissetr=1.Sothenu_0(theta),thisgiventhing,hastomatchthis,whenrisone.Soit'sa_0plusa_1,nowwhatdoIwritehere?risone,soit'sjustcos(theta).Now,b_1,risone,soIjusthavesin(theta).AndIhaveana_2*cos(2theta),andab_2*sin(2theta),andsoon.Here,justletmeputittogethernow.I'mgivenanytemperaturedistributionaroundtheboundary.It'sinequilibrium,thetemperature,whereifthetemperature'shighnearthatpointandlowoverherethetemperatureinsidewillgraduallygofromthathighpoint,dot dot dot dot,tothelowerone.Bymatchingontheboundary.Andthisisthematchontheboundary.

Now,thisisreallyaleadintothelastpartofthiscourse.Sowhosenameisassociatedwithaserieslikethat?Fourier.Yourecognizethataswhat'scalledaFourierseries.Sotheideais,I'mgiventheseboundaryvalues.Ifindtheirexpansioninsinesandcosines,andthat'swhatwe'lldoinNovember.AndthenI'vegotit.ThenIknowthea'sandtheb's.AndthenbasicallyIjustputinther's.randrsquaredsandrcubedsandsoon.SothenI'vegottheanswerinside.Inprincipleit'ssoeasy.So,whyisiteasy,though?First,it'seasybecauseit'sacirclewe'reworkingin.IfIwasinanellipseorastrangeshape,forgetit.Imean,sothisisquitespecial.Andsecondly,it'seasybecausethesefunctionsaresonice.Fourierworkswiththebestfunctionsever.Thesesinesandcosines.SoI'llfindawaytofindthosecoefficients,thea'sandtheb's.Eventhoughthere arelotsofthem,I'llbeabletopickthemoffoneatthetime,thea'sandb's.OnceIknowthea'sandb's,Iknowtheanswer.Sodoyouseethisisinprincipleagreatwaytosolveit?Infact,it'sthewayweusedoverhere,whenmyu_0wassin(3theta),thentheonlyterminitsFourierserieswas1*sin(3theta).Andthenthesolutionwas1rcubedsin(3theta).Soyoucanlearnthingsfromthis.Forexample,oh,whatcanyoulearn?OnethingInoticed,animportantfeatureofLaplace'sequationisthatthissolutioninsidethecirclegetsverysmooth.Theboundaryconditionscouldbelikeadeltafunction.Icouldsaythatontheboundary,thetemperatureiszeroeverywhereexceptatthatpointitspikes.SoIcouldtakeu_0--

Soexample2,andIwon'tdoitin full,wouldbeu_0ontheboundaryequaladeltafunction.Aspikeatthatonepoint.Soalltheheatiscomingfromthesourceatthatonepoint.LikeI'vegot afiregoingthere.Keepingtherestoftheboundaryfrozen,theheat'skind ofgoingtocomeinside.SothenhowwouldIproceed?Well,ifIhavethisboundaryvalueasadeltafunction,IlookforitsFourierseries,andit'saveryimportant,beautiful,Fourierseriesfora deltafunction.Wouldyouwanttoknowit?Imean,we'llknowitwellinNovember.Wouldyouwant toknowitinOctober?This isHalloween,Iguess,sodelta--I'lltellyouwhatitis.Sinceyouinsist.delta(theta)Ithinkwill--Ithinkthere's a1/(2pi)orsomething.Ah,shoot.We'llgetitexactlyright.It'ssomethinglike1and2cos(theta)and2cos(2theta),I'mnotsureaboutthe2pi.2cos(2theta),and2cos(3theta),andsoon.We'llknowitwellwhenwegetthere.WhatInoticeaboutthisdeltafunction--Ofcourseyou'regoingtoexpectthedeltafunctionbeingsomehowalittlebitstrange.Attheta=0,whatdoesthatseriesaddupto?Just so youbegintogetahangofFourierseries.Attheta=0,whatdoesthatserieslooklike?Well,allthesecosinethetasare?One.Sothisseriesattheta=0is1+2+2+2+2...It'sinfinite.Andthat'swhatwewant.Thedeltafunctionisinfiniteattheta=0.Andit'speriodic,ofcourse,sothatifIgoaroundtotheta=2piI'llcomebacktozeroagain.Attheta=pi,youcouldsortofsee,well,yeah,theta=pi isasortofinterestingpoint.Atthetaispi,what'sthecosine?Isnegativeone,right?Butthenthecos(2pi)willbeplusone.Soattheta=pi,IthinkI'mgettingaoneminusatwoplusatwo,minusatwo,plusatwo.Yousee,it'sdoingitsbesttocancelitselfoutandgivemethezerothatIwant,thetheta=pioverontheleftsideofthecircle.

