1 Data Compression Engineering Math Physics (EMP) Steve Lyon Electrical Engineering.
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TableofContents
ComplexNumbersIComplexNumbersIIMatrixSolutionofSimultaneousEquationDifferentialCalculusIDifferentialCalculusIIIntegralCalculusIIntegralCalculusIIOrdinaryDifferentialEquationsIOrdinaryDifferentialEquationsIIOrdinaryDifferentialEquationsIIIFourierSeriesIFourierSeriesIILaplaceTransformVectorCalculusIVectorCalculusIIDisclaimerAboutUsHelpAndSupport
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Chapter:1ComplexNumbersI
Topicscoveredinthissnack-sizedchapter:
ComplexNumbersImaginaryNumbersArgandDiagramComplexConjugateAddingandSubtractingComplexNumbersMultiplyingComplexNumbersDividingComplexNumbersTheComplexPlane
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ComplexNumbers
Acomplexnumberisanumberconsistingofarealandimaginarypart.
Itcanbewrittenintheform a+bi.oWhereaandbarerealnumbers
i(calledasIota)isthestandardimaginaryunitwiththepropertyi2=-1.
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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
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ArgandDiagram
ThegraphicalrepresentationofthecomplexnumberfieldiscalledanArganddiagram.Anycomplexnumberz=a+ibcanberepresentedbyanorderedpair(a,b)andhenceplottedonxy-axiswiththerealpartmeasuredalongthex-axisandtheimaginarypartalongthey-axis.
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ComplexConjugate
Inmathematics,complexconjugatesareapairofcomplexnumbers,bothhavingthesamerealpart,butwithimaginarypartsofequalmagnitudeandoppositesigns.
Theconjugateofthecomplexnumberz=a+ib,whereaandbarerealnumbers,is.
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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
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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
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DividingComplexNumbers
Fordividingtwocomplexnumbersmultiplytopandbottombythecomplexconjugateofthedenominator.
Thedenominator isnowarealnumber.
Example:
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TheComplexPlane
Thecomplexplaneorz-planeisageometricrepresentationofthecomplexnumbers.ItcanbemodifiedasaCartesianplane.Therealpartofacomplexnumberrepresentedbyadisplacementalongthex-axisandtheimaginarypartbyadisplacementalongthey-axis.
Themultiplicationoftwocomplexnumberscanbeexpressedeasilyinpolarcoordinates.Themagnitudeormodulusoftheproductistheproductofthetwoabsolutevalue,ormoduli.
Theangleorargumentoftheproductisthesumofthetwoangles,orarguments.Inparticular,multiplicationbyacomplexnumberofmodulus1actsasarotation.
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Geometricrepresentationof anditsconjugate inthecomplexplane
Thedistancealongtheredlinefromtheorigintothepointzisthemodulusorabsolutevalueofz.
Theangle istheargumentofz.
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Chapter:2ComplexNumbersII
Topicscoveredinthissnack-sizedchapter:
PolarFormofaComplexNumberEulersFormulaDeMoivresTheoremPowersofComplexNumbers
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PolarFormofaComplexNumber
Thepolarformofacomplexnumberisanotherwaytorepresentacomplexnumber.Theformz=a+biiscalledtherectangularcoordinateformofacomplexnumber.
Thehorizontalaxisistherealaxisandtheverticalaxisistheimaginaryaxis.
Therealandcomplexcomponentsintermsofrand whereristhelengthofthevectorand istheanglemadewiththerealaxis.
FromPythagoreanTheorem:
Byusingthebasictrigonometricratios:
Multiplyingeachsidebyr:
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Therectangularformofacomplexnumberisgivenby
Substitutethevaluesofaandb.
Inthecaseofacomplexnumber,rrepresentstheabsolutevalueormodulusandtheangle iscalledtheargumentofthecomplexnumber.
Thepolarformofacomplexnumber
is
Where,
and
and
For
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Or
Or
For
Example:
Expressthecomplexnumberinpolarform.
Thepolarformofacomplexnumber is .So,firstfindtheabsolutevalueofr.
Nowfindtheargument .
Sincea>0,usetheformula
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Thathere ismeasuredinradians.
Therefore,thepolarformof isabout .
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EulersFormula
Eulersformulaestablishestherelationshipbetweenthetrigonometricfunctionsandcomplexexponentialfunction.
Itstatesthatforanyrealnumberx:eix=cosx+isinx
whereiistheimaginaryunit.
