Handbook_of_Mathematics_Physics_and_Astr

Handbook of Mathematics, Physics and Astronomy Data School of Physical and Geographical Sciences University of Keele Keele University c/ci∇cleco√y∇t2013 Contents 1 Reference Data 1 1.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Astrophysical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Electron Configurations of the Elements . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Greek Alphabet and SI Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Mathematics 7 2.1 Mathematical Constants and Notation . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Trigonometrical Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Standard Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Standard Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.9 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.10 Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.11 Vectors and Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.13 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.14 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.15 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.16 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.17 Determinants and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.18 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.19 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.20 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Selected Physics Formulae 47 3.1 Equations of Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Equations of Relativistic Kinematics and Mechanics . . . . . . . . . . . . . . . . 49 3.3 Thermodynamics and Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . 50 i Reference Data 1.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Astrophysical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Electron Configurations of the Elements . . . . . . . . . . . . . . . . . . . . 5 1.5 Greek Alphabet and SI Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 1.1 Physical Constants Symbol Quantity Value c Speed of light in free space 2.998×108m s−1 h Planck constant 6.626×10−34J s ¯h h/ 2π 1.055×10−34J s G Universal gravitation constant 6.674×10−11N m2kg−2 e Electron charge 1.602×10−19C me Electron rest mass 9.109×10−31kg mp Proton rest mass 1.673×10−27kg mn Neutron rest mass 1.675×10−27kg u Atomic mass unit (1 12mass of12C) = 1.661×10−27kg NA Avogadro’s constant 6.022×1023mol−1 = 6.022×1026(kg-mole)−1 kB Boltzmann constant 1.381×10−23J K−1 R Gas constant = Nk 8.314×103J K−1(kg-mole)−1 8.314 J K−1mol−1 θB Bohr magneton 9.274×10−24J T−1(or A m2) θN Nuclear magneton 5.051×10−27J T−1 R∞ Rydberg constant 10973732 m−1 a0 Bohr radius 5.292×10−11m σ Stefan-Boltzmann constant 5.670×10−8J K−4m−2s−1 α Fine structure constant 1/137.04 σT Thomson cross-section 6.652×10−29m2 θ0 Permeability of free space 4π×10−7H m−1 ǫ0 Permittivity of free space 1/(θ0c2) = 8.854×10−12F m−1 eV Electron volt 1.602×10−19J g Standard acceleration of gravity 9.807 m s−2 atm Standard atmosphere 101325 N m−2= 101325 Pa 2 1.2 Astrophysical Quantities Symbol Quantity Value M⊙ Mass of Sun 1.989×1030kg R⊙ Radius of Sun 6.955×108m L⊙ Bolometric luminosity of Sun 3.846×1026W M⊙ bol Absolute bolometric magnitude of Sun +4.75 M⊙ vis Absolute visual magnitude of Sun +4.83 T⊙ Effective temperature of Sun 5778 K Spectral type of Sun G2 V MJ Mass of Jupiter 1.899×1027kg RJ Equatorial radius of Jupiter 71492 km M⊕ Mass of Earth 5.974×1024kg R⊕ Equatorial radius of Earth 6378 km M/leftmoon Mass of Moon 7.348×1022kg R/leftmoon Equatorial radius of Moon 1738 km Sidereal year 3.156×107s AU Astronomical Unit 1.496×1011m ly Light year 9.461×1015m pc Parsec 3.086×1016m Jy Jansky 10−26W m−2Hz−1 H0 Hubble constant 72±5 km s−1Mpc−1 3 1.3 Periodic Table 4 1.4 Electron Configurations of the Elements Z Element Electron configuration 1s2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 1 H 1 2 He 2 3 Li 21 4 Be 22 5 B 22 1 6 C 22 2 7 N 22 3 8 O 22 4 9 F 22 5 10 Ne 22 6 11 Na 22 6 1 12 Mg 22 6 2 13 Al 22 6 2 1 14 Si 22 6 2 2 15 P 22 6 2 3 16 S 22 6 2 4 17 Cl 22 6 2 5 18 Ar 22 6 2 6 19 K 22 6 2 6 - 1 20 Ca 22 6 2 6 - 2 21 Sc 22 6 2 6 1 2 22 Ti 22 6 2 6 2 2 23 V 22 6 2 6 3 2 24 Cr 22 6 2 6 5 1 25 Mn 22 6 2 6 5 2 26 Fe 22 6 2 6 6 2 27 Co 22 6 2 6 7 2 28 Ni 22 6 2 6 8 2 29 Cu 22 6 2 6 10 1 30 Zn 22 6 2 6 10 2 31 Ga 22 6 2 6 10 2 1 32 Ge 22 6 2 6 10 2 2 33 As 22 6 2 6 10 2 3 34 Se 22 6 2 6 10 2 4 35 Br 22 6 2 6 10 2 5 36 Kr 22 6 2 6 10 2 6 37 Rb 22 6 2 6 10 2 6 - - 1 38 Sr 22 6 2 6 10 2 6 - - 2 39 Y 22 6 2 6 10 2 6 1 - 2 40 Zr 22 6 2 6 10 2 6 2 - 2 41 Nb 22 6 2 6 10 2 6 4 - 1 42 Mo 22 6 2 6 10 2 6 5 - 1 43 Tc 22 6 2 6 10 2 6 6 - 1 44 Ru 22 6 2 6 10 2 6 7 - 1 45 Rh 22 6 2 6 10 2 6 8 - 1 46 Pd 22 6 2 6 10 2 6 10 - - 47 Ag 22 6 2 6 10 2 6 10 - 1 48 Cd 22 6 2 6 10 2 6 10 - 2 49 In 22 6 2 6 10 2 6 10 - 2 1 50 Sn 22 6 2 6 10 2 6 10 - 2 2 51 Sb 22 6 2 6 10 2 6 10 - 2 3 52 Te 22 6 2 6 10 2 6 10 - 2 4 53 I 22 6 2 6 10 2 6 10 - 2 5 54 Xe 22 6 2 6 10 2 6 10 - 2 6 5 1.5 Greek Alphabet and SI Prefixes The Greek alphabet Aα alpha N ν nu Bβ beta Ξ ξ xi Γγ gamma O o omicron ∆δ delta Π π pi Eǫ,ε epsilon P ρ,̺rho Zζ zeta Σ σ,ς sigma Hη eta T τ tau Θθ,ϑ theta Y υ upsilon Iι iota Φ φ,ϕ phi Kκ kappa X χ chi Λλ lambda Ψ ψ psi Mθ mu Ω ω omega SI Prefixes Name Prefix Factor yotta Y 1024 zetta Z 1021 exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deca da 101 deci d 10−1 centi c 10−2 milli m 10−3 micro θ 10−6 nano n 10−9 pico p 10−12 femto f 10−15 atto a 10−18 zepto z 10−21 yocto y 10−24 6 Mathematics 2.1 Mathematical Constants and Notation . . . . . . . . . . . . . . . . . . . . .8 2.