Anyway,sothat'sanextremeexample.Butnow,what'sthetemperatureinside?Canyoujustfollowthesamerule?Whatwillbethetemperatureinside?Ifthat'sthedeltafunction,ifthat'stherightseries,whatever,itmightbea4pi,I'mnotsure,forthat.Now,youcantellmewhat'sthesolution,what'sthetemperaturedistributioninsideacirclewhenonepointontheboundaryhasaheatsource,adeltafunction.WhatdoIdo?HowdoImatchthiswiththisguy?Ijustputinther's,right?If thisiswhatit'ssupposedtomatchwhenrisone,thenwhenris--SomaybeI'llputitunderhere.Sotheu(r,theta),fromthedeltaguy,isjustputinther's.1+2r*cos(theta),and2rsquaredcos(2theta),andsoon.OK,andeventually2rtothe100thcos(100theta),andmore.OK,Iwritethisout,youcouldsaywhydidhewritethisdown?IwantedtomakethispointthattheimportantfeatureofthesolutiontoLaplace'sequationishowsmoothitgetswhenyougoinsidetheregion.Andwhyisthat?Becauseatr=1/2,thistermispracticallygone,right?IfIgohalfwayintothecircle,thistermispracticallygone.1/2tothehundredthpower.AndifIgotothecenterofthecircle,it'scompletelygone.Infact,what'sthevalueatthecenterofthecircle?What'sthetemperatureatthecenter?1/2pi.Thisistheonlytermthat'sremaining.Andit'stheaverage,aroundthecircle.Thatmakesphysicalsense,Iguess.Sincethewholething'scompletelyisotropic,we'vegotaperfectcircle,thevalueatthecenterofthecircleisalwaystheaveragegoingaround.TheconstanttermintheFourierseries,thisguy.We'llgettoknowthatoneverywell.That'stheaverage.You'rejustseeingalittlebitofFourierseriesearly,here.Butmypointisthatyoucouldhavehighoscillationaroundtheboundary,thatdampsoutbecauseofthesepowersofr.Andinsidethecircleit'sonlythelowordertermsthatbegintotakeover.

