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ComplexNumbersIComplexNumbersIIMatrixSolutionofSimultaneousEquationDifferentialCalculusIDifferentialCalculusIIIntegralCalculusIIntegralCalculusIIOrdinaryDifferentialEquationsIOrdinaryDifferentialEquationsIIOrdinaryDifferentialEquationsIIIFourierSeriesIFourierSeriesIILaplaceTransformVectorCalculusIVectorCalculusIIDisclaimerAboutUsHelpAndSupport

Chapter:1ComplexNumbersI

Topicscoveredinthissnack-sizedchapter:

ComplexNumbersImaginaryNumbersArgandDiagramComplexConjugateAddingandSubtractingComplexNumbersMultiplyingComplexNumbersDividingComplexNumbersTheComplexPlane

ComplexNumbers

Acomplexnumberisanumberconsistingofarealandimaginarypart.

Itcanbewrittenintheform a+bi.oWhereaandbarerealnumbers

i(calledasIota)isthestandardimaginaryunitwiththepropertyi2=-1.

ImaginaryNumbers

Imaginarynumberisthenumberwhichhasimaginarypart.Wecansplitthenegativenumbersintopositivenumberand1.Wearedefining(-1)=i

oi2=-1oi3=-ioi4=1

Thisisknownasimaginarynumber.

Example:

Findtheimaginarynumberofthefollowing(-5)Solution:

Weknow-5=-15So(-5)=(-15)=(-1)5

=2.23i

Sotheimaginarynumberis2.23i

ArgandDiagram

ThegraphicalrepresentationofthecomplexnumberfieldiscalledanArganddiagram.Anycomplexnumberz=a+ibcanberepresentedbyanorderedpair(a,b)andhenceplottedonxy-axiswiththerealpartmeasuredalongthex-axisandtheimaginarypartalongthey-axis.

ComplexConjugate

Inmathematics,complexconjugatesareapairofcomplexnumbers,bothhavingthesamerealpart,butwithimaginarypartsofequalmagnitudeandoppositesigns.

Theconjugateofthecomplexnumberz=a+ib,whereaandbarerealnumbers,is.

AddingandSubtractingComplexNumbers

Addorsubtracttwocomplexnumbers

and

Theruleistoaddtherealandimaginarypartsseparately:

z1+z2=a+ib+c+id

=a+c+i(b+d)

z1-z2=a+ibc-id

=ac+i(b-d)

Example1:

(1+i)+(3+i)

=1+3+i(1+1)

=4+2i

Example2:

(2+5i)-(1-4i)

=2+5i-1+4i

=1+9i

MultiplyingComplexNumbers

Wemultiplytwocomplexnumbersjustaswemultiplyexpressionsoftheform(x+y)together.

(a+ib)(c+id)

=ac+a(id)+(ib)c+(ib)(id)

=ac+iad+ibc-bd

=acbd+i(ad+bc)

Example:

(2+3i)(3+2i)

=23+22i+3i3+3i2i(Bysubstitutingi2=-1)

=6+4i+9i6=13i

DividingComplexNumbers

Fordividingtwocomplexnumbersmultiplytopandbottombythecomplexconjugateofthedenominator.

Thedenominator isnowarealnumber.

Example:

TheComplexPlane

Thecomplexplaneorz-planeisageometricrepresentationofthecomplexnumbers.ItcanbemodifiedasaCartesianplane.Therealpartofacomplexnumberrepresentedbyadisplacementalongthex-axisandtheimaginarypartbyadisplacementalongthey-axis.

Themultiplicationoftwocomplexnumberscanbeexpressedeasilyinpolarcoordinates.Themagnitudeormodulusoftheproductistheproductofthetwoabsolutevalue,ormoduli.

Theangleorargumentoftheproductisthesumofthetwoangles,orarguments.Inparticular,multiplicationbyacomplexnumberofmodulus1actsasarotation.

Geometricrepresentationof anditsconjugate inthecomplexplane

Thedistancealongtheredlinefromtheorigintothepointzisthemodulusorabsolutevalueofz.

Theangle istheargumentofz.

Chapter:2ComplexNumbersII


PolarFormofaComplexNumberEulersFormulaDeMoivresTheoremPowersofComplexNumbers

PolarFormofaComplexNumber

Thepolarformofacomplexnumberisanotherwaytorepresentacomplexnumber.Theformz=a+biiscalledtherectangularcoordinateformofacomplexnumber.

Thehorizontalaxisistherealaxisandtheverticalaxisistheimaginaryaxis.

Therealandcomplexcomponentsintermsofrand whereristhelengthofthevectorand istheanglemadewiththerealaxis.

FromPythagoreanTheorem:

Byusingthebasictrigonometricratios:

Multiplyingeachsidebyr:

Therectangularformofacomplexnumberisgivenby

Substitutethevaluesofaandb.

Inthecaseofacomplexnumber,rrepresentstheabsolutevalueormodulusandtheangle iscalledtheargumentofthecomplexnumber.

