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3.MatrixAlgebra

Unitmatrices

zero,i.e.,(I)ij=

Products

(AB)ij=lå k=1A ikBkj

IngeneralAB6=BA.

Transposematrices

Inversematrices

(A1)ij=transposeofcofactorofAij jAj

Determinants

j

Aj=å

i,j,k,... ijk...A1iA2jA3k...

22matrices

IfA=ab

cd then, j

Aj=adbcAT=ac

bd A 1=1 jAj db ca

Productrules

(AB...N)T=NT...BTAT j

Orthogonalmatrices

matrixQ, Q

1=QT,jQj=1,QTisalsoorthogonal.

5

Solvingsetsoflinearsimultaneousequations

x=ATb.

Hermitianmatrices

Eigenvaluesandeigenvectors

Theneigenvalues

)=jAIj.IfAisHermitianthentheeigenvalues matrixA.

TrA=å

i i,alsojAj=Õ ii.

IfSisasymmetricmatrix,

U TSU= andS=UUT. correspondingeigenvalue.

Commutators

[A,B]ABBA [A,B]=[B,A] [A,B]y=[By,Ay] [A+B,C]=[A,C]+[B,C] [AB,C]=A[B,C]+[A,C]B [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0

Hermitianalgebra

b y=(b 1,b

2,...)

MatrixformOperatorformBra-ketform

HermiticitybAc=(Ab)cZ

O=Z (O )h jOji

Eigenvalues,

realAui=(i)uiO i=(i) iOjii=ijii

Orthogonalityuiuj=0Z

i j=0hijji=0(i6=j)

Completenessb=å

iu i(uib) i i Z i ij iihiji

Rayleigh-Ritz

Lowesteigenvalue

0bAbbb0Z

O Z h jOj i h j i 6

Paulispinmatrices

x=01 10 ,y=0i i0 ,z=10 01 xy=iz,yz=ix,zx=iy,xx=yy=zz=I

4.VectorCalculus

Notation

polarcoordinates =(r,,');incaseswithradialsymmetry=(r). areindependentfunctionsofx,y,z. 6 6 6 6 6 6 6 7 7 7 7 7 7 7 5 grad =r,divA=rA,curlA=rA

Identities

grad(

1+2)grad1+grad2div(A1+A2)divA1+divA2

grad(

12)1grad2+2grad1

curl(A1+A2)curlA1+curlA2 div(

A)divA+(grad)A,curl(A)curlA+(grad)A

div(A1A2)A2curlA1A1curlA2 div(curlA)0,curl(grad )0 7

Grad,Div,CurlandtheLaplacian

Conversionto

Cartesian

z=rcos

Divergence

1 1 +1rsin

CurlrA

ijk A xAyAz 1 bb'1 bz A A'Az 1 r2sinbr1rsinb1 rb'

ArrArA'sin

Laplacian

r 2 1 +1 1 r +1r2sin sin 1 r2sin2

Transformationofintegrals

S=asurfacearea

=avolumecontainedbyaspeciedsurface b t=theunittangenttoCatthepointP b n=theunitoutwardpointingnormal

A=somevectorfunction

dL=thevectorelementofcurve(=btdL) dS=thevectorelementofsurface(=b ndS) ThenZ C

AbtdL=Z

C AdL andwhenA=r Z C (r )dL=Z C d

Gauss'sTheorem(DivergenceTheorem)

WhenSdenesaclosedregionhavingavolume

Z (rA)d=Z S (Ab n)dS=Z S AdS also Z (r)d=Z SdSZ (rA)d=Z S (b nA)dS 8

Stokes'sTheorem

WhenCisclosedandboundstheopensurfaceS,Z

S (rA)dS=Z C AdL also Z S (b nr )dS=Z CdL

Green'sTheorem

Z S rdS=Z r( r)d =Z r2+(r )(r)d

Green'sSecondTheorem

Z ( r2r2 )d=Z S [ (r)(r )]dS

5.ComplexVariables

Complexnumbers

Thecomplexnumberz=x+iy=r(cos

realquantityristhemodulusofzandtheangle r(cos isin)=rei;zz=jzj2=x2+y2

DeMoivre'stheorem

(cos +isin)n=ein=cosn+isinn

Powerseriesforcomplexvariables.

e z=1+z+z2

2!++znn!+convergentforallnitez

sinz=zz3

3!+z55!convergentforallnitez

cosz=1z2

2!+z44!convergentforallnitez

ln(1+z)=zz2

2+z33principalvalueofln(1+z)

tan1z=zz3 3+z55 (1+z)n=1+nz+n(n1)

2!z2+n(n1)(n2)3!z3+

9

6.TrigonometricFormulae

cos2A+sin2A=1sec2Atan2A=1cosec2Acot2A=1

1tan2A.

2 2 tan(AB)=tanAtanB

1tanAtanBsinAcosB=sin(A+B)+sin(AB)2

sinA+sinB=2sinA+B

2cosAB2

sinAsinB=2cosA+B

2sinAB2

cosA+cosB=2cosA+B

2cosAB2

cosAcosB=2sinA+B

2sinAB2cos2A=1+cos2A

2 sin2A=1cos2A 2 cos3A=3cosA+cos3A 4 sin3A=3sinAsin3A 4 a a

2=b2+c22bccosA

a=bcosC+ccosB cosA=b2+c2a2 2bc tanAB

2=aba+bcotC2

area=1 sina sinA=sinbsinB=sincsinC cosa=cosbcosc+sinbsinccosA cosA=cosBcosC+sinBsinCcosa 10

7.HyperbolicFunctions

coshx=1

2(ex+ex)=1+x22!+x44!+validforallx

sinhx=1

2(exex)=x+x33!+x55!+validforallx

coshix=cosxcosix=coshx sinhix=isinxsinix=isinhx tanhx=sinhx coshxsechx=1coshx cothx=coshx sinhxcosechx=1sinhx cosh2xsinh2x=1

Forlargepositivex:

coshxsinhx!ex 2 tanhx!1

Forlargenegativex:

coshxsinhx!ex 2 tanhx!1

Relationsofthefunctions

sinhx=sinh(x)sechx=sech(x) coshx=cosh(x)cosechx=cosech(x) tanhx=tanh(x)cothx=coth(x) sinhx=2tanh(x=2)

1tanh2(x=2)=tanhxq

1tanh2xcoshx=1+tanh2(x=2)1tanh2(x=2)=1q

1tanh2x

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