Thisisthekindoftrickyouhave,ornottrickbutthekindofmethodthatyoucanuseforsolvingLaplace'sequationbyaninfiniteseries.Ofcourse,apersonwhowantsanumbercancomplainthat,waitaminute,howdoIusethatinfiniteseries?Well,ofcourse,ifyouwantedtoknowthetemperatureat aparticularpointyou'dhavetopluginthatvalueofr,thatvalueoftheta,addupthetermsuntilyouhopethattheybecomesosmallthatyoucanignorethem.Soinfiniteseriesisoneformofasolution.Andsomehowtheseareexamples--Ishouldusethewordsseparationofvariables.Separationofvariablesisthegoldenideainthisanalysisstuff.SeparationofvariablesmeansIgottherpartseparatedfromthethetapart.Andthatworkedgreat,workedwellforacircle.Let'ssee,maybeforasquareIcouldtrytoseparatex fromy.Maybethere'sahomeworkproblem,asolutionthatseparatesxfromy,Ithinkissomethinglike--Sothiswouldbeanotherfamily.Goodforsquares,somethinglikesin(kx)sinhtimes,sothisseparationissomethinginxtimessomethinginy.AgainI'mjustmentioningthings.IthinkthatthatsolvesLaplace'sequationbecauseifItaketwoxderivatives,that'llbringdownksquared,butit'llflipthesign,right?Thesetwoderivativesofthesinewillbeaminus.AndifItake--Ineedakythere.AndifItooktwoderivativesofthishyperbolicsine,yourememberthat'sthee^(ky)andthee^(-ky).Thetwoderivativesofthatwillbringoutaksquaredwithaplussign.Sotwoxderivativesbringouttheminusksquared,twoyderivativesbringoutaplusksquaredandtogetherthatsolvesLaplace'sequation.We'llcheckthatinourhomeworkproblem.Sotherewouldbeanexample,goodforasquare.So,there'shopetodoanexactsolutioninaspecialregion.Now,what'sthisGreen'sfunctionidea?OK,that'snowthisisanotherthing.

Solasttimeweappreciatedthatthiscombinationx+iywasmagic.Theideawasthatwecouldtakeanyfunctionofx+iy,anditsolvesLaplace'sequation.Canwejustsee,sortofverycrudelywhythatis?Wesawthepattern,wesawx+iytothe nth.Sortof,wewentasfarasn=3,checkeditallout.Butnow,reallyifIwanttobeableto--whydoesthatsolveLaplace'sequationforanyn?ShouldIjustplugthatintoLaplace'sequation?WhathappensifItakethetwoxderivativesofthisthing?Sothisgoingtobeatypicalfunctionofx+iy,typicallyniceone.IfItaketwoxderivatives,IwanttoplugitinandseethatitreallydoessolveLaplace'sequation.Sotwoxderivativesofthatwillgivemewhat?Thefirstxderivativewillbringdownanntimesthisthingtothen-1.Andthenthenextxderivativewillbringdownann-1timesthisthingtothen-2.Sothat'llbetheu_xx.Andwhataboutu_yy?Thisismyu.I'msort ofjustcheckingthatyes,this--Seeagain,seeifitstillworksFridaywhatworkedWednesday.Thatthisx+iyismagicandfunctionsofitlikepowers,exponentials,logarithms,whatever,allsolveLaplace'sequation.OK,sowedidu_xx,andwegot--easy.Now,whathappenswithu_yy?Doyouseethepoint?

AUDIENCE: [INAUDIBLE]

PROFESSOR STRANG: Sorryoppositesign.Andwhydoesthesigncomeoutopposite?Becauseofthatguy.Yeah,it'sthechainrule,right?Thederivativeofthiswithrespecttoywillgivemeanntimesthisthingtoonelowerpower.Timesthederivativeofwhat'sinside.Andthederivativeofwhat'sinsideisani.Andthenthesecondderivativewillbringdownann-1,thisguywillbedown ton-2,anotheriwillcomeoutand--Justwhatyouwant,right?Becausetheisquaredisminusone,thosecancel.Whenthoseareequaloppositesigns.Andwegetu_xx+u_yyequaling0.Sothatworks.And,actually,thesameideawouldworkforanyfunctionofx+iy.Thetwoxderivativesjustgivef'.Twoyderivativeswillgivef'butthechainrulewillbringoutibothtimesandwe'vegotit.OK,Ithinkwejustneedanothercoupleofexamples.Andthisofcoursecouldbeinpolarcoordinates,fofre^(i*theta).That'sjust,everybodyrecognizesre^(i*theta)isthesameasx+iy?Betterjustbesurewe'vegotthat.x issomepointhereinthecomplexplane.iytakesusup tohere.Sothere'sx+iy.That'sx+iythere,butit'salso--Soletmeputthoseinbetter.So there'sx andthere'sy.Everybodyknowsthispicture,right?Thisxandthisy,nowifIwant togotopolarcoordinates,thatangleistheta,thisxisr*cos(theta),thisyisr*sin(theta),andthisguyisre^(i*theta).re^(i*theta).r*cos(theta)plusi*r*sin(theta)isthesameasre^i*theta.That'sutterlyfundamental.Everybody'sresponsibleforthatpictureofputtingthecomplexnumbersintotheirbeautifulpolarform.That'swhatmadeourrto thenthcos(n*theta)allsosimple.Now,whatwasIaimingtodo?Giveaparticularf.