Thepolarformofacomplexnumber

is

Where,

and

and

For

Or

Or

For

Example:

Expressthecomplexnumberinpolarform.

Thepolarformofacomplexnumber is .So,firstfindtheabsolutevalueofr.

Nowfindtheargument .

Sincea>0,usetheformula

Thathere ismeasuredinradians.

Therefore,thepolarformof isabout .

EulersFormula

Eulersformulaestablishestherelationshipbetweenthetrigonometricfunctionsandcomplexexponentialfunction.

Itstatesthatforanyrealnumberx:eix=cosx+isinx

whereiistheimaginaryunit.

Proof:

Forrealvaluesofx:

Itcanalsobedemonstratedusingacomplexintegral.

Let

ln

So

DeMoivresTheorem

Theorem

Inpolarform,if and thentheproduct iseasityobtained:

Inparticular,if and (i.e. ),

Multiplyingeachsideby gives

Onaddingtheargumentsofthetermsintheproduct.Similarly

Aftercompletingnsuchproductswehave:

wherenisapositiveinteger.

Thisresultcanbeshowntobetrueforthosecasesinwhichnisanegativeintegerand

whennisarationalnumbere.g.

Note:

If isarationalnumber:

oThisresultisknownasDeMoivresTheorem.

InexponentialformDeMoivrestheorem,inthecasewhenpisapositiveinteger,issimplyastatementofthelawsofindices:

Example:

UseDeMoivrestheoremtoobtainanexpressionfor intermsofpowersofalone.

Solution:

FromDeMoivrestheoremwehave

However,expandingthelefthandside(using: )

Andthen,equatingtherealpartsofbothsides,givestherelation:

Replacing by ;

Finally:

istherequiredrelation.

PowersofComplexNumbers

Definition:If isasequenceofcomplexnumberssuchthatthelimits

and

Exist,thenwesaythat isthelimitof andiswrittenas:

Theorem2:If then

oIf then .

oIf then .

oIf then existsifandonlyif .

Corollary: existsifonly or .

oIf thenthepowers spiralinto .oIf thenthepowers spiraloutto .

oIf and thenthepowers runaroundontheunitcircle.

Example:

and

Chapter:3Matrix


MatrixTypesofMatricesPropertiesofMatrixAdditionPropertiesofMatrixMultiplicationTheTransposeofaMatrixDeterminantofaMatrixMinorsofaMatrixCo-factorofaMatrixAdjointofaMatrixInverseofaMatrixWaystofindtheInverseofMatrix

Matrix

AMatrixisarectangulararrayofnumbersenclosedbyapairofbracket.Asetofmnnumbersarrangedintheformofanorderedsetofmrowsandncolumnsiscalledmnmatrix(tobereadasmbynmatrix).

mnmatrixAiswrittenas:

A= oi=1,2,moj=1,2,n

Where representstheelementattheintersectionof throwand thcolumn.

TypesofMatrices

SquareMatrix

Amatrixinwhichthenumberofrowsisequaltothenumberofcolumnsiscalledasquarematrix.

oThusmnmatrixAwillbeasquarematrixifm=n

DiagonalElements

Inasquarematrixallthoseelements forwhich i.e.allthoseelementswhichoccurinthesamerowandsamecolumnnamely arecalledtheDiagonalElements.

DiagonalMatrix

AsquarematrixAissaidtobeadiagonalmatrixifallitsnon-diagonalelementsbezero.

Example:

ScalarMatrix

AdiagonalmatrixwhoseallthediagonalelementsareequaliscalledaScalarMatrix.

Example:

UnitorIdentityMatrix

Asquarematrix allofwhosenon-diagonalelementsarezeroandeachofthediagonalelementisunity.

Example:

and

Ingeneralforaunitmatrix, for and for

ZeroMatrixorNullMatrix

AnymnmatrixinwhichalltheelementsarezeroiscalledaZeromatrixorNullmatrixofthetypemnandisdenotedby

,

,

SymmetricMatrix

Asquarematrix willbecalledSymmetricifforallvaluesof and ,

Aisasymmetricmatrixinwhich ,

PropertiesofMatrixAddition

Matrixadditioniscommutative:A+B=B+A

Matrixadditionisassociative:A+(B+C)=(A+B)+C

PropertiesofMatrixMultiplication

MultiplicationofMatricesisdistributivewithrespecttoadditionofmatrices.oA(B+C)=AB+AC

MatrixMultiplicationisassociativeifconformabilityisassured.oA(BC)=(AB)C

ThemultiplicationofMatricesisnotalwayscommutative.oABisnotalwaysequaltoBA

MultiplicationofaMatrixAbyanullmatrixconformablewithAformultiplicationisanullmatrix.

oA0=0

IfAB=0thenitdoesnotnecessarilymeanthatA=0orB=0orbothare0.Example:

MultiplicationofMatrix byaUnitMatrixoLetAbeamnmatrixandIbeasquareunitmatrixofordern,sothatAandIareconformableformultiplicationthen

TheTransposeofaMatrix

If beagivenmatrixofthetypemnthenthematrixobtainedbychangingtherowsofAintocolumnsandcolumnsofAintorowsiscalledTransposeofmatrixAandisdenotedby

AstherearemrowsinAthereforetherewillbemcolumnsin andsimilarlyastherearencolumnsinAtherewillbenrowsin

Example:

Then

PropertiesofTranspose:

DeterminantofaMatrix

ADeterminantisarealnumberassociatedwitheverysquarematrix.