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Trigonometrical Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 2.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Standard Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Standard Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .16 2.9 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.10 Curvilinear Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.11 Vectors and Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.13 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.14 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .30 2.15 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.16 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . .35 2.17 Determinants and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.18 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.19 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.20 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 2.1 Mathematical Constants and Notation Constants π= 3.141592654 ... (N.B. π2≃10) e= 2.718281828 ... ln 10 = 2 .302585093 ... log10e= 0.434294481 ... lnx= 2.302585093 log10x 1 radian = 180 /π≃57.2958 degrees 1 degree = π/180≃0.0174533 radians Notation Factorial n=n! = n×(n−1)×(n−2)×...×2×1 (N.B. 0! = 1) Stirling’s approximation n!≃/parenleftbiggn e/parenrightbiggn (2πn)1/2(n≫1) lnn!≃nlnn−n (Error <∼4% for n≥15) Double Factorial n!! =  n×(n−2)× ≤≤≤ × 5×3×1 for n >0 odd n×(n−2)× ≤≤≤ × 6×4×2 for n >0 even 1 for n=−1,0 exp(x) = ex lnx= logex arcsin x= sin−1x arccos x= cos−1x arctan x= tan−1x n/summationdisplay i=1Ai=A1+A2+A3+≤≤≤+An= sum of nterms n/productdisplay i=1Ai=A1×A2×A3× ≤≤≤ × An= product of nterms Sign function: sgn x=  +1 if x >0 0 if x= 0 −1 if x <0 8 2.2 Algebra Polynomial expansions (a+b)2=a2+ 2ab+b2 (ax+b)2=a2x2+ 2abx+b2 (a+b)3=a3+ 3a2b+ 3ab2+b3 (ax+b)3=a3x3+ 3a2bx2+ 3ab2x+b3 Quadratic equations ax2+bx+c= 0 x=−b±√ b2−4ac 2a Logarithms and Exponentials Ify=axthen y=exlnaand logay=x a0= 1 a−x= 1/ax ax×ay=ax+y ax/ay=ax−y (ax)y= (ay)x=axy ln 1 = 0 ln(1/x) = −lnx ln(xy) = ln x+ lny ln(x/y) = ln x−lny lnxy=ylnx Change of base: logay=logby logba and in particular ln y=log10y log10e≃2.303 log10y 9 2.3 Trigonometrical Identities Trigonometric functions  θ bc a sinθ=a ccosθ=b ctanθ=a b cscθ=1 sinθsecθ=1 cosθcotθ=1 tanθ Basic relations (sinθ)2+ (cos θ)2≡sin2θ+ cos2θ= 1 1 + tan2θ= sec2θ 1 + cot2θ= csc2θ sinθ cosθ= tan θ Sine and Cosine Rules TTTTTTTT  γ αβ ba c Sine Rulea sinα=b sinβ=c sinγ Cosine Rule a2=b2+c2−2bccosα 10 Expansions for compound angles sin(A+B) = sin AcosB+ cosAsinB sin(A−B) = sin AcosB−cosAsinB cos(A+B) = cos AcosB−sinAsinB cos(A−B) = cos AcosB+ sinAsinB tan(A+B) =tanA+ tan B 1−tanAtanB tan(A−B) =tanA−tanB 1 + tan AtanB sin/parenleftbigg θ+π 2/parenrightbigg = + cos θ sin/parenleftbiggπ 2−θ/parenrightbigg = + cos θ cos/parenleftbigg θ+π 2/parenrightbigg =−sinθ cos/parenleftbiggπ 2−θ/parenrightbigg = + sin θ sin(π+θ) = −sinθ sin(π−θ) = + sin θ cos(π+θ) = −cosθ cos(π−θ) =−cosθ cosAcosB=1 2[cos(A+B) + cos( A−B)] sinAsinB=1 2[cos(A−B)−cos(A+B)] sinAcosB=1 2[sin(A+B) + sin( A−B)] cosAsinB=1 2[sin(A+B)−sin(A−B)] sin 2A= 2 sin AcosA cos 2A= cos2A−sin2A= 2 cos2A−1 = 1−2 sin2A tan 2A=2 tanA 1−tan2A Factor formulae sinA+ sinB= +2 sin/parenleftbiggA+B 2/parenrightbigg cos/parenleftbiggA−B 2/parenrightbigg sinA−sinB= +2 cos/parenleftbiggA+B 2/parenrightbigg sin/parenleftbiggA−B 2/parenrightbigg cosA+ cosB= +2 cos/parenleftbiggA+B 2/parenrightbigg cos/parenleftbiggA−B 2/parenrightbigg cosA−cosB=−2 sin/parenleftbiggA+B 2/parenrightbigg sin/parenleftbiggA−B 2/parenrightbigg 11 2.4 Hyperbolic Functions Definitions and basic relations sinhx=ex−e−x 2 coshx=ex+e−x 2 tanhx=sinhx coshx=e2x−1 e2x+ 1 sechx= 1/coshx cosech x= 1/sinhx cothx= 1/tanhxcosh2x−sinh2x= 1 1−tanh2x= sech2x coth2x−1 = cosech2x sinh−1x= loge[x+√ x2+ 1] cosh−1x=±loge[x+√ x2−1] tanh−1x=1 2loge/parenleftbigg1 +x 1−x/parenrightbigg (x2<1) Expansions for compound arguments sinh(A±B) = sinh AcoshB±coshAsinhB cosh(A±B) = cosh AcoshB±sinhAsinhB tanh(A±B) =tanhA±tanhB (1 + tanh AtanhB) sinh 2A= 2 sinh AcoshA cosh 2 A= cosh2A+ sinh2A= 2 cosh2A−1 = 1 + 2 sinh2A tanh 2 A=2 tanh A (1 + tanh2A) Factor formulae sinhA+ sinh B= 2 sinh/parenleftbiggA+B 2/parenrightbigg cosh/parenleftbiggA−B 2/parenrightbigg sinhA−sinhB= 2 cosh/parenleftbiggA+B 2/parenrightbigg sinh/parenleftbiggA−B 2/parenrightbigg coshA+ cosh B= 2 cosh/parenleftbiggA+B 2/parenrightbigg cosh/parenleftbiggA−B 2/parenrightbigg coshA−coshB= 2 sinh/parenleftbiggA+B 2/parenrightbigg sinh/parenleftbiggA−B 2/parenrightbigg 12 2.5 Differentiation Definition f′(x)≡d dxf(x) = lim δx→0/bracketleftiggf(x+δx)−f(x) δx/bracketrightigg f′′(x)≡d2 dx2f(x) =d dxf′(x) fn(x)≡dn dxnf(x) = the nthorder differential, obtained by taking nsuccessive differentiations of f(x). The overdot notation is often used to indicate a derivative ta ken with respect to time: ˙y≡dy dt,¨y≡d2y dt2,etc. Rules of differentiation Ifu=u(x) and v=v(x) then: sum ruled dx(u+v) =du dx+dv dx factor ruled dx(ku) =kdu dxwhere kis any constant product ruled dx(uv) =udv dx+vdu dx quotient ruled dx/parenleftbiggu v/parenrightbigg =/parenleftigg vdu dx−udv dx/parenrightigg/slashig v2 chain ruledy dx=dy du×du dx Leibnitz’ formula Leibnitz’ formula for the nthderivative of a product of two functions u(x) and v(x): [uv]n=unv+nun−1v1+n(n−1) 2!un−2v2+n(n−1)(n−2) 3!un−3v3+≤≤≤+uvn, where un=dnu/dxnetc. 