NowIwant togiveaparticularfunctionf,ormaybeacoupleofchoices.Acoupleoffunctionsf,andseethattheirrealpartsandtheirimaginarypartssolveLaplace'sequation.Letmetakefirstaonethatworkscompletely.Taketherealpartand theimaginarypart--Letmetakee^(x+iy).It'safunctionofx+iy,extremelynicefunctionofx+iy,andwecanfigureoutitsrealandimaginaryparts,andwegettwosolutionstoLaplace'sequation.Thegoodwayistowritethisthingase^xtimese^(iy).Andagainwe'llwrite itase^xtimescos(y)+i*sin(y).SonowIcanseethattherealpart--Icanseewhattherealpartis,andIcanseewhattheimaginarypartis.Therealpartwillbe,that'sreal.Andthat'sreal.sothiswillsogivemee^x*cos(y).Andtheimaginarypartwillbee^x*sin(y).Youseeit.AndthosewillsolveLaplace'sequation.CanIgiveanametothiswholefieldofanalysis?Thise^zisananalytic --Ishouldjustusethatword --ananalyticfunction.Andtheseguys,therealandimaginaryparts,aretwoharmonicfunctions.Maybeit'snotsoimportanttoknowthewordharmonicfunction.Butanalyticfunction,yeah,Iwouldsaythat'sanimportantword.Actually,whatdoesit mean?It'safunctionofz.Sowe'reinthecomplexplaneherenow.It'safunctionofz,e^z,anditcanbewrittenasapowerseries,of course,onepluszplus1 over2 factorialzsquaredandallthoseguys.Soithasapowerseries.Thatmakesitacombinationofourspecialones.Thegreatthingaboutthatseriesisitconverges.Soananalyticfunction,ananalyticfunctionisthesumofapowerseriesthatconverges.Andthisonedoes.Sothere'sanexample.Yeah,sothewholetheoryof analyticfunctionsisactually,that'sChapter5ofthetextbook.Andwewon'tgetbeyondthispoint,Ithink,inonesemesterwithanalyticfunctions.

SowhatamIsaying,though?I'msayingthatthetheoryofanalyticfunctionsiscloselytiedtoLaplace'sequation.Becausetherealandtheimaginarypartsgivemethispairuandsthatsatisfy,theyeachsatisfyLaplace'sequation.Andthey'reconnectedbytheCauchy-Riemannequations.Boy,it'salotofmathematicscomingrealfasthere.NowI'dliketotakeonemoreexample.Insteadof theexponential,canwetakethelogarithm.Iwanttotakethelogofx+iy,andIwantyoutosplititintoitsrealandimaginaryparts,andgettheuandthesthatgowiththat.Sothiswaslikethenicestpossible.Wegotaseriesof,e^zisgoodforeveryz,theseriesconverges,fantastic.It'san analyticfunctioneverywhere.Bestpossible.Nowwegotoonethat'snotbestpossiblebutneverthelesshighlyvaluable.OK,soe^z,I'vedone.Letmeerasee^z,takelogz.OK,sonowI'mnotdoinge^zanymore.AndIwant tofindthelogarithm,OK.So,what'sthedealwiththelogarithm?Realandimaginaryparts.NowI'mgoingtotakethelogofx+iy.Thatisafunctionofx+iy,exceptatonepointithasaproblem,right?There'sapointwherethisisnotgoing tobeanalytic,andthere'sgoing tobeaspecialpointintheflowwhichissingularsomehow.Butawayfromthatpoint,wehaveanice-lookingfunction,thelogarithmofx+iy,andnowI'dliketogetitsrealandimaginaryparts.I'dliketoknowtheuandthes.Butnobodyintheirrightmindwantstotakethelogarithmofasum,right?That'saveryfoolishthingtotrytodo,thelogofasum.What'sthegoodwaytogetsomewherewiththis?Realandimaginarypart.Icantakethelogofaproduct.Sothepolariswaybetteragain.Iwant towritethisasalogofretothe--Iwanttowriteitthatway.Andnowwhat'sthelogofaproduct?Thesumofthetwopieces.SoIhavelogr,andthelogofe^(i*theta),whichis?Whichisi*theta.Boy,look,thisisfantastic.Fantasticexcept atzero.Imean,it'sfantasticbutit'sgotabigproblematzero.Butit'sanextremelyimportantexample.