Thedeterminantofasquarematrix isdenotedby or

Determinantofa matrices

MinorsofaMatrix

AMinorforanyelementisthedeterminantthatresultswhentherowandcolumnofthatelementaredeleted.

Forthematrixshownbelow(Notethat isrow and iscolumn )

C1

C2

C3

R1

1

4

3

R2

0

5

2

R3

3

6

1

Minorfor ( , ,deleted)is :

C2

C3

R1

4

3

R3

6

1

Co-factorofaMatrix

Co-factorforanyelementiseithertheminorortheoppositeoftheminor,basedontheelementspositionintheoriginalDeterminant.

oIftherowandcolumnoftheelementadduptoanevennumber,theco-factoristhesameastheminor.oIftherowandcolumnoftheelementadduptoanoddnumbertheco-factoristheminorwithoppositesign.

AdjointofaMatrix

Thematrixformedbytakingthetransposeofthecofactormatrixofagivenmatrix.TheAdjointofmatrixisoftenwrittenasadjA.Example:

Findtheadjointforthematrix

Solution:

FirstdeterminethecofactormatrixoCofactorof5=3oCofactorof7=-4oCofactorof4=-7oCofactorof3=4

Cofactormatrix=

AdjA=Transposeofcofactormatrix

InverseofaMatrix

ForasquarematrixA,theinverseiswrittenas .WhenAismultipliedby theresultistheidentitymatrixI.

Non-squarematricesdonothaveinverses.

WaystofindtheInverseofMatrix

Adjointmethod:

or

Example:

Considerthematrix

ThecofactormatrixforAis

Sotheadjointis

SincedetA=22,weget

AugmentedMatrixMethod:

AnAugmentedMatrixisamatrixobtainedbyappendingthecolumnsoftwogivenmatrices,usuallyforthepurposeofperformingthesameelementaryrowoperationsoneachofthegivenmatrices.

Example:

GiventhematricesAandB,where:

,B=

Then,theaugmentedmatrix iswrittenas:

Chapter:4SolutionofSimultaneousEquation


SolutionofLinearEquationsCramersRuleSolutionofSimultaneousEquationbyGaussianEliminationmethodEigenvaluesandEigenvectorsCayleyHamiltonTheorem

SolutionofLinearEquations

Considerthesetofequations:

or

Theabovesetsofequationscanbeconvenientlywritteninmatrixformasunder:

or

Intheabove,thematrixAiscalledCoefficientMatrix.Iftheaboveequationshaveasolutionwesaythattheyareconsistentandhaveeitherauniquesolutionorinfinitesolutions.

Incasetheydonothaveanysolution,weshallsaythatthesystemsofequationsareinconsistent.

Example:

haveauniquesolution

x=1,y=2ascanbeverifiedbysolvingthem.

Herethecoefficientmatrixis

and

oMatrixAisnon-singularanditsinverseexists.oInthiscase,wewillhaveauniquesolution.Theaboveequationcanbewritteninmatrixformas:

oWhere isanon-singularmatrixas and

oMultiplyingbothsidesofequation by ,weget:

or

CramersRule

Givenasetoflinearequations:

Considerthedeterminant:

NowmultiplyDbyx,andusethepropertyofdeterminantsthatmultiplicationbyaconstantisequivalenttomultiplicationofeachentryinasinglecolumnbythatconstant,so

Anotherpropertyofdeterminantsenablesustoaddaconstanttimesanycolumntoanycolumnandobtainthesamedeterminant,soaddytimescolumn2andztimescolumn3tocolumn1,

If ,thenreducesto ,sothesystemhasnondegeneratesolutions(i.e.,solutionsotherthan )onlyif (inwhichcasethereisafamilyofsolutions).

If and ,thesystemhasnouniquesolution.

Ifinstead and ,thensolutionsaregivenby

andsimilarlyfor

Thisprocedurecanbegeneralizedtoasetofnequationsso,givenasystemofnlinearequations

Let

If ,thennondegeneratesolutionsexistonlyif .

If and ,thesystemhasnouniquesolution.Otherwise,compute

Then for Inthethree-dimensionalcase,thevectoranalogofCramersruleis

Example:

UseCramersRuletosolvethesystem:5x4y=2

6x5y=1

Solution:

Webeginbysettingupandevaluatingthethreedeterminants :

=(5)(-5)-(6)(-4)

=-25+24=-1

=(2)(-5)-1(-4)

=-10+4=-6

=(5)(1)-(6)(2)

=512=-7

FromCramersRule,wehave

Thesolutionis(6,7).