13 2.6 Standard Derivatives d dx(xn) = nxn−1 d dx(exp[ax]) = aexp[ax] d dx(ax) = axlna d dx(lnx) = x−1 d dx(ln(ax+b)) =a (ax+b) d dx(logax) = x−1logae d dx(sin(ax+b)) = acos(ax+b) d dx(cos(ax+b)) = −asin(ax+b) d dx(tan(ax+b)) = asec2(ax+b) d dx(sinh( ax+b)) = acosh(ax+b) d dx(cosh( ax+b)) = asinh(ax+b) d dx(tanh( ax+b)) = asech2(ax+b) d dx(arcsin( ax+b)) = a[1−(ax+b)2]−1/2 d dx(arccos( ax+b)) = −a[1−(ax+b)2]−1/2 d dx(arctan( ax+b)) = a[1 + (ax+b)2]−1 d dx(exp [axn]) = anx(n−1)exp [axn] d dx(sin2x) = 2 sin xcosx d dx(cos2x) = −2 sinxcosx 14 2.7 Integration Definitions The area ( A) under a curve is given by A= lim dxi→0/summationdisplay f(xi)dxi=/integraldisplay f(x)dx The Indefinite Integral is/integraldisplay f(x)dx=F(x) +C where F(x) is a function such that F′(x) =f(x) and Cis the constant of integration. The Definite Integral is /integraldisplayb af(x)dx=F(b)−F(a) =/bracketleftbigg F(x)/bracketrightbiggb a where ais the lower limit of integration and bthe upper limit of integration. Rules of integration sum rule/integraldisplay (f(x) +g(x))dx=/integraldisplay f(x)dx+/integraldisplay g(x)dx factor rule/integraldisplay kf(x)dx=k/integraldisplay f(x)dxwhere kis any constant substitution/integraldisplay f(x)dx=/integraldisplay f(x)dx duduwhere u=g(x) is any function of x N.B. for definite integrals you must also substitute the values of uinto the limits of the integral. Integration by parts An integral of the form/integraldisplay u(x)q(x)dxcan sometimes be solved if q(x) can be integrated andu(x) differentiated. So if we let q(x) =dv dx, sov=/integraldisplay q(x)dx, then the Integration by Parts formula is/integraldisplay udv dxdx=uv−/integraldisplay vdu dxdx Note that if you pick uanddv dxthe wrong way round you will end up with an integral that is even more complex than the initial one. The aim is to pi ckuanddv dxsuch that du dxis simplified. 15 2.8 Standard Indefinite Integrals In the following table Cis the constant of integration. /integraldisplay xndx=xn+1 n+ 1+C(n/ne}ationslash=−1) /integraldisplay x−1dx= ln|x|+C /integraldisplay ln|x|dx=xlnx−x+C /integraldisplay sinxdx =−cosx+C /integraldisplay cosxdx = sin x+C /integraldisplay tanxdx =−ln|cosx|+C /integraldisplay cotxdx = ln|sinx|+C /integraldisplay sec2xdx = tan x+C /integraldisplay csc2xdx =−cotx+C /integraldisplay cos2xdx =1 2x+1 2sinxcosx+C /integraldisplay sin2xdx =1 2x−1 2sinxcosx+C /integraldisplay sinnxdx =−sinn−1xcosx n+(n−1) n/integraldisplay sinn−2xdx+C /integraldisplay cosnxdx =cosn−1xsinx n+(n−1) n/integraldisplay cosn−2xdx+C /integraldisplay sinxcosxdx =1 2sin2x+C /integraldisplay cosmxcosnxdx =sin(m−n)x 2(m−n)+sin(m+n)x 2(m+n)+C(m2/ne}ationslash=n2) /integraldisplay sinmxsinnxdx =sin(m−n)x 2(m−n)−sin(m+n)x 2(m+n)+C(m2/ne}ationslash=n2) /integraldisplay sinmxcosnxdx =−cos(m−n)x 2(m−n)−cos(m+n)x 2(m+n)+C(m2/ne}ationslash=n2) /integraldisplay x2cosxdx = (x2−2) sinx+ 2xcosx+C /integraldisplay x2sinxdx = (2−x2) cosx+ 2xsinx+C /integraldisplay xcosnxdx =xsinnx n+cosnx n2+C /integraldisplay xsinnxdx =−xcosnx n+sinnx n2+C /integraldisplay eaxdx=eax a+C 16 /integraldisplay xeaxdx=eax(x−1/a)/a+C /integraldisplay xe−inxdx=1 n/parenleftbigg1 n+ix/parenrightbigg e−inx+C /integraldisplay eaxsinkxdx =eax(asinkx−kcoskx) (a2+k2)+C /integraldisplay eaxcoskxdx =eax(acoskx+ksinkx) (a2+k2)+C /integraldisplay sinhxdx = cosh x+C /integraldisplay coshxdx = sinh x+C /integraldisplay tanhxdx = ln cosh x+C /integraldisplay sech2xdx = tanh x+C /integraldisplay csch2xdx = coth x+C /integraldisplay1 a2+x2dx=1 aarctan/parenleftbiggx a/parenrightbigg +C /integraldisplay1 a2−x2dx=1 atanh−1/parenleftbiggx a/parenrightbigg =1 2aln/parenleftbigga+x a−x/parenrightbigg +C /integraldisplay1 (a2−x2)1/2dx= arcsin/parenleftbiggx a/parenrightbigg +C =−arccos/parenleftbiggx a/parenrightbigg +C /integraldisplay1 (x2−a2)1/2dx= cosh−1/parenleftbiggx a/parenrightbigg +C /integraldisplayx2 (a2+x2)dx=x−aarctan/parenleftbiggx a/parenrightbigg +C /integraldisplay1 (a2+x2)1/2dx= ln[ x+ (x2+a2)1/2] +C = sinh−1/parenleftbiggx a/parenrightbigg +C /integraldisplay1 (a2+x2)3/2dx= sin/bracketleftbigg arctan/parenleftbiggx a/parenrightbigg/bracketrightbigg /a2+C =1 a2x (a2+x2)1/2+C /integraldisplayx1/2 (a−x)1/2dx=aarcsin (√(x/a))−a/radicalig x/a−(x/a)2+C /integraldisplayx (a2+x2)1/2dx= (a2+x2)1/2+C /integraldisplay1 (a+bx2)2dx=x 2a(a+bx2)+1 2a√(ab)arctan[ x√(b/a)] +C 17 2.9 Definite Integrals /integraldisplay∞ 0x1/2e−xdx=√π 2/integraldisplay∞ 0x1/2 (ex−1)dx=2.61√π 2 /integraldisplay∞ 0xne−xdx=/integraldisplay1 0/parenleftbigg ln1 x/parenrightbiggn = Γ(n+ 1) the Gamma function Note: for nan integer greater than 0, Γ( n+ 1) = n!, Γ(1) = 0! = 1 2√π/integraldisplayu 0e−x2dx= erf( u) the Error function Note that erf( ∞) = 1, so that2√π/integraldisplay∞ 0e−ax2dx=1√a. /integraldisplay∞ −∞e−ax2dx=/radicalbiggπ a /integraldisplay∞ 0x2ne−ax2dx=1×3×5× ≤≤≤ × (2n−1) 2n+1an/radicalbiggπ a≡S(n= 1,2,3...) /integraldisplay∞ −∞x2ne−ax2dx= 2S /integraldisplay∞ 0x2n+1e−ax2dx=n! 2an+1(a >0;n= 0,1,2...) /integraldisplay+∞ −∞x2n+1e−ax2dx= 0 /integraldisplay∞ 0x2ln(1−e−x)dx=−π4 45 /integraldisplay∞ 0e−axcos(kx)dx=a a2+k2 /integraldisplay∞ 01 1 +x2ndx=π/2n sin(π/2n) 18 2.10 Curvilinear Coordinate Systems Definitions Spherical Coordinates x=rsinθcosφ y=rsinθsinφ z=rcosθ Cylindrical Coordinates x=rcosφ y=rsinφ z=z Elements of area and volume Elements of Area Cartesian ( x,y)dS=dxdy Plane polar ( r,θ)dS=r dr dθ Elements of Volume Cartesian ( x,y,z )dV=dxdy dz Spherical polar ( r,θ,φ)dV=r2sinθ dr dθ dφ Cylindrical polar ( r,φ,z)dV=r dr dφdz miscellaneous Area of elementary circular annulus, width dr, centred on the origin: dS= 2πrdr Volume of elementary cylindrical annulus of height dzand thickness dr:dV= 2πrdrdz Volume of elementary spherical shell of thickness dr, centred on the origin: dV= 4πr2dr 19 Z XY r/0/0/0/0 /1/1/1/1 direction Cylindrical polar co-ordinatesz direction φ directionx = r cos y = r sin z = zφφ P(r, ,z)φ Volume elementZ XY dz dr r dφ Z XYφ Spherical Polar co-ordinatesr φθP(r,θ,φ)x = r sin θcosφ y = r sin θsinφ z = r cos θ Z XY Volume elementdr r dr sin d θφ Figure 2.1: Coordinate Systems and Elements of volume. 