Proof:
Forrealvaluesofx:
Itcanalsobedemonstratedusingacomplexintegral.
Let
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ln
So
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DeMoivresTheorem
Theorem
Inpolarform,if and thentheproduct iseasityobtained:
Inparticular,if and (i.e. ),
Multiplyingeachsideby gives
Onaddingtheargumentsofthetermsintheproduct.Similarly
Aftercompletingnsuchproductswehave:
wherenisapositiveinteger.
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Thisresultcanbeshowntobetrueforthosecasesinwhichnisanegativeintegerand
whennisarationalnumbere.g.
Note:
If isarationalnumber:
oThisresultisknownasDeMoivresTheorem.
InexponentialformDeMoivrestheorem,inthecasewhenpisapositiveinteger,issimplyastatementofthelawsofindices:
Example:
UseDeMoivrestheoremtoobtainanexpressionfor intermsofpowersofalone.
Solution:
FromDeMoivrestheoremwehave
However,expandingthelefthandside(using: )
Andthen,equatingtherealpartsofbothsides,givestherelation:
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Replacing by ;
Finally:
istherequiredrelation.
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PowersofComplexNumbers
Definition:If isasequenceofcomplexnumberssuchthatthelimits
and
Exist,thenwesaythat isthelimitof andiswrittenas:
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Theorem2:If then
oIf then .
oIf then .
oIf then existsifandonlyif .
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Corollary: existsifonly or .
oIf thenthepowers spiralinto .oIf thenthepowers spiraloutto .
oIf and thenthepowers runaroundontheunitcircle.
Example:
and
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Chapter:3Matrix
Topicscoveredinthissnack-sizedchapter:
MatrixTypesofMatricesPropertiesofMatrixAdditionPropertiesofMatrixMultiplicationTheTransposeofaMatrixDeterminantofaMatrixMinorsofaMatrixCo-factorofaMatrixAdjointofaMatrixInverseofaMatrixWaystofindtheInverseofMatrix
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Matrix
AMatrixisarectangulararrayofnumbersenclosedbyapairofbracket.Asetofmnnumbersarrangedintheformofanorderedsetofmrowsandncolumnsiscalledmnmatrix(tobereadasmbynmatrix).
mnmatrixAiswrittenas:
A= oi=1,2,moj=1,2,n
Where representstheelementattheintersectionof throwand thcolumn.
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TypesofMatrices
SquareMatrix
Amatrixinwhichthenumberofrowsisequaltothenumberofcolumnsiscalledasquarematrix.
oThusmnmatrixAwillbeasquarematrixifm=n
DiagonalElements
Inasquarematrixallthoseelements forwhich i.e.allthoseelementswhichoccurinthesamerowandsamecolumnnamely arecalledtheDiagonalElements.
DiagonalMatrix
AsquarematrixAissaidtobeadiagonalmatrixifallitsnon-diagonalelementsbezero.
Example:
ScalarMatrix
AdiagonalmatrixwhoseallthediagonalelementsareequaliscalledaScalarMatrix.
Example:
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UnitorIdentityMatrix
Asquarematrix allofwhosenon-diagonalelementsarezeroandeachofthediagonalelementisunity.
Example:
and
Ingeneralforaunitmatrix, for and for
ZeroMatrixorNullMatrix
AnymnmatrixinwhichalltheelementsarezeroiscalledaZeromatrixorNullmatrixofthetypemnandisdenotedby
,
,
SymmetricMatrix
Asquarematrix willbecalledSymmetricifforallvaluesof and ,
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Aisasymmetricmatrixinwhich ,
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PropertiesofMatrixAddition
Matrixadditioniscommutative:A+B=B+A
Matrixadditionisassociative:A+(B+C)=(A+B)+C
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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
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TheTransposeofaMatrix
If beagivenmatrixofthetypemnthenthematrixobtainedbychangingtherowsofAintocolumnsandcolumnsofAintorowsiscalledTransposeofmatrixAandisdenotedby
AstherearemrowsinAthereforetherewillbemcolumnsin andsimilarlyastherearencolumnsinAtherewillbenrowsin
Example:
Then
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PropertiesofTranspose:
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DeterminantofaMatrix
ADeterminantisarealnumberassociatedwitheverysquarematrix.