SolutionofSimultaneousEquationbyGaussianEliminationmethod

EquivalentSystems:

Twosystemsoflinearequationsareequivalentiftheyhaveidenticalsolutions.Echelonform:

Asystemofthreelinearequationsinvariablesx,y,zissaidtobeinechelonformifitcanbewrittenas

Wherethecoefficientsa,b,canddaregivennumbers,someofwhichmaybezero.

GaussianElimination:

ThesystematiceliminationofvariablestochangeasystemoflinearequationsintoanequivalentsysteminechelonformfromwhichwecanreadthesolutioniscalledGaussianElimination.

ElementaryOperationsandEquivalentSystems:

ThekeytoGaussianeliminationistheideaofelementaryoperation,thereplacementofoneequationinasystembyanothergivesanequivalentsysteminawaythatleavesthesolutionunchanged.

Let denotesthe equationofthesystemand-2 + iswhatwegetwhenwemultiplybothsidesofequation by-2andaddtheresulttoequation .

Operation

NotationandMeaning

Interchangetwoequations

meansinterchangeequation

and .

Multiplybyanonzeroconstant

meansreplaceequation

with .

Addamultipleofoneequationtoanotherequation

meansreplace with

.

Performinganyoftheelementaryoperationsonasystemoflinearequationsgivesanequivalentsystem.

Example:

SolvethefollowingsystemofequationbyGaussianElimination:

Solution:

Thefollowingelementaryoperationsleadtoanechelonform,fromwhichwefindx,yandz.

Wenowhaveanechelonforminwhich ,0.z=0,issatisfiedbyanynumberz.Thereforewehaveinfinitesolutions.

Letz=t,wheretisanynumber. impliesy=z+5=t+5.x=32y+2z=32(t+5)+2t=32t10+2t=-7Therefore,x=-7,y=t+5,z=t,wheretisanynumber.

EigenvaluesandEigenvectors

LetAbeann nmatrix.Thenumber isaneigenvalueofAifthereexistsanon-zerovector suchthat

Inthiscase,vector iscalledaneigenvectorofAcorrespondingto .

WecanrewritetheconditionAv= vas

=0

whereIisthen nidentitymatrix.Nowinorderofanon-zerovectorvtosatisfythis

equation,thedeterminantof mustbeequaltozero.Thatis,

ThisequationisknownascharacteristicequationofAand isthecharacteristicpolynomialofA.

Example:

LetA= .Then

p( )=det

=(2- )(-1- )(-4)(-1)

=

=( -3)( +2)

Thus, and aretheeigenvaluesofA.

Letsfindtheeigenvectorscorrespondingto

Letv= .Then(A3I)v=0givesus

= ,fromwhich

weobtaintheduplicateequations.

- -4 =0

- -4 =0

Ifwelet =t,then =-4t.Alleigenvectorscorrespondingto aremultiplesof

.

Repeatingthisfor ,wefindthat

4 -4 =0

- + =0

Ifwelet =t,then =t.Alleigenvectorscorrespondingto aremultiplesof

.

CayleyHamiltonTheorem

IfAisagivennnmatrixandInisthennidentitymatrix,thenthecharacteristicpolynomialofAisdefinedas:

p( )=det(A- I)

TheCayley-HamiltontheoremstatesthateverysquarematrixAsatisfiesitsowncharacteristicequation.

p(A)=0

Example:

ConsiderthematrixA=Itscharacteristicpolynomialis:

p( )=det(A- )

=

= 2

p(A)=(1-A)2-2

=A22A-1

Chapter:5DifferentialCalculusI


TaylorSeriesMaclaurinSeriesPartialDerivativeMaximaandMinima

TaylorSeries

ATaylorseriesisaseriesexpansionofafunctionaboutapoint.Aone-dimensionalTaylorseriesisanexpansionofarealfunctionf(x)aboutapointx=aisgivenby

Example:

FindtheTaylorseriesforf(x)=Solution:

Therefore,

MaclaurinSeries

AMaclaurinseriesisaTaylorseriesexpansionofafunctionabout0.

Example:

FindtheMaclaurinExpansionof .Solution:

Here,of(x)=cosx,of(x)=-sinx,of(x)=-cosx,of(x)=sinx,

Thenevaluatingeachoftheseatx=0of(0)=1of(0)=0of(0)=-1of(0)=0

Nowsubstitutingthesevaluesinto

PartialDerivative

Letudenotethefunctionofindependentvariablesxandy;i.e.,u=u(x,y).Atpoint(x,y)thepartialderivativeofuwithrespecttoxandyaredefinedby

Providethelimitexist.Wemayusethefollowingnotation:

Ifu=u(x,y)arecontinuous,then

Also,

Sincepartialdifferentiationissameastheordinarydifferentiationwithothervariablesregardedasconstants;thefollowingresultholdforpartialdifferentiation:

oDifferentialcoefficientofaSum:Letz(x,y)=u(x,y)+v(x,y).Thenwehave

oDifferentialcoefficientoftheproduct:Letz(x,y)=u(x,y).v(x,y)i.e.,z=uv.Thenwehave

oDifferentialcoefficientoftheQuotient:Let

Then

oFunctionofaFunction:Letz=f(u)andu=u(x,y).Then

oHeredz/duisusedandnotz/u,aszisthefunctionofsinglevariableu.