20 Solid angle Elementary solid angle d Ωda steradiansr θNormal to elementary area da d = da cosΩ θ r2 Figure 2.2: Solid angle. 1. The solid angle subtended by any closed surface at any point insi de the surface is 4π; 2. The solid angle subtended by any closed surface at a point outsid e the surface is zero. 21 2.11 Vectors and Vector Algebra Vectors are quantities with bothmagnitude and direction; they are combined by the triangle rule (see Fig. 2.3). A+B=B+A=C θBC A Figure 2.3: Vector addition. Vectors may be denoted by bold type A, by putting a little arrow over the symbol /vectorA, or by underlining the symbol A. Unit vectors are usually denoted by a circumflex accent (e.g.ˆi). Magnitude etc. |A|=√(A≤A) =√(A2 x+A2 y+A2 z) The angle θbetween two vectors AandBis given by cosθ=A≤B |A||B|=AxBx+AyBy+AzBz/radicalig (A2 x+A2 y+A2 z)(B2 x+B2 y+B2 z) Unit vectors Unit vector in the direction of A=A/|A| Cartesian co-ordinates: ˆi,ˆj,ˆkare unit vectors in the directions of the x,y,z cartesian axes respectively. IfAx,Ay,Azare the cartesian components of Athen A=ˆiAx+ˆjAy+ˆkAz Addition and subtraction A+B=B+A(Commutative law) (A+B) +C=A+ (B+C) (Associative law) 22 p AθB Figure 2.4: Vector (or Cross) Product; the vector pis directed out of the page. Products Scalar product A≤B≡ |A||B|cosθ=B≤A(ascalar ) ˆi≤ˆi=ˆj≤ˆj=ˆk≤ˆk= 1 ˆi≤ˆj=ˆj≤ˆk=ˆk≤ˆi= 0 A≤B=AxBx+AyBy+AzBz A≤(B+C) =A≤B+A≤C Vector (or Cross) product See Fig. 2.4 A×B=−B×A= (|A||B|sinθ)ˆ p(avector ) where ˆ pis aunitvector perpendicular to both AandB. Note that the vector product is non-commutative. ˆi׈j=ˆk ˆj׈k=ˆiˆk׈i=ˆj ˆi׈i=ˆj׈j=ˆk׈k= 0 Also, in cartesian co-ordinates, A×B= (AyBz−AzBy)ˆi+ (AzBx−AxBz)ˆj+ (AxBy−AyBx)ˆk =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆiˆjˆk AxAyAz BxByBz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle 23 Scalar triple product (A×B)≤C= (B×C)≤A= (C×A)≤B(ascalar ) (Note the cyclic order: A→B→C) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleAxAyAz BxByBz CxCyCz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle =Ax(ByCz−BzCy) +Ay(BzCx−BxCz) +Az(BxCy−ByCx) Vector triple product A×(B×C) = (A≤C)B−(A≤B)C (avector ) A×(B×C) +B×(C×A) +C×(A×B) = 0 (Note the cyclic order: A→B→C) 24 2.12 Complex Numbers z=a+ibis a complex number where a,barerealandi=√−1 (N.B. sometimes jis used instead of i). ais the realpart of zandbis the imaginary part. Sometimes the real part of a complex quantity zis denoted by ℜ(z), the imaginary part by ℑ(z). Ifa1+ib1=a2+ib2thena1=a2andb1=b2. Modulus and argument The modulus of z≡ |z|=√ a2+b2. The argument of z=θ= arctan( b/a) Complex conjugate To form the complex conjugate of any complex number simply rep laceiby−iwherever it occurs in the number. Thus if z=a+ibthen the complex conjugate is z∗=a−ib. Ifz=Ae−ixthenz∗=A∗e+ix. Note:|z|=√ zz∗=/radicalig (a+ib)(a−ib) =√ a2+b2 Rationalization Ifz=A/B, where AandBare both complex numbers, then the quotient can be ‘ratio- nalized’ as follows: z=A B=AB∗ BB∗=AB∗ |B|2 and the denominator is now real. Polar form See Fig. 2.5. If z=a+ibthen|z|=√ a2+b2andθ= arctan( b/a). Note when evaluating arctan( b/a),θmust be put in the correct quadrant (see Fig 2.6). eiθ= cos θ+isinθ(Euler’s identity) sinθ=eiθ−e−iθ 2icosθ=eiθ+e−iθ 2 z=a+ib =|z|cosθ+i|z|sinθ =|z|exp[iθ] Ifz1=|z1|exp[iθ1] and z2=|z2|exp[iθ2] then z1z2=|z1||z2|expi[θ1+θ2] andz1 z2=|z1| |z2|expi[θ1−θ2] 25 ############# b aθ|z|Imaginary part ofz Real part ofza=|z|cosθ b=|z|sinθ Figure 2.5: Argand diagram. Ifzn=wwhere w=|w|exp[iθ] then z=|w|1/nexp[i(θ+ 2kπ)/n] where k= 0,1,2...(n−1) |zn|=|z|n |z|m|z|n=|z|m+n /vextendsingle/vextendsingle/vextendsingle/vextendsinglez1 z2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|z1| |z2| DeMoivre’s theorem einθ= (cos θ+isinθ)n= cos nθ+isinnθwhere n is an integer Trigonometric and hyperbolic functions sinh(iθ) = isinθ cosh(iθ) = cos θ tanh(iθ) = itanθsin(iθ) = isinhθ cos(iθ) = cosh θ tan(iθ) = itanhθ Figure 2.6: Selecting the correct quadrant for θ= arctan( b/a) 26 2.13 Series Arithmetic progression (A.P.) S=a+ (a+d) + (a+ 2d) + (a+ 3d) +≤≤≤+ (a+ [n−1]d) Sum over nterms is Sn=n 2[2a+ (n−1)d] Geometric progression (G.P.) S=a+ar+ar2+ar3+≤≤≤+arn−1 Sum over nterms is Sn=a(1−rn) (1−r) If|r|<1 the sum to infinity is S∞=a (1−r) Binomial theorem (a+b)n=an+nan−1b+n(n−1) 2!an−2b2+n(n−1)(n−2) 3!an−3b3+≤≤≤ Ifnis apositive integer the series contains ( n+ 1) terms. If nis anegative integer or a positive or negative fraction the series is infinite. The series converges if |b/a|<1. Special cases: (1±x)n= 1±nx+n(n−1) 2!x2±n(n−1)(n−2) 3!x3+≤≤≤Valid for all n. (1±x)−1= 1∓x+x2∓x3+x4∓ ≤≤≤ (1±x)−2= 1∓2x+ 3x2∓4x3+ 5x4∓ ≤≤≤ (1±x)1 2= 1±x 2−x2 8±x3 16−5x4 128± ≤≤≤ (1±x)−1 2= 1∓x 2+3x2 8∓5x3 16+35x4 128∓ ≤≤≤ Maclaurin’s theorem f(x) =f(0) +xdf dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0+x2 2!d2f dx2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0+x3 3!d3f dx3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0+≤≤≤+xn n!dnf dxn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0+≤≤≤ where, for example d2f/dx2|x=0means the result of forming the second derivative of f(x) with respect to xandthensetting x= 0. 