Sowhat'stherealpart?It'ssittingthere.Thisismyu.Thisismyu(r,theta),myu(x,y),whateveryouwant,isthelogofr.Thelogofthesquarerootofxsquared plusysquared.Iclaimthatagainbythismagiccombination,thislog,this--risthesquare rootofxsquaredplus ysquared.IclaimifyousubstitutethatintoLaplace'sequationyouget zero.Itworks.Andwhat'stheimaginarypart,thes?Thetwinistheimaginarypart,whichistheta.Oh,whatisthetainx, ifIwanteditinxandy?Whatwouldthetabe?It'sthearctan,it's the anglewhosetangentissomething.y/x,soifIreallywantit inrectangularxystuff,it'stheanglewhosetangentisy/x.Andagain,ifyourememberincalculushowtotakederivativesofthisthingandyouplugitintoLaplace'sequationyougetzero.It works.Sothat'sagreatsolutionexceptwhere?Atzero.Except atzero.Andthisdoesn'ttelluswhat'shappeningatzero.It'sanexcellentsolution.What'sthepicture?SobyWednesday'sexamI'mnotexpectingyoutobeanexpertonthetheoryofanalyticfunctions.Idon'texpectyoutoknowanyconformalmappings.ByWednesday,God,that's--But,Idoexpectyou tohavethesepicturesinmind.

SowhenIdrawthoseaxes,whatpictureisit thatI'mplanningon?I'mplanningontheequipotentialsuequalconstant,andthe,whoaretheotherguys?Thestreamlines.Theplaceswherethestreamfunctions--Sohereisthepotentialfunction.Sowhataretheequipotentialcurves?Forthatguy?Circles.Thisisaconstantwhenrisaconstant,sotheequipotentialfunctionswouldbecircles.Idon'twant todrawthatcirclewithradiuszero,though.I'mnervousaboutthatone.Butalltheothersaregreat.Andwhatarethestreamlines,now?Thestreamlinesare,well,whatwillthestreamlinesbe?IfI'vedrawnonefamily,youcantellmetheotherfamily.Thestreamlineswillbe?Radiallines.Becausethey'regoing tobeperpendiculartothis.AndsowhatdoIget,thisisthestreamfunction,theta.Sowhat'sastreamline?Thestreamfunctionshouldbeaconstant.Theta'saconstant.ThatmeansI'mgoingoutonrays.Thoseareallstreamlines.Again,everythingfantastic.Ifyoulookinalittleregionhereyouseejustabeautifulpictureofequipotentialsandstreamlinescrossingthematrightangles.Everythinggreat.Justthatpointisobviouslyaproblem.

