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ContentsCHAPTER99.19.29.39.4CHAPTER1010.110.210.310.410.5CHAPTER1111.111.211.311.411.5CHAPTER1212.112.212.312.4CHAPTER1313.113.213.313.413.513.613.7PolarCoordinatesandComplexNumbersPolarCoordinates348PolarEquationsandGraphs351Slope,Length,andAreaforPolarCurves356ComplexNumbers360InfiniteSeriesTheGeometricSeriesConvergenceTests:PositiveSeriesConvergenceTests:AllSeriesTheTaylorSeriesforex,sinx,andcosxPowerSeriesVectorsandMatricesVectorsandDotProductsPlanesandProjectionsCrossProductsandDeterminantsMatricesandLinearEquationsLinearAlgebrainThreeDimensionsMotionalongaCurveThePositionVector446PlaneMotion:ProjectilesandCycloids453TangentVectorandNormalVector459PolarCoordinatesandPlanetaryMotion464PartialDerivativesSurfacesandLevelCurves472PartialDerivatives475TangentPlanesandLinearApproximations480DirectionalDerivativesandGradients490TheChainRule497Maxima,Minima,andSaddlePoints504ConstraintsandLagrangeMultipliers514CHAPTER11VectorsandMatricesThischapteropensupanewpartofcalculus.Itismultidimensionalcalculus,becausethesubjectmovesintomoredimensions.Inthefirsttenchapters,allfunctionsdependedontimetorpositionx-butnotboth.Wehadf(t)ory(x).Thegraphswerecurvesinaplane.Therewasoneindependentvariable(xort)andonedependentvariable(yorf).Nowwemeetfunctionsf(x,t)thatdependonbothxandt.Theirgraphsaresurfacesinsteadofcurves.Thisbringsustothecalculusofseveralvariables.Startwiththesurfacethatrepresentsthefunctionf(x,t)orf(x,y)orf(x,y,,t).Iemphasizefunctions,becausethatiswhatcalculusisabout.EXAMPLE1f(x,t)=cos(x-t)isatravelingwave(cosinecurveinmotion).Att=0thecurveisf=cosx.Atalatertime,thecurvemovestotheright(Figure11.1).Ateachtwegetacross-sectionofthewholex-tsurface.Forawavetravelingalongastring,theheightdependsonpositionaswellastime.Asimilarfunctiongivesawavegoingaroundastadium.Eachpersonstandsupandsitsdown.Somehowthewavetravels.EXAMPLE2f(x,y)=3x+y+1isaslopingroof(fixedintime).Thesurfaceistwo-dimensional-youcanwalkaroundonit.Itisflatbecause3x+y+1isalinearfunction.Intheydirectionthesurfacegoesupat45.Ifyincreasesby1,sodoesf.Thatslopeis1.Inthexdirectiontheroofissteeper(slope3).Thereisadirectioninbetweenwheretheroofissteepest(slopefi).EXAMPLE3f(x,y,t)=cos(x-y-t)isanoceansurfacewithtravelingwaves.Thissurfacemoves.Ateachtimetwehaveanewx-ysurface.Therearethreevariables,xandyforpositionandtfortime.Ican'tdrawthefunction,itneedsfourdimensions!Thebasecoordinatesarex,y,tandtheheightisf.Thealternativeisamoviethatshowsthex-ysurfacechangingwitht.Attimet=0theoceansurfaceisgivenbycos(x-y).Thewavesareinstraightlines.Thelinex-y=0followsacrestbecausecos0=1.Thetopofthenextwaveisontheparallellinex-y=2n,becausecos2n=1.Figure11.1showstheoceansurfaceatafixedtime.Thelinex-y=tgivesthecrestattimet.Thewatergoesupanddown(likepeopleinastadium).Thewavegoestoshore,butthewaterstaysintheocean.11VectorsandMatricesFig.11.1Movingcosinewithasmallopticalillusion-thedarkerFig.11.2Linearfunctionsgiveplanes.bandsseemtogofromtoptobottomasyouturn.Ofcoursemultidimensionalcalculusisnotonlyforwaves.Inbusiness,demandisafunctionofpriceanddate.Inengineering,thevelocityandtemperaturedependonpositionxandtimet.Biologydealswithmanyvariablesatonce(andstatisticsisalwayslookingforlinearrelationslikez=x+2y).Aseriousjobliesahead,tocarryderivativesandintegralsintomoredimensions.Inaplane,everypointisdescribedbytwonumbers.Wemeasureacrossbyxandupbyy.Startingfromtheoriginwereachthepointwithcoordinates(x,y).Iwanttodescribethismovementbyavector-thestraightlinethatstartsat(0,O)andendsat(x,y).Thisvectorvhasadirection,whichgoesfrom(0,O)to(x,y)andnottheotherway.Inapicture,thevectorisshownbyanarrow.Inalgebra,visgivenbyitstwocomponents.Foracolumnvector,writexabovey:v=[,I(xandyarethecomponentsofv).Notethatvisprintedinboldface;itscomponentsxandyareinlightface.?Thevector-vintheoppositedirectionchangessigns.Addingvto-vgivesthezerovector(differentfromthezeronumberandalsoinboldface):X-Xandvv=[-0.]=[:IY-YNoticehowvectoradditionorsubtractionisdoneseparatelyonthex'sandy's:?Anotherwaytoindicateavectoris2Youwillrecognizevectorswithoutneedingarrows.11.IVectorsandDotProductsFig.11.3Parallelogramforv+w,stretchingfor2v,signsreversedfor-v.Thevectorvhascomponentsv,=3andv,=1.(Iwritev,forthefirstcomponentandv,forthesecondcomponent.Ialsowritexandy,whichisfinefortwocom-ponents.)Thevectorwhasw,=-1andw,=2.Toaddthevectors,addthecom-ponents.Todrawthisaddition,placethestartofwattheendofv.Figure11.3showshowwstartswherevends.VECTORSWITHOUTCOORDINATESInthathead-to-tailadditionofv+w,wedidsomethingnew.Thevectorwwasmovedawayfromtheorigin.Itslengthanddirectionwerenotchanged!Thenewarrowisparalleltotheoldarrow-onlythestartingpointisdifferent.Thevectoristhesameasbefore.Avectorcanbedefinedwithoutanoriginandwithoutxandyaxes.Thepurposeofaxesistogivethecomponents-theseparatedistancesxandy.Thosenumbersarenecessaryforcalculations.Butxandycoordinatesarenotnecessaryforhead-to-tailadditionv+w,orforstretchingto2v,orforlinearcombinations2v+3w.Someapplicationsdependoncoordinates,othersdon't.Generallyspeaking,physicsworkswithoutaxes-itiscoordinate-free.