27 Taylor’s theorem f(x) =f(a) + (x−a)df dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a+(x−a)2 2!d2f dx2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a+(x−a)3 3!d3f dx3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a+≤≤≤ ≤≤≤+(x−a)n n!dnf dxn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=a+≤≤≤ where, for example d2f/dx2|x=aagain means the result of forming the second derivative off(x) with respect to xandthensetting x=a. Series expansions of trigonometric functions sinθ=θ−θ3 3!+θ5 5!−θ7 7!+≤≤≤ cosθ= 1−θ2 2!+θ4 4!−θ6 6!+≤≤≤ Forθinradians and small (i.e. θ≪1): sinθ≃θ Error <∼4 % for θ<∼30◦≃0.52 radians tanθ≃θ Error <∼4 % for θ<∼30◦≃0.52 radians cosθ≃1 Error <∼4 % for θ<∼16◦≃0.28 radians Series expansions of exponential functions e±x= 1±x+x2 2!±x3 3!+x4 4!+≤≤≤ Convergent for all values of x ln(1 + x) = x−x2 2+x3 3−x4 4+≤≤≤ Convergent for −1< x≤1 Forxsmall (i.e. x≪1): e±x≡exp[±x]≃1±x≤≤≤ ln(1±x)≃ ±x≤≤≤ Series expansions of hyperbolic functions sinhx≡1 2(ex−e−x) =x+x3 3!+x5 5!+≤≤≤ coshx≡1 2(ex+e−x) = 1 +x2 2!+x4 4!+≤≤≤ 28 L’Hˆ opital’s rule If two functions f(x) and g(x) are both zero or infinite at x=athe ratio f(a)/g(a) is undefined. However the limit of f(x)/g(x) asxapproaches amay exist. This may be found from limx→af(x) g(x)=f′(a) g′(a) where f′(a) means the result of differentiating f(x) with respect to xandthenputting x=a. Convergence Tests D’Alembert’s ratio test In a series,∞/summationdisplay n=1an, let the ratio R= limn→∞/parenleftbiggan+1 an/parenrightbigg . •IfR <1 the series is convergent •IfR >1 the series is divergent •IfR= 1 the test fails. The Integral Test A sum to infinity of anconverges if/integraldisplay∞ 1andnis finite. This can only be applied to series where anis positive and decreasing as ngets larger. 29 2.14 Ordinary Differential Equations General points 1. In general, finding a function which is ‘a solution of’ (i.e. sa tisfies) any particular differential equation is a trial and error process. It involves i nductive not deductive reasoning, comparable with integration as opposed to different iation. 2. If the highest differential coefficient in the equation is the nththen the general solution must contain narbitrary constants. 3. The known physical conditions–the boundary conditions –may enable one particular solution or a set of solutions to be selected from the infinite famil y of possible mathematical solutions; that is boundary conditions may allo w specific values to be assigned to the arbitrary constants in the general solution. 4. Virtually all the ordinary differential equations met in basi c physics are linear, that is the differential coefficients occur to the first power only. Definitions order of a differential equation Theorder of a differential equation is the order of the highest differentia l coefficient it contains. degree of a differential equation Thedegree of a differential equation is the power to which the highest order differential coefficient is raised. dependent and independent variables Ordinary differential equations involve only two variables, o ne of which is referred to as thedependent variable and the other as the independent variable. It is usually clear from the nature of the physical problem which is the independent an d which is the dependent variable. 30 First Order Differential Equations Direct Integration The equationdy dx=f(x) has the solution y=/integraldisplay f(x)dx. Thus it can be solved (in principle) by direct integration. Separable Variables First order differential equations of the formdy dx=f(x)g(y), where f(x) is a function of xonly and g(y) is a function of yonly. Dividing both sides by g(y) and integrating gives/integraldisplaydy g(y)=/integraldisplay f(x)dx+C, which can be used to obtain the solution of y(x). The linear equation A general first order linear equation of the formdy dx+P(x)y=Q(x) This can be solved by multiplying through by an ‘integrating f actor’ eI, where I=/integraltextP(x)dx, so that the original equation can be rewritten as d dx/parenleftig yeI/parenrightig =eIQ(x) Since QandIare only functions of xwe can integrate both sides to obtain yeI=/integraldisplay Q(x)eIdx Second Order Differential Equations Direct Integration Equations of the formd2y dx2=f(x), can be solved by integrating twice: y=/integraldisplay/bracketleftbigg/integraldisplay f(x)dx+C/bracketrightbigg dx=/integraldisplay/bracketleftbigg/integraldisplay f(x)dx/bracketrightbigg dx+Cx+D Note that there are two arbitrary constants, CandD. 31 Homogeneous Second Order Differential Equations ad2y dx2+bdy dx+cy= 0 where a,bandcare constants. Letting y=Aeαx, gives the auxiliary equation aα2+bα+c= 0 This is solved for αusing the quadratic equation, which gives two values for α,α1and α2. The general solution is the combination of the two, y=Aeα1x+Beα2x. The auxiliary equation has real roots When b2>4ac, both α1andα2are real. The general solution is y=Aeα1x+Beα2x. The auxiliary equation has complex roots When b2<4ac, both α1andα2are com- plex. Using Euler’s Equation, substituting C=A+BandD=i(A−B), the general solution can be written as y=eαx(Ccos(βx) +Dsin(βx)) where α=−b/(2a) and β=/radicalig (4ac−b2)/(2a). The auxiliary equation has equal roots When b2= 4ac, there is only one α. The general solution is given by y= (A+Bx)eαx Non-homogeneous Second Order Differential Equations Non-homogeneous second order differential equations are of the form ad2y dx2+bdy dx+cy=f(x) To