Now,andI'msuspectingthatthere'sasourcehere.Ithinkthisflow,whichisgivenbytheseguys,comesfromsomekindofadeltafunctionrightthere.Andtheflowgoesoutwards.SoIknowu,Iknowvisthegradientofu,right?Icouldtakethexandyderivatives,I'dknowthevelocity.Iknowthestreamfunction,thedivergencewouldbezero.Everythinggreat,exceptattheorigin.Ithinkwe'vegotsomeactionattheorigin.Because,here'sthewaytotestit.Iwant toseewhat'shappeningattheorigin.AndI'mgoingtousethedivergencetheorem.Yeah.Yeah.I'mgoingtousethedivergencetheorem.Sothedivergencetheoremsays--Whatisthedivergencetheorem?Sothisisthekeythingthatconnectsdoubleintegrals.LetmetakeacircleofradiusR.Sothat'sthecircleofradiusR.Rcouldbebig,orlittle.SoIintegrateoverthecircleofradiusR.Sowhat'sthedeal?visthesameasw.Whatdoesthedivergencetheoremtell me?IttellsmethatifIintegrate,whatdoIintegrate,thedivergenceofw?dx/dy,orr*dr*d theta.ThenIgettheflux.Sothisisakeyidentity.Fundamentally,morethanjustthekeyidentity,it'scentralhere.Thetotalflowoutoftheregionmustmakeitthroughtheboundary.SoIintegratethisboundary,andthisboundaryisacircleofradiusR,andwhatdoIintegratealongthatcircle?What'stheothersideofthedivergencetheorem?wdotn.wdotn,aroundtheboundary.Andremember,Ihavethisnice--mycurvehereisthisnicecircle.SoI'mgoingtointegratearoundthatcircle.Firstofall,whatisn?Bydefinition,nisthenormalthatpointsoutward,straightout.Soit'sactuallygoingoutthatway.Ateverypointit'spointingstraightout.Andds--Yeah,Ithinkwecanfigureoutexactlywhatthatright-handsideis.HowdoIgetthatright-handside?I'mlookingforw,andthenIhavetointegrate.OK,hereismyu.Myuislogr.Sowhat'sthegradientoflogr?Itpointsoutwards.Andhowlargeisthederivative?Sothederivativeofthislogris1/r.Ithinkthatthiscomesdownto,this istheintegral.Aroundthecircle.Ithinkthatthisthingis1/R.Iwentprettyquicklythere,soI'llaskyoutolookinthebookbecausethisissuchanimportantexampleit'sdonethereinmoredetail.SoI'mclaimingthatthederivativeis1/R,andthatitpointsdirectlyout.Sothegradientpointsout.Thenormalpointsout,sothatIjustgetexactly1/R.Now,whatisds?Forintegratingaroundthecirclewhat'salittletinypieceofarconacircle?OfradiusR?Rd theta.Goodman.Rd theta.Nowthat'sanintegralIcando,right?AndwhatdoIget?2pi.RcancelsR,I'mintegratingdthetaaroundfromzeroto2pi.Theansweris2pi.SowhatdoIlearnfromthat?Ilearnthatsomehowthissourcein theinsidehasstrength2pi.What'ssittinginthereis2pitimesadeltafunction.ThisisthesolutiontoLaplace'sequationexceptatthatsourceterm,soIreallyshouldsayPoisson'sequation.ThishasturnedouttobethesolutiontoPoissonwithadelta,orwith2pitimesadelta.Wehavejustsolvedthisimportantequation.Poisson'sequationwith apointsource.And,ofcourse,that'simportantbecausewhenyoucan solvewithapointsource,youcanputtogetherallsortsofsources.