Thedeterminantofasquarematrix isdenotedby or
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Determinantofa matrices
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Determinantofa matrices
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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
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Co-factorofaMatrix
Co-factorforanyelementiseithertheminorortheoppositeoftheminor,basedontheelementspositionintheoriginalDeterminant.
oIftherowandcolumnoftheelementadduptoanevennumber,theco-factoristhesameastheminor.oIftherowandcolumnoftheelementadduptoanoddnumbertheco-factoristheminorwithoppositesign.
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AdjointofaMatrix
Thematrixformedbytakingthetransposeofthecofactormatrixofagivenmatrix.TheAdjointofmatrixisoftenwrittenasadjA.Example:
Findtheadjointforthematrix
Solution:
FirstdeterminethecofactormatrixoCofactorof5=3oCofactorof7=-4oCofactorof4=-7oCofactorof3=4
Cofactormatrix=
AdjA=Transposeofcofactormatrix
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InverseofaMatrix
ForasquarematrixA,theinverseiswrittenas .WhenAismultipliedby theresultistheidentitymatrixI.
Non-squarematricesdonothaveinverses.
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WaystofindtheInverseofMatrix
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Adjointmethod:
or
Example:
Considerthematrix
ThecofactormatrixforAis
Sotheadjointis
SincedetA=22,weget
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AugmentedMatrixMethod:
AnAugmentedMatrixisamatrixobtainedbyappendingthecolumnsoftwogivenmatrices,usuallyforthepurposeofperformingthesameelementaryrowoperationsoneachofthegivenmatrices.
Example:
GiventhematricesAandB,where:
,B=
Then,theaugmentedmatrix iswrittenas:
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Chapter:4SolutionofSimultaneousEquation
Topicscoveredinthissnack-sizedchapter:
SolutionofLinearEquationsCramersRuleSolutionofSimultaneousEquationbyGaussianEliminationmethodEigenvaluesandEigenvectorsCayleyHamiltonTheorem
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SolutionofLinearEquations
Considerthesetofequations:
or
Theabovesetsofequationscanbeconvenientlywritteninmatrixformasunder:
or
Intheabove,thematrixAiscalledCoefficientMatrix.Iftheaboveequationshaveasolutionwesaythattheyareconsistentandhaveeitherauniquesolutionorinfinitesolutions.
Incasetheydonothaveanysolution,weshallsaythatthesystemsofequationsareinconsistent.
Example:
haveauniquesolution
x=1,y=2ascanbeverifiedbysolvingthem.
Herethecoefficientmatrixis
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and
oMatrixAisnon-singularanditsinverseexists.oInthiscase,wewillhaveauniquesolution.Theaboveequationcanbewritteninmatrixformas:
oWhere isanon-singularmatrixas and
oMultiplyingbothsidesofequation by ,weget:
or
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CramersRule
Givenasetoflinearequations:
Considerthedeterminant:
NowmultiplyDbyx,andusethepropertyofdeterminantsthatmultiplicationbyaconstantisequivalenttomultiplicationofeachentryinasinglecolumnbythatconstant,so
Anotherpropertyofdeterminantsenablesustoaddaconstanttimesanycolumntoanycolumnandobtainthesamedeterminant,soaddytimescolumn2andztimescolumn3tocolumn1,
If ,thenreducesto ,sothesystemhasnondegeneratesolutions(i.e.,solutionsotherthan )onlyif (inwhichcasethereisafamilyofsolutions).
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If and ,thesystemhasnouniquesolution.
Ifinstead and ,thensolutionsaregivenby
andsimilarlyfor
Thisprocedurecanbegeneralizedtoasetofnequationsso,givenasystemofnlinearequations
Let
If ,thennondegeneratesolutionsexistonlyif .
If and ,thesystemhasnouniquesolution.Otherwise,compute
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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
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Thesolutionis(6,7).
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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
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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.
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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
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= ,fromwhich
weobtaintheduplicateequations.
- -4 =0
- -4 =0
Ifwelet =t,then =-4t.Alleigenvectorscorrespondingto aremultiplesof
.
Repeatingthisfor ,wefindthat
4 -4 =0
- + =0
Ifwelet =t,then =t.Alleigenvectorscorrespondingto aremultiplesof
.
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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
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Chapter:5DifferentialCalculusI
Topicscoveredinthissnack-sizedchapter:
TaylorSeriesMaclaurinSeriesPartialDerivativeMaximaandMinima
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TaylorSeries
ATaylorseriesisaseriesexpansionofafunctionaboutapoint.Aone-dimensionalTaylorseriesisanexpansionofarealfunctionf(x)aboutapointx=aisgivenby
Example:
FindtheTaylorseriesforf(x)=Solution:
Therefore,
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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
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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
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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
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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.