solve, first solve the homogeneous equation (i.e. for right-ha nd side = 0), ad2y dx2+bdy dx+cy= 0 using the method given above to get the solution y=Aeα1x+Beα2x which is known as the complementary function (CF) . Then we find a particular solution (PS)for the whole equation. The general solution is CF + PS. The particular solution is taken to be the same form as the funct ionf(x). f(x) =kassume y=C f(x) =kxassume y=Cx+D f(x) =kx2assume y=Cx2+Dx+E f(x) =ksinxorkcosxassume y=Ccosx+Dsinx f(x) =ekxassume y=Cekx 32 2.15 Partial Differentiation Definition Iff=f(x,y) with xandyindependent, then /parenleftigg∂f ∂x/parenrightigg y≡lim δx→0f(x+δx,y)−f(x,y) δx = derivative with respect to xwithykept constant /parenleftigg∂f ∂y/parenrightigg x≡lim δy→0f(x,y+δy)−f(x,y) δy = derivative with respect to ywithxkept constant The rules of partial differentiation are the same as differentia tion, always bearing in mind which term is varying and which are constant. Convenient notation fx=∂f ∂x,fxx=∂2f ∂x2, fxy=∂2f ∂x∂y, fy=∂f ∂y, fyy=∂2f ∂y2, fyx=∂2f ∂y∂x Note that for functions with continuous derivatives fxy=∂2f ∂x∂y=∂2f ∂y∂x=fyx Total Derivatives Total change in fdue to infinitesimal changes in both xandy: df=/parenleftigg∂f ∂x/parenrightigg ydx+/parenleftigg∂f ∂y/parenrightigg xdy df dx=/parenleftigg∂f ∂x/parenrightigg y+/parenleftigg∂f ∂y/parenrightigg xdy dxis the total derivative offwith respect to x. df dy=/parenleftigg∂f ∂y/parenrightigg x+/parenleftigg∂f ∂x/parenrightigg ydx dyis the total derivative offwith respect to y. For a function where each variable depends upon a third param eter, such as f(x,y) where xandydepend on time ( t): df dt=/parenleftigg∂f ∂x/parenrightigg ydx dt+/parenleftigg∂f ∂y/parenrightigg xdy dt 33 Maxima and Minima with two or more variables Iffis a function of two or more variables we can still find the maximu m and minimum points of the function. Consider a 3-d surface given by f=f(x,y). We can identify the following types of stationary points where gradients are zero: peak – alocal maximum pit– alocal minimum pass orsaddle point – minimum in one direction, maximum in the other. -10 -5 0 5 10-10-5 0 5 10-200-180-160-140-120-100-80-60-40-20 0 -10 -5 0 5 10-10-5 0 5 10-100-80-60-40-20 0 20 40 60 80 100 Figure 2.7: Surface plots showing a peak (left) and a saddle Poi nt (right) At each peak, pit or pass, the function fis stationary, i.e. ∂f ∂x=∂f ∂y= 0 Letf(x0,y0) be a stationary point and define the second derivative test discriminant as D=/parenleftigg∂2f ∂x2/parenrightigg /parenleftigg∂2f ∂y2/parenrightigg −/parenleftigg∂2f ∂x∂y/parenrightigg2 =fxxfyy−f2 xy which is evaluated at ( x0,y0) and, ifD >0 and fxx>0 we have a pit (minimum) ifD >0 and fxx<0 we have a peak (maximum) ifD <0 we have a pass (saddle point) ifD= 0 we do not know, have to test further comparing f(x0,y0),f(x0±dx,y 0), f(x0,y0±dy), i.e. compare with values close to f(x0,y0). 34 2.16 Partial Differential Equations The following partial differential equations are basic to physi cs: One dimension −− −− ∂2φ ∂x2=D∂φ ∂t ∂2φ ∂x2=1 c2∂2φ ∂t2Three dimensions ∇2φ= 0 Laplace’s equation ∇2φ= constant Poisson’s equation ∇2φ=D∂φ ∂tDiffusion equation ∇2φ=1 c2∂2φ ∂2tWave equation In general a partial differential equation can be satisfied by a wid e variety of different functions, i.e. if φ=f(x,t) orφ=f(x,y,z ),fmay take many different forms which are notequivalent ways of representing the same set of surfaces. For exam ple,anycontinuous, differentiable function of ( x±ct) will fit the one-dimensional wave equation. ‘Solving’ these partial differential equations in a particula r physical context therefore involves choosing not just constants but also the functions whic h fit the boundary condi- tions. Equations involving three or four independent variab les, e.g. ( x,y,t) or (x,y,z,t ) can be solved only when the ‘boundaries’ are surfaces of some simpl e co-ordinate sys- tem, such as rectangular, polar, cylindrical polar, spherica l polar. The partial differential equations can then be separated into a number of ordinary differential equations in the separate co-ordinates, and solutions can be expressed as expansio ns of various classical mathematical functions. This is analogous to the general rep resentation of the solution f(x±ct) of the one-dimensional wave equation by a Fourier series of sin e and cosine functions. 35 2.17 Determinants and Matrices Determinants The general set of simultaneous linear equations may be writte n as: a11x1+a12x2+a13x3+≤≤≤+a1nxn=y1 a21x1+a22x2+a23x3+≤≤≤+a2nxn=y2 a31x1+a32x2+a33x3+≤≤≤+a3nxn=y3 ... am1x1+am2x2+am3x3+≤≤≤+amnxn=ym The solutions of these equations are: xk=1 Dn/summationdisplay j=1yjDjk (Cramer’s rule) where D=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea11a12a13≤≤≤a1n a21a22a23≤≤≤a2n ... am1am2am3≤≤≤amn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle is the determinant of the coefficients of the xiand where Djk= (−1)j+k×[determinant obtained by suppressing the jth row and the kthcolumn of D] Djkis called the co-factor ofajk. The determinant Dcan be expanded, and ultimately evaluated, as follows: D=a11D11+a12D12+≤≤≤+a1nD1n(‘expansion by the first row’) or D=a11D11+a21D21+≤≤≤+am1Dm1(‘expansion by the first column’) The expansion procedure is repeated for D1netc. until the remaining determinants have dimensions 2 ×2. If D=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea b c d/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglethenD= (ad−bc) Note: this method is very tedious for m,n > 3 and it may be better to use