Example:

Findf/xandf/yforthefunctionf(x,y)=x3y+ex?Solution:

f(x,y)=x3y+ex

MaximaandMinima

Thediagrambelowshowspartofafunctiony=f(x).

ThepointAisalocalmaximumandthepointBisalocalminimum.Thetermlocalisusedsincethesepointsarethemaximumandminimuminthisparticularregion.

Theremaybeothersoutsidethisregion.Ateachofthesepointsthetangenttothecurveisparalleltothex-axissothederivativeofthefunctioniszero.

AboutthelocalmaximumpointAthegradientchangesfrompositive,tozero,tonegative.Thegradientisthereforedecreasing.

AboutthelocalminimumpointBthegradientchangesfromnegative,tozero,topositive.Thegradientisthereforeincreasing.

Therateofchangeofafunctionismeasuredbyitsderivative.oWhenthederivativeispositive,thefunctionisincreasing.oWhenthederivativeisnegative,thefunctionisdecreasing.

Thustherateofchangeofthegradientismeasuredbyitsderivative,whichisthesecondderivativeoftheoriginalfunction.Inmathematicalnotationthisisasfollows.

Atthepoint(a,b)

oIf and thenthepoint(a,b)isalocalmaximum

oIf and thenthepoint(a,b)isalocalminimum.

Example:

Findthestationarypointofthefunctiony= andhencedeterminethenatureofthispoint.

Solution:

Ify= then,

Now whenx=1.

Thefunctionhasonlyonestationarypointwhenx=1(andy=2).

Since(d2y)/(dx2)=2>0forallvaluesofx,thisstationarypointisalocalminimum.

Thusthefunctiony= hasalocalminimumatthepoint(1,2).

Chapter:6DifferentialCalculusII


TangentandNormalSub-tangentandSubnormalRadiusofCurvature

TangentandNormal

Theequationofthetangentatpoint(x1,y1)tothecurvey=f(x)is

Thenormalatpoint(x1,y1)tothecurvey=f(x)is

Example:

Tangentsaredrawnfromtheorigintothecurvey=sinx.Provethatpointsofcontactlieonx2y2=x2y2.

Solution:

If(x1,y1)isthepointofcontactthen

Differentiatey=sinxw.r.t.x,weget

Thetangentequationis:

Sincegiventhattangent(2)passesthrough(0,0),then

Fromequations(1)and(3),weget

Hencepointofcontactlieonthecurveis

Sub-tangentandSubnormal

LetthetangentPTandnormalPGatanypointP(x,y)ofanycurvemeettheaxisofxinTandGrespectively.

FromPdrawPMperpendicularonx-axis,thenthelengthTMiscalledCartesiansub-tangentatPandthelengthMGiscalledCartesiansubnormalatPofthecurve.

If betheanglewhichthetangentatPmakeswithaxisofx,thenslope

oSub-tangent=

oSubnormal=

oLengthoftangentatpointP(x,y)=PT

oLengthofnormalatpointP(x,y)=PG

oInterceptonx-axis

oInterceptony-axis

Example:

Provethatthecurveay2=(x+b)3.Thesubnormalvariesasthesquareofthesub-tangent.

Solution:

Givenay2=(x+b)3

Therefore,

Weknowthat

Therefore,

Hence,subnormal (subtangent)2.

RadiusofCurvature

LetPbeanypointonthecurveC.DrawthetangentatPtothecircle.

OsculatingCircle:

ThecirclehavingthesamecurvatureasthecurveatPtouchingthecurveatP,iscalledthecircleofcurvatureorOsculatingCircle.

RadiusofCurvature:

TheradiusoftheOsculatingcircleiscalledtheradiusofcurvatureandisdenotedby .

CentreofCurvature:

ThecenterofcurvatureforapointP(x,y)ofacurveisthecenterCofthecircleofcurvatureatP.Thecoordinates( )ofthecenterofcurvaturearegivenby

Example:

Forthecurvey=c ,showthat

Solution:

y=c

Chapter:7IntegralCalculusI


TheDefiniteIntegralPropertiesofDefiniteIntegralsDefiniteIntegralastheLimitofaSumSummationofSeriesUsingDefiniteIntegralsastheLimitasSum

TheDefiniteIntegral

Let betheprimitiveorantiderivativeofafunctionf(x)definedoninterval[a,b]

i.e., .

Thenthedefiniteintegraloff(x)over[a,b]isdenotedby andisdefinedas.

Thenumbersaandbarecalledthelimitsofintegration,aiscalledthelowerlimitandbtheupperlimit.