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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).
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Chapter:6DifferentialCalculusII
Topicscoveredinthissnack-sizedchapter:
TangentandNormalSub-tangentandSubnormalRadiusofCurvature
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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:
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Sincegiventhattangent(2)passesthrough(0,0),then
Fromequations(1)and(3),weget
Hencepointofcontactlieonthecurveis
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Sub-tangentandSubnormal
LetthetangentPTandnormalPGatanypointP(x,y)ofanycurvemeettheaxisofxinTandGrespectively.
FromPdrawPMperpendicularonx-axis,thenthelengthTMiscalledCartesiansub-tangentatPandthelengthMGiscalledCartesiansubnormalatPofthecurve.
If betheanglewhichthetangentatPmakeswithaxisofx,thenslope
oSub-tangent=
oSubnormal=
oLengthoftangentatpointP(x,y)=PT
oLengthofnormalatpointP(x,y)=PG
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oInterceptonx-axis
oInterceptony-axis
Example:
Provethatthecurveay2=(x+b)3.Thesubnormalvariesasthesquareofthesub-tangent.
Solution:
Givenay2=(x+b)3
Therefore,
Weknowthat
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Therefore,
Hence,subnormal (subtangent)2.
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RadiusofCurvature
LetPbeanypointonthecurveC.DrawthetangentatPtothecircle.
OsculatingCircle:
ThecirclehavingthesamecurvatureasthecurveatPtouchingthecurveatP,iscalledthecircleofcurvatureorOsculatingCircle.
RadiusofCurvature:
TheradiusoftheOsculatingcircleiscalledtheradiusofcurvatureandisdenotedby .
CentreofCurvature:
ThecenterofcurvatureforapointP(x,y)ofacurveisthecenterCofthecircleofcurvatureatP.Thecoordinates( )ofthecenterofcurvaturearegivenby
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Example:
Forthecurvey=c ,showthat
Solution:
y=c
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Chapter:7IntegralCalculusI
Topicscoveredinthissnack-sizedchapter:
TheDefiniteIntegralPropertiesofDefiniteIntegralsDefiniteIntegralastheLimitofaSumSummationofSeriesUsingDefiniteIntegralsastheLimitasSum
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TheDefiniteIntegral
Let betheprimitiveorantiderivativeofafunctionf(x)definedoninterval[a,b]
i.e., .
Thenthedefiniteintegraloff(x)over[a,b]isdenotedby andisdefinedas.
Thenumbersaandbarecalledthelimitsofintegration,aiscalledthelowerlimitandbtheupperlimit.
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PropertiesofDefiniteIntegrals
Example:
Evaluate
Solution:
Put,
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Weget
Nowintegratingbyparts,weget
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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.
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Since
sothat
Hence
Where
Example:
Evaluate asthelimitofsum.
Solution:
Let
Bydefinition,wehave,
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SummationofSeriesUsingDefiniteIntegralsastheLimitasSum
Iff(x)isanintegrablefunctiondefinedoninterval[a,b],thenwehave
Where
or
Puttinga=0,b=1,then
in(1),weget
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Thisformulaisveryusefulinfindingthesummationofinfiniteserieswhichareexpressibleintheform
WorkingRule:
StepI:writethegivenseriesintheform
StepII:Replace by byxand bydx.
StepIII:Obtainlowerandupperlimitsbycomputing fortheleastandgreatestvaluesofrrespectively.
StepIV:Evaluatetheintegralobtainedinpreviousstep.Thevaluesoobtainedistherequiredsumofthegivenseries.
Example:
Evaluate:
Solution:
Thegivenlimit
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Chapter:8IntegralCalculusII
Topicscoveredinthissnack-sizedchapter:
BetaFunctionGammaFunctionRelationbetweenBetaandGammaFunctionsEvaluationofDoubleIntegralsEvaluationofTripleIntegrals
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BetaFunction
BetaFunction:TheBetafunctiondenotedbyB(m,n)withparameterm,nisdefinedas
Properties:
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GammaFunction
GammaFunction:TheGammafunctionisdefinedasthedefiniteintegral.
ThisfunctionisalsocalledEulerianintegralofsecondkind.