AndthisiscalledtheGreen'sfunction.TheGreen'sfunctionisthesolutionwhenthesource isadelta.SoifIdivideby2pi,nowI'vegotit.Idividethis by2piandthereistheGreen'sfunction.Ihavetoputthatinboldletters.Green'sfunction.It'sthesolutiontotheequationwhenthesource isadeltaandtheanswerisuisthelog ofrover2pi.So that'stheGreen'sfunctionin2-D.Physicists,youknow,theyliveanddiewiththeseGreen'sfunction.Live, let's say,withGreen'sfunction.AndtheywouldwanttoknowtheGreen'sfunctionin3-D.SotheGreen'sfunctioninthreedimensionsalsoturnsoutbeautifully.Thisisin,theywouldsay,infreespace.ThisistheGreen'sfunctionwhenthere'snoothercharges.Nothingishappening,exceptforthechargerightatthecenter.AndifI'mintwodimensionstheGreen'sfunctionisthislogr.Soitgrowsmoreslowly.Itbehaveslikelogr.Andin3-DIthinktheansweris1/(4pi*r).It'sjustamazingthatthoseGreen'sfunctions,whentherightsideisadelta,havesuchniceformulas.OK,letmetakeonemomenthere.I'lltellyouwhatconformalmappingisabout.Butwhat'syourtake-homefromthislecture?Yourtake-homeistwomethodsthatwecanreallyusetogetaformulafortheanswer.OnemethodwasforLaplace'sequationinacircle.Gettheboundaryconditionsinaseriesofsinesandcosines,andthenjustputinther'sthatweneed.That'sasimple,simplemethod.ProvidedwecangetstartedwiththeFourierseries.Thesecondmethodis,lookatfunctionsofx+iy,andtrytopickonethatmatchesyourproblem.Andifyourproblemhasapointsource,attheorigin,wefoundtheone.Sotheliteratureforhundredsofyearsisaimedatsolvingotherproblems.If thepointsource issomewhereelse,whathappens?That'snothard.Ifit'snotapointsourcebutsomeotherkindofsource,oriftheregionisnotacircle.

CanIsayinonefinalsentencejustwhattodo,thisconformalmappingidea,whentheregionisnotacircle.Well,Icansayit inoneword,makeitacircle.Imean,that'swhatRiemannsaid,youcoulddoit.Youcouldthinkofafunction,soRiemannsaidthatthere'salwayssomefunctionofx+iy,letmecallthisRiemann'sfunctioncapitalFofx,y.Sothisisnowtheideaofconformalmapping.Changevariables.Conformalmappingisachangeofvariables.HepickedsomefunctionandletitsrealpartbeX andletitsimaginarypartbeY.CapitalY.OK,thisistotallyridiculoustoputconformalmappingin30seconds.But,nevermind,let'sjustdoit.Thebookdescribesconformalmappingsandclassicalappliedmathcoursesdomuchmorewithconformalmapping.Butthetruthis,computationallythey'renotanythinglikeasmuchusedasthese.Sowhat'stheidea?Theideaistofindaneatfunctionofx+iy,sothatyourcrazyboundarybecomesacircle.Inthecapital X,capitalYvariables.Soyou'remappingtheregion,ellipse,whateveritlookslike,bychangingfromlittlex,littley,whereitwasanellipse,tocapital X,capitalY,whereit'sacircle.AndthepointisLaplace'sequationstaysLaplace'sequation.ThatchangeofvariablesdoesnotmessupLaplace'sequation.Sothatthenyou'vegotitinacircle.Yousolveitinacircle,fortheseguys.Andthenyougoback.Inaword,you'reabletosolveLaplace'sequationinthiscrazyregionbecauseyouneverleavethemagicx+iy.Youfindacombinationwiththatmagicx+iythatmakesyourregionintoacircle.InthecirclewenowknowhowtousecapitalXplusicapital Y.You'restayingwiththatmagiccombinationandgettingtheregiontobewhatyoulike.

Sopeopleknowalotoftheseconformalmappings.AfamousoneistheJoukowskione,thattakessomethingthatlooksverylikeanairfoil,andyoucangetacircleoutofit.SoI'llputdownJoukowski'sname.Sothat'sonethatItrustCourse16stillfindsvaluable.It'satransformationthattakescertainshapesandtheyincludeshapesthatlooklikeairfoils,andproducecircles.OK,sosorryaboutsuchaquickpresentationofsuchabasicsubject.Conformalmapping,notonanyexam,that'dbeimpossible.It'sreallythisstuffthatyou'renumberoneresponsiblefor.