PropertiesofDefiniteIntegrals

Example:

Evaluate

Solution:

Put,

Weget

Nowintegratingbyparts,weget

DefiniteIntegralastheLimitofaSum

Letf(x)beacontinuousrealvaluedfunctiondefinedontheclosedinterval[a,b]whichisdividedontheclosedinterval[a,b]whichisdividedintonequalpartseachofwidthhbyintersecting(n1)pointsa+h,a+2h,a+3h,.,a+(n1)hbetweenaandb.

LetSndenotethesumoftheareasofnrectangles.ThenSn=h.f(a)+h.f(a+h)+h.f(a+2h)++h.f(a+(n-1)h)

Sn=h[f(a)+f(a+h)+f(a+2h)++f(a+(n-1)h)]

Sndenotestheareawhichisclosetotheareaoftheregionboundedthecurvey=f(x)x-axisandtheordinatesx=a,x=b.

Since

sothat

Hence

Where

Example:

Evaluate asthelimitofsum.

Solution:

Let

Bydefinition,wehave,

SummationofSeriesUsingDefiniteIntegralsastheLimitasSum

Iff(x)isanintegrablefunctiondefinedoninterval[a,b],thenwehave

Where

or

Puttinga=0,b=1,then

in(1),weget

Thisformulaisveryusefulinfindingthesummationofinfiniteserieswhichareexpressibleintheform

WorkingRule:

StepI:writethegivenseriesintheform

StepII:Replace by byxand bydx.

StepIII:Obtainlowerandupperlimitsbycomputing fortheleastandgreatestvaluesofrrespectively.

StepIV:Evaluatetheintegralobtainedinpreviousstep.Thevaluesoobtainedistherequiredsumofthegivenseries.

Example:

Evaluate:

Solution:

Thegivenlimit

Chapter:8IntegralCalculusII


BetaFunctionGammaFunctionRelationbetweenBetaandGammaFunctionsEvaluationofDoubleIntegralsEvaluationofTripleIntegrals

BetaFunction

BetaFunction:TheBetafunctiondenotedbyB(m,n)withparameterm,nisdefinedas

Properties:

GammaFunction

GammaFunction:TheGammafunctionisdefinedasthedefiniteintegral.

ThisfunctionisalsocalledEulerianintegralofsecondkind.

Properties:

RelationbetweenBetaandGammaFunctions

Toprovethat,

Proof:

Weknowthat

Putting weget

Similarly

Therefore,

Changingtopolarcoordinates,weput

weget

Hence,

Example:

Provethat:

Solution:

Weknowthat:

Puttingm+n=1i.e.,m=1nintheequation(1)weget

Since,weknowthat

Puttingthesevaluesin(2),weget

EvaluationofDoubleIntegrals

IfthegivenregionRbeboundedbytheinequalities and

Intheaboveintegral iscalculatedfirst.Duringthisintegrationxisregardedasconstant.

Example:

Solution:

Example:

Findtheareabetweentheparabolasy2=4axandx2=4ay.Solution:

Solvingequation(1)and(2)wegetthepoint(4a,4a)

TakingstripPQparalleltox-axissothat

limits to andyvaries0to4a

EvaluationofTripleIntegrals

IftheregionVbeboundedbythe

inequalities thenintegral

oIfthelimitsofzaregivenfunctionsofxandylimitsyasfunctionsofxwhilextaketheconstantvalues,then

oWeintegratefirstw.r.t.zkeepingxandyconstantsandthentheremainingintegrationdoneasinthecaseofdoubleintegrals.IftheareaSisboundedbythecurvesy=y1(x),y=y2(x),xandx=b.

Example:

Evaluatethefollowingintegral.

Solution:

Chapter:9OrdinaryDifferentialEquationsI


DifferentialEquationOrderandDegreeofaDifferentialEquationLinearandNon-linearDifferentialEquationsSolutionofaDifferentialEquationSolutionofFirstOrderandFirstDegreeDifferentialEquationsDifferentialEquationswhereVariablesareSeparableEquationsReducibletoVariableSeparableHomogeneousDifferentialEquationsLeibnitzsLinearDifferentialEquations

DifferentialEquation

Anequationwhichcontainsderivativesofoneormoreindependentvariablesiscalleddifferentialequation.

Example:

OrderandDegreeofaDifferentialEquation

Order:

Theorderofadifferentialequationistheorderofthehighestorderderivativeappearingintheequation.

Example:

isoforder2,becausethehighestorderderivativeis2.

DegreeofDifferentialEquation:

Thedegreeofadifferentialequationisthedegreeofhighestderivativewhichoccursinit.

Example:

Thisdifferentialequationisofdegreetwo.

LinearandNon-linearDifferentialEquations

SolutionofaDifferentialEquation

Asolutionorintegralofadifferentialequationisarelationbetweenthevariables,andconstantwhichsatisfiesthegivendifferentialequation.

GeneralSolution:

Thesolutionwhichcontainsasmanyasarbitraryconstantsastheorderofthedifferentialequationiscalledthegeneralsolutionofthedifferentialequation.