Properties:
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RelationbetweenBetaandGammaFunctions
Toprovethat,
Proof:
Weknowthat
Putting weget
Similarly
Therefore,
Changingtopolarcoordinates,weput
weget
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Hence,
Example:
Provethat:
Solution:
Weknowthat:
Puttingm+n=1i.e.,m=1nintheequation(1)weget
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Since,weknowthat
Puttingthesevaluesin(2),weget
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EvaluationofDoubleIntegrals
IfthegivenregionRbeboundedbytheinequalities and
Intheaboveintegral iscalculatedfirst.Duringthisintegrationxisregardedasconstant.
Example:
Solution:
Example:
Findtheareabetweentheparabolasy2=4axandx2=4ay.Solution:
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Solvingequation(1)and(2)wegetthepoint(4a,4a)
TakingstripPQparalleltox-axissothat
limits to andyvaries0to4a
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EvaluationofTripleIntegrals
IftheregionVbeboundedbythe
inequalities thenintegral
oIfthelimitsofzaregivenfunctionsofxandylimitsyasfunctionsofxwhilextaketheconstantvalues,then
oWeintegratefirstw.r.t.zkeepingxandyconstantsandthentheremainingintegrationdoneasinthecaseofdoubleintegrals.IftheareaSisboundedbythecurvesy=y1(x),y=y2(x),xandx=b.
Example:
Evaluatethefollowingintegral.
Solution:
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Chapter:9OrdinaryDifferentialEquationsI
Topicscoveredinthissnack-sizedchapter:
DifferentialEquationOrderandDegreeofaDifferentialEquationLinearandNon-linearDifferentialEquationsSolutionofaDifferentialEquationSolutionofFirstOrderandFirstDegreeDifferentialEquationsDifferentialEquationswhereVariablesareSeparableEquationsReducibletoVariableSeparableHomogeneousDifferentialEquationsLeibnitzsLinearDifferentialEquations
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DifferentialEquation
Anequationwhichcontainsderivativesofoneormoreindependentvariablesiscalleddifferentialequation.
Example:
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OrderandDegreeofaDifferentialEquation
Order:
Theorderofadifferentialequationistheorderofthehighestorderderivativeappearingintheequation.
Example:
isoforder2,becausethehighestorderderivativeis2.
DegreeofDifferentialEquation:
Thedegreeofadifferentialequationisthedegreeofhighestderivativewhichoccursinit.
Example:
Thisdifferentialequationisofdegreetwo.
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LinearandNon-linearDifferentialEquations
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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
Differentiating(2)w.r.t.x,weget
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Wehave
=0
satisfiesthegivendifferentialequation.
Hence,itisasolutionofthegivendifferentialequation.
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SolutionofFirstOrderandFirstDegreeDifferentialEquations
Adifferentialequationoffirstorderandfirstdegreeinvolvesx,yand Soitcanbeputinanyoneofthefollowingforms:
Wheref(x,y)andg(x,y)areobviouslythefunctionofxandy.
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DifferentialEquationswhereVariablesareSeparable
Ifthedifferentialequationscanbeputintheform ,wesaythatthevariablesareseparableandsuchequationscanbesolvedbyintegratingonbothsides.
Thesolutionisgivenby
WhereCisanarbitraryconstant.
Example:
Solve
Solution:
Integratingbothsides,weget
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EquationsReducibletoVariableSeparable
Differentialequationsoftheform canbereducedtovariableseparableformbythesubstitution .
Example:
Solve
Solution:
Givendifferentialequationis
Put
Henceequation(1)becomes,
Integratingbothsides,weget
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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
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Onintegration,
WhereCisanarbitraryconstantofintegration.
Example:
Solve:
Solution:
Thegivenequationcanbewrittenas
Put
Hence(1)becomes,
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Onintegrating,weget
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LeibnitzsLinearDifferentialEquations
LinearDifferentialEquationiny:Thegeneralformofalineardifferentialequationis
oWherePandQarefunctionsofxonlyorconstants.
Tosolvetheequation,whentheyaremultipliedbyafactor,whichiscalledintegralfactor(I.F.).
Multiplyingbothsidesof(1)byI.F. ,weget
Whichistherequiredsolution,whereCistheconstantofintegration.