Example:

y=Acosx+Bsinx,isthegeneralsolutionofthedifferentialequation

ParticularSolution:

Solutionobtainedbygivingparticularvaluestothearbitraryconstantsinthegeneralsolutionofadifferentialequationiscalledaparticularsolution.

Example:

y=3cosx+2sinx,isaparticularsolutionofthedifferentialequation

Example:

Showthatthefunctiony=(A+B)e3xisasolutionoftheequation

Solution:

Given

Differentiating(1)w.r.t.x,weget


Wehave

=0

satisfiesthegivendifferentialequation.

Hence,itisasolutionofthegivendifferentialequation.

SolutionofFirstOrderandFirstDegreeDifferentialEquations

Adifferentialequationoffirstorderandfirstdegreeinvolvesx,yand Soitcanbeputinanyoneofthefollowingforms:

Wheref(x,y)andg(x,y)areobviouslythefunctionofxandy.

DifferentialEquationswhereVariablesareSeparable

Ifthedifferentialequationscanbeputintheform ,wesaythatthevariablesareseparableandsuchequationscanbesolvedbyintegratingonbothsides.

Thesolutionisgivenby

WhereCisanarbitraryconstant.

Example:

Solve

Solution:

Integratingbothsides,weget

EquationsReducibletoVariableSeparable

Differentialequationsoftheform canbereducedtovariableseparableformbythesubstitution .

Example:

Solve

Solution:

Givendifferentialequationis

Put

Henceequation(1)becomes,

Integratingbothsides,weget

HomogeneousDifferentialEquations

Afunctionf(x,y)iscalledahomogeneousfunctionofdegreenif

Alternatively,ahomogeneousfunctionf(x,y)ofdegreencanalwaysbewrittenas

Example:

f(x,y)=x2y2+3xyisahomogeneousfunctionofdegree2,because

Thefirstorderfirstdegreedifferentialequationisoftheform

oWheref(x,y)andg(x,y)arehomogeneousfunctionsofthesamedegree,then(1)iscalledahomogeneousdifferentialequation.

Thegivendifferentialequationcanbewrittenas

Ify=vx,then .

Substitutingthesevaluesin weget

Onintegration,

WhereCisanarbitraryconstantofintegration.

Example:

Solve:

Solution:

Thegivenequationcanbewrittenas

Put

Hence(1)becomes,

Onintegrating,weget

LeibnitzsLinearDifferentialEquations

LinearDifferentialEquationiny:Thegeneralformofalineardifferentialequationis

oWherePandQarefunctionsofxonlyorconstants.

Tosolvetheequation,whentheyaremultipliedbyafactor,whichiscalledintegralfactor(I.F.).

Multiplyingbothsidesof(1)byI.F. ,weget

Whichistherequiredsolution,whereCistheconstantofintegration.

Example:

Solve:

Solution:

Hereyisalone,soitmaybelinearinyTherefore,

Here

Therefore,

Hencesolution:

Putting

Puttingthevalueoftintheaboveequation

Chapter:10OrdinaryDifferentialEquationsII


BernoullisDifferentialEquationorReducibletoLinearDifferentialEquationsExactDifferentialEquationsEquationsofFirstorderandHigherDegreeEquationSolvableforpEquationsSolvableforyEquationsSolvableforxClairautsEquation

BernoullisDifferentialEquationorReducibletoLinearDifferentialEquations

Adifferentialequationsoftheform

issaidtobeaBernoullisEquation,wherePandQarefunctionsofxofconstants.Dividingbothsidesof(1),byyn,weget

Put

Hence(2)becomes:

oWhichisalineardifferentialequationint.

Example:

Solution:

Rewritegivenequation:

Putting

Hence(1)becomes:

oWhichisalinearD.Einit.

Therefore,

Thecompletesolutionis

ExactDifferentialEquations

Afirstorderdifferentialequationoftheform

issaidtobeexactdifferentialequation,ifandonlyifitsatisfythefollowingnecessarycondition:

oWhereMandNarefunctionsofxandy.

Thesolutionofanexactequation(1)willbe

Example:

Solution:

Comparingwith

Here

Therefore,wehave

Therefore,Clearly

Hencegivenequation(1)isanexact.

TheonlynewtermobtainedonintegratingNwithrespecttoyisyasthetermsarealreadypresentintheintegrationofM.

Hencethegeneralsolutionofthegivendifferentialequationis:

EquationsofFirstorderandHigherDegree

Adifferentialequationofthefirstorderandnthdegreeis

Where,

isrepresentedbyp

arefunctionsofxandyorconstants.