Example:
Solve:
Solution:
Hereyisalone,soitmaybelinearinyTherefore,
Here
Therefore,
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Hencesolution:
Putting
Puttingthevalueoftintheaboveequation
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Chapter:10OrdinaryDifferentialEquationsII
Topicscoveredinthissnack-sizedchapter:
BernoullisDifferentialEquationorReducibletoLinearDifferentialEquationsExactDifferentialEquationsEquationsofFirstorderandHigherDegreeEquationSolvableforpEquationsSolvableforyEquationsSolvableforxClairautsEquation
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BernoullisDifferentialEquationorReducibletoLinearDifferentialEquations
Adifferentialequationsoftheform
issaidtobeaBernoullisEquation,wherePandQarefunctionsofxofconstants.Dividingbothsidesof(1),byyn,weget
Put
Hence(2)becomes:
oWhichisalineardifferentialequationint.
Example:
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Solution:
Rewritegivenequation:
Putting
Hence(1)becomes:
oWhichisalinearD.Einit.
Therefore,
Thecompletesolutionis
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ExactDifferentialEquations
Afirstorderdifferentialequationoftheform
issaidtobeexactdifferentialequation,ifandonlyifitsatisfythefollowingnecessarycondition:
oWhereMandNarefunctionsofxandy.
Thesolutionofanexactequation(1)willbe
Example:
Solution:
Comparingwith
Here
Therefore,wehave
Therefore,Clearly
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Hencegivenequation(1)isanexact.
TheonlynewtermobtainedonintegratingNwithrespecttoyisyasthetermsarealreadypresentintheintegrationofM.
Hencethegeneralsolutionofthegivendifferentialequationis:
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EquationsofFirstorderandHigherDegree
Adifferentialequationofthefirstorderandnthdegreeis
Where,
isrepresentedbyp
arefunctionsofxandyorconstants.
Example:
Themethodofsolutionofaboveequationdependsupon,whetheritisoSolvableforp,oSolvablefory,oSolvableforx,oClairautsform
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EquationSolvableforp
Example:
Solve:
Solution:
Given:
Hencegeneralsolutionofgivenequationis
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EquationsSolvablefory
Thegivenequationbesolvablefory,sothatitcanbeputintotheform
Differentiatingwithrespecttox,wehave
Whichisadifferentialequationinthevariablesxandp.letitspossiblesolutionbe
Example:
Solve
Solution:
Differentiatingthegivenequationwithrespecttox,weget
Neglectingthesecondfactornotcontainingthederivativesofp,
Integrating,
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Thegivenequationmaybewrittenas
Eliminatingpfromtheaboveequation
oWhichistherequiredsolutionofthegivenequation.
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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
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or
Rejectingthesecondfactor,wehave
Integrationgives,
Substitutingthisvalueofpinthegivenequation,wehave
oWhichistherequiredsolutionofthegivendifferentialequation.
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ClairautsEquation
Thedifferentialequationoftheform
isknownasClairautsequation.
Differentiating(1)w.r.t.x,weget
Onintegrating,weget
Eliminatingpfromequations(1),weget
Example:
Solvethefollowing
Solution:
Givendifferentialequationis
Takingsubstitution and
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Therefore
Puttingvalueofpin(1),weget
WhichisClairautsform
Henceputting wegetthesolution
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Chapter:11OrdinaryDifferentialEquationsIII
Topicscoveredinthissnack-sizedchapter:
LinearHigherOrderDifferentialEquationsLinearNon-HomogenousEquationswithConstantCoefficientsParticularIntegralShortMethodsoffindingParticularIntegrals
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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.
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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.
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oIfA.E.hasonepairofirrationalroots,say, where ispositivei.e.,
,thenfortheseroots
Or
Or
oIfA.E.hastwoequalpairsofirrationalroots,say i.e.,
,thenfortheseroots.
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LinearNon-HomogenousEquationswithConstantCoefficients
Thegeneralsolutionofthelinearnon-homogeneousdifferentialequation.
i.e.,of isgivenbyy=C.F.+P.I.
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ParticularIntegral
TheP.Iofthedifferentialequationf(D)y=Qisdefinedby
TofindtheParticularIntegral
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ShortMethodsoffindingParticularIntegrals
TofindP.I.oftheform ,when
If where ThentoFindP.I.oftheform
ToFindP.I.oftheforms when
TofindP.I.oftheform ,WhereVisafunctionofx
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TofindP.I.theforms and
Tofind ,whereVissomefunctionofx
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Chapter:12FourierSeriesI
Topicscoveredinthissnack-sizedchapter:
FourierSeriesDifferentformsofEulerFormulaeDirichletsConditionsParsevalsIdentityforFourierSeriesEvenandOddFunctionHalfRangeSeries(HalfRangeExpansion)
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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