Example:

Themethodofsolutionofaboveequationdependsupon,whetheritisoSolvableforp,oSolvablefory,oSolvableforx,oClairautsform

EquationSolvableforp

Example:

Solve:

Solution:

Given:

Hencegeneralsolutionofgivenequationis

EquationsSolvablefory

Thegivenequationbesolvablefory,sothatitcanbeputintotheform

Differentiatingwithrespecttox,wehave

Whichisadifferentialequationinthevariablesxandp.letitspossiblesolutionbe

Example:

Solve

Solution:

Differentiatingthegivenequationwithrespecttox,weget

Neglectingthesecondfactornotcontainingthederivativesofp,

Integrating,

Thegivenequationmaybewrittenas

Eliminatingpfromtheaboveequation

oWhichistherequiredsolutionofthegivenequation.

EquationsSolvableforx

Ifgivendifferentialequationofthefrom

Thenitcanbesolvedforx.Differentiating(1)w.r.t.yweget

Nowonsolvingtheaboveequation(2),wegetthesolutionof(2),say

Theneliminatingpfrom(1)wegettherequiredsolutionofgivenequation(1).

Example:

Solvethefollowing:

Solution:

Thegivenequationcanbewrittenas

Differentiatingw.r.t.y,weget

or

Rejectingthesecondfactor,wehave

Integrationgives,

Substitutingthisvalueofpinthegivenequation,wehave

oWhichistherequiredsolutionofthegivendifferentialequation.

ClairautsEquation

Thedifferentialequationoftheform

isknownasClairautsequation.


Onintegrating,weget

Eliminatingpfromequations(1),weget

Example:

Solvethefollowing

Solution:

Givendifferentialequationis

Takingsubstitution and

Therefore

Puttingvalueofpin(1),weget

WhichisClairautsform

Henceputting wegetthesolution

Chapter:11OrdinaryDifferentialEquationsIII


LinearHigherOrderDifferentialEquationsLinearNon-HomogenousEquationswithConstantCoefficientsParticularIntegralShortMethodsoffindingParticularIntegrals

LinearHigherOrderDifferentialEquations

Alineardifferentialequationofnthorderisoftheform

Wherea1,a2,.anandQarefunctionsofxorconstants.Ifa1,a2,anareallconstants,thenaboveequationiscalledalineardifferentialequationofnthorderwithconstantcoefficients.

ThesymbolD

ThesymbolsDandDnareusedfor and respectively.

Theexpression

iscalledadifferentialoperatorofthenthorder.

ComplementaryFunction(C.F.)

CaseI:WhentherootsofA.E.areallrealanddistinct.

owherec1,c2,.,cnarearbitraryconstants.

CaseII:WhensomerootsofA.E.arerealandequal.

oIftworootsareequal,m1=m2=m,say,thenfortheseequalroots

oIfthreerootsareequal,m1=m2=m3=m,saythenfortheseequalroots

oSimilarly,wecanwriteC.F.whenfour,five,,rootsareequal.

CaseIII:WhensomerootsofA.E.areimaginary.Remembertheimaginaryrootsoccurinpairs.

oIfA.E.hasonepairofimaginaryrootsi.e.,tworootsareimaginary,say,i.e., thenfortheseroots

Or

Or

oIfA.E.hastwoequalpairsofimaginaryrootsi.e.,fourrootsareimaginary,sayi.e. thenfortheseroots

andsoon.

CaseIV:WhensomerootsofA.E.areirrational.Rememberthatirrationalrootsoccurinpairs.

oIfA.E.hasonepairofirrationalroots,say, where ispositivei.e.,

,thenfortheseroots

Or

Or

oIfA.E.hastwoequalpairsofirrationalroots,say i.e.,

,thenfortheseroots.

LinearNon-HomogenousEquationswithConstantCoefficients

Thegeneralsolutionofthelinearnon-homogeneousdifferentialequation.

i.e.,of isgivenbyy=C.F.+P.I.

ParticularIntegral

TheP.Iofthedifferentialequationf(D)y=Qisdefinedby

TofindtheParticularIntegral

ShortMethodsoffindingParticularIntegrals

TofindP.I.oftheform ,when

If where ThentoFindP.I.oftheform

ToFindP.I.oftheforms when

TofindP.I.oftheform ,WhereVisafunctionofx

TofindP.I.theforms and

Tofind ,whereVissomefunctionofx

Chapter:12FourierSeriesI


FourierSeriesDifferentformsofEulerFormulaeDirichletsConditionsParsevalsIdentityforFourierSeriesEvenandOddFunctionHalfRangeSeries(HalfRangeExpansion)

FourierSeries

PeriodicFunctions:

Afunctionf(x)issaidtobeperiodicfunctionT>0ifforallx,f(x+T)=f(x)andTistheleastofsuchvalues.

Example:

sinx,cosxareperiodicfunctionswithperiod2 .

tanx,cotxareperiodicfunctionswithperiod .

FourierSeries:

FourierSeriesisaninfiniteseriesrepresentationofperiodicfunctionintermsofthetrigonometricsineandcosinefunctions.

Fourierseriesistobeexpressedintermsofperiodicfunctions-sinesandcosines.ThestudyofFourierseriesisabranchofFourieranalysis.

Eulersformulae:

Fourierseriesforthefunctionsf(x)intheinterval

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