a ‘condensation’ procedure (see text books). example/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea11a12a13 a21a22a23 a31a32a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a11/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea22a23 a32a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−a12/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea21a23 a31a33/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+a13/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea21a22 a31a32/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle =a11(a22a33−a32a23)−a12(a21a33−a31a23) +a13(a21a32−a22a31) Note that the value of a determinant is unaltered if the rows an d columns are interchanged. See Sections on Vectors and Vector Calculus, where vector pro duct and the curl of a vector are expressed as determinants. 36 Matrices The general set of simultaneous linear equations above can be w ritten as  a11a12≤≤≤a1n a21a22≤≤≤a2n ... am1am2≤≤≤amn  (an [m×n] matrix) x1 x2 ... xn = y1 y2 ... ym  (column vectors) where the arrays of ordered coefficients are called matrices . One of the coefficients, or terms, is called an ‘element’ and a matrix is often denoted by t he general element [ aij], where iindicates the row and jthe column. Addition of matrices If two matrices are of the same order m×nthen [aij] + [bij] = [aij+bij] Scalar multiplication Ifλis ascalar number then λ[aij] = [λaij] Matrix multiplication Multiplication of two matrices [ aij],[bij] is possible onlyif the number of columns in [aij] is the same as the number of rowsin [bij]. The product [ cij] is given by [cij] =n/summationdisplay k=1aikbkj Note 1. Matrix multiplication is not defined unless the two matrices have the appropriate number of rows and columns. 2. Matrix multiplication is generally non-commutative: AB/ne}ationslash=BA The unit matrix The unit (or identity) matrix, denoted by I, is a square ( n×n) matrix with its diagonal elements equal to unity and all other elements zero. For exam ple the 3 ×3 unit matrix is I= 1 0 0 0 1 0 0 0 1  If we have a square matrix Aof order nand the unit matrix of the same order then IA=AI=A and in general, provided the matrix product is defined (see abov e), multiplying anymatrix Aby a unit matrix leaves Aunchanged. 37 The transpose of a matrix If the rows and columns of a matrix are interchanged, a new mat rix, called the transposed matrix , is obtained. The transpose of a matrix Ais denoted by AT. For example if A= a11a12 a21a22 a31a32  then AT=/bracketleftigg a11a21a31 a12a22a32/bracketrightigg The adjoint matrix The adjoint of a matrix (denoted by adj A) is defined as the transpose of the matrix of the cofactors , where the cofactors are as defined above (see Section on Determ inants, p. 36). The inverse of a matrix The inverse A−1of a matrix Ahas the property that A−1A=AA−1=I, the unit matrix. It is evaluated as follows: A−1=adjA |A|, where |A|is the determinant of A. Hermitian and unitary matrices If a matrix Acontains complex elements then the complex conjugate of Ais found by tak- ing the complex conjugate of the individual elements. A matri xAis said to be Hermitian if /tildewiderA∗=A Aunitary matrix is defined by the condition A/tildewiderA∗=I 38 2.18 Vector Calculus Differentiation of vectors (non-rotating axes) dA dt=ˆidAx dt+ˆjdAy dt+ˆkdAz dt d(A≤B) dt=/parenleftigg A≤dB dt/parenrightigg +/parenleftiggdA dt≤B/parenrightigg d(A×B) dt=/parenleftigg A×dB dt/parenrightigg +/parenleftiggdA dt×B/parenrightigg Gradient of a scalar function ^z rφφ r z^^ θ φrφ^r^ θ^ Figure 2.8: Cylindrical (left) and Spherical (right) polar s. cartesian co-ordinates ∇ ≡ˆi∂ ∂x+ˆj∂ ∂y+ˆk∂ ∂z(avector operator). ∇U= grad U, where Uis a scalar. ∇Uis a vector. ∇(U+V) =∇U+∇V U,V scalars. ∇(UV) =V(∇U) + (∇U)V cylindrical co-ordinates ∇ ≡ˆ r∂ ∂r+ˆφ1 r∂ ∂φ+ˆ z∂ ∂z spherical polar co-ordinates ∇ ≡ˆ r∂ ∂r+ˆθ1 r∂ ∂θ+ˆφ1 rsinθ∂ ∂φ 39 Divergence of a vector function cartesian co-ordinates ∇ ≤A=∂Ax ∂x+∂Ay ∂y+∂Az ∂z ≡divA ascalar cylindrical polar co-ordinates ∇ ≤A=1 r∂ ∂r(rAr) +1 r∂Aφ ∂φ+∂Az ∂z spherical polar co-ordinates ∇ ≤A=1 r2∂ ∂r(r2Ar) +1 rsinθ∂ ∂θ(Aθsinθ) +1 rsinθ∂Aφ ∂φ Curl of a vector function cartesian co-ordinates ∇ ×A≡curlA=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆiˆjˆk ∂ ∂x∂ ∂y∂ ∂z AxAyAz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≡/parenleftigg∂Az ∂y−∂Ay ∂z/parenrightigg ˆi+/parenleftigg∂Ax ∂z−∂Az ∂x/parenrightigg ˆj+/parenleftigg∂Ay ∂x−∂Ax ∂y/parenrightigg ˆk cylindrical polar co-ordinates ∇ ×A=1 r/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ rrˆφˆ z ∂ ∂r∂ ∂φ∂ ∂z ArrAφAz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≡/parenleftigg1 r∂Az ∂φ−∂Aφ ∂z/parenrightigg ˆ r+/parenleftigg∂Ar ∂z−∂Az ∂r/parenrightigg ˆφ+1 r/parenleftigg∂ ∂r(rAφ)−∂Ar ∂φ/parenrightigg ˆ z spherical polar co-ordinates ∇ ×A=1 r2sinθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ rrˆθrsinθˆφ ∂ ∂r∂ ∂θ∂ ∂φ ArrAθrAφsinθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≡1 rsinθ/parenleftigg∂ ∂θ(Aφsinθ)−∂Aθ ∂φ/parenrightigg ˆ r+1 r/parenleftigg1 sinθ∂Ar ∂φ−∂ ∂r(rAφ)/parenrightigg ˆθ +1 r/parenleftigg∂ ∂r(rAθ)−∂Ar ∂θ/parenrightigg ˆφ 40 Relations ∇ ×(∇U)≡0 ∇ ≤(∇ ×A)≡0 ∇ ≤(∇U)≡ ∇2U=∂2U ∂x2+∂2U ∂y2+∂2U ∂z2 ∇ ×(∇ ×A) = ∇(∇ ≤A)− ∇2A The Laplacian operator ∇2 cartesian co-ordinates ∇2=∂2 ∂x2+∂2 ∂y2+∂2 ∂z2 cylindrical polar co-ordinates ∇2=1 r/bracketleftigg∂ ∂r/parenleftigg r∂ ∂r/parenrightigg +∂ ∂φ/parenleftigg1 r∂ ∂φ/parenrightigg +∂ ∂z/parenleftigg r∂ ∂z/parenrightigg/bracketrightigg =∂2 ∂r2+1 r∂ ∂r+1 r2∂2 ∂φ2+∂2 ∂z2 spherical polar co-ordinates ∇2=1 r2sinθ/bracketleftigg∂ ∂r/parenleftigg r2sinθ∂ ∂r/parenrightigg +∂ ∂θ/parenleftigg sinθ∂ ∂θ/parenrightigg +∂ ∂φ1 sinθ∂ ∂φ/bracketrightigg =∂2 ∂r2+2 r∂ ∂r+1 r2∂2 ∂θ2+cotθ r2∂ ∂θ+1 r2sin2θ∂2 ∂φ2 Integral theorems Divergence/Gauss’ theorem /dispoiint SA≤ds=/dispiiint V(∇ ≤A)dV Stokes’ theorem /contintegraldisplay LA≤dl=/dispiint S(∇ ×A)≤ds Green’s theorem /dispoiint S(θ∇φ−φ∇θ)≤ds=/dispiiint V(θ∇2φ−φ∇2θ)dV 41 2.19 Fourier Series If a function f(t) is periodic in twith period T, (i.e. f(t+nT) =f(t) for any integer n and all t), then f(t) =a0 2+∞/summationdisplay n=1/bracketleftbigg ancos/parenleftbigg2πnt T/parenrightbigg +bnsin/parenleftbigg2πnt T/parenrightbigg/bracketrightbigg where a0=2 T/integraldisplayT 0f(t)dt an=2 T/integraldisplayT 0f(t) cos/parenleftbigg2πnt T/parenrightbigg dt bn=2 T/integraldisplayT 0f(t) sin/parenleftbigg2πnt T/parenrightbigg dt Notes: 1.tcan be any continuous variable, not necessarily time . 2. The function f(t) is a continuous function from t=−∞tot= +∞. For some functions t= 0 may be so chosen as to produce a simpler series in which either all an= 0 or allbn= 0, e.g. a ‘square’ wave. See Fig. 2.9. t = T t = 0f(t) t t = T t = 0f(t) t Figure 2.9: Even function (left), f(+t) =f(−t) sobn= 0. Odd function (right), f(+t) = −f(−t) soan= 0 42 Complex form of the Fourier series f(t) =∞/summationdisplay n=−∞Cnexp/parenleftbigg i2πnt T/parenrightbigg where Cn=1 T/integraldisplayT 0f(t) exp/parenleftbigg −i2πnt T/parenrightbigg dt =1 2(an−ibn) forn >0 =1 2(an+ibn) forn <0 ℜ[Cn] =1 T/integraldisplayT 0f(t) cos/parenleftbigg2πnt T/parenrightbigg dt ℑ[Cn] =−1 T/integraldisplayT 0f(t) sin/parenleftbigg2πnt T/parenrightbigg dt Average value of the product of two periodic functions f1(t)f2(t) =∞/summationdisplay n=−∞(C1)n(C2)n where the ‘bar’ means ‘averaged over a complete period’. {f(t)}2=∞/summationdisplay n=−∞CnC−n=/summationdisplay nCnC∗ n=/summationdisplay n|Cn|2 =a2 0 4+∞/summationdisplay n=11 2(a2 n+b2 n) Fourier transforms For non-periodic functions: F(ω) =/integraldisplay∞ −∞f(t) exp[−iωt]dt Fourier transform f(t) =1 2π/integraldisplay∞ −∞F(ω) exp[iωt]dω inverse Fourier transform The functions f(t) and F(ω) are called a Fourier transform pair. Some examples are given in Figure 2.10. 43 TOP HAT FUNCTION f(t) h a tF( ) ωω 0ha 2= aah sinωa2 ωa2 = sincωa2 π 4π a GAUSSIAN FUNCTIONf(t) th eh f(t) = h exp [-t / 2σ2]F( )ω ωσ h ωeσ π πh 2σ0 F( )= hσ πexp[-σ ω2 24] DELTA FUNCTIONf(t) t f(t) dt = 1 f(t) = 0 except for t = 0 and 8 -8F( ) F( ) = 1 1 0ωω ωω Figure 2.10: Examples of Fourier transforms 44 2.20 Statistics Mean and RMS Ifx1,x2,≤≤≤xnarenvalues of some quantity, then Thearithmetic mean is x=1 nn/summationdisplay ixi=1 n(x1+x2+≤≤≤+xn) Thegeometric mean is xGM=/parenleftiggn/productdisplay ixi/parenrightigg1/n =n√x1×x2× ≤≤≤ × xn Theroot-mean-square (RMS) is RMS =/radicaltp/radicalvertex/radicalvertex/radicalbt1 n/summationdisplay ix2 i=/radicaligg x2 1+x2 2+≤≤≤+x2 n n Permutations Permutations of nthings taken rat a time =n(n−1)(n−2)≤≤≤(n−r+ 1) =n! (n−r)!≡nPr Combinations Combinations of nthings taken rat a time =n! r!(n−r)!≡nCr Note that in a permutation the order in which selection is made is significant. Thus ABC/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipuprightDEFG is a different permutation , but the same combination , asACB/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipuprightDEFG . Binomial distribution Random variables can have twovalues A or B (e.g. heads or tails in the case of a toss of a coin). Let the probability of A occurring = p, the probability of B occurring = (1 −p) =q. The probability of A occurring mtimes in ntrials is pm(A) =n! m!(n−m)!pmqn−mN.B. 0! = 1 This is the mthterm in the binomial expansion of ( p+q)n. 45 Poisson distribution Events occurring with average frequency νbut randomly distributed, in time for example (e.g. radioactive decay of nuclei, shot noise, goals, floods and h orse kicks!). Probability ofmevents occurring in time interval Tis pm(T) =(νT)m m!exp(−νT) Normal distribution (Gaussian) For a continuous variable which is randomly distributed abou t a mean value θwith standard deviation σ, (e.g. random experimental errors of measurement), the prob ability that a measurement lies between xandx+dxis p(x)dx=1 σ√ 2πexp/parenleftigg −(x−θ)2 2σ2/parenrightigg dx. The quantity σ2is also known as the variance . Given a sample of Nmeasurements, the mean value of the sample, ¯ x, is an unbiased estimator of θ, and the sample standard deviation ,sN−1=/radicalbigg/summationtext i(xi−¯x)2 N−1is an unbiased estimator of σ. Thestandard error on the mean (SEM) is: σ¯x=sN−1√ N. For large N, the difference ¯ x−θis itself a normal distribution with mean 0 and standard deviation σ¯x. 46 Selected Physics Formulae 3.1 Equations of Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Equations of Relativistic Kinematics and Mechanics . . . . . . . . . . . . 49 3.3 Thermodynamics and Statistical Physics . . . . . . . . . . . . . . . . . . . 50 47 3.1 Equations of Electromagnetism Definitions B=θrθ0H= magnetic field D=ǫrǫ0E= electric displacement E= electric field J= conduction current density ρ= charge density where ǫrandθrare the relative permittivity and permeability respectivel y. The definitions forBandDare for linear, isotropic, homogeneous media. Biot-Savart law dB=θ0 4πIdl×r r3 Maxwell’s equations These are four differential equations linking the space- and tim e-derivatives of the elec- tromagnetic field quantities: ∇ ≤B= 0 ∇ ≤D=ρ∇ ×E+∂B ∂t= 0 ∇ ×H−∂D ∂t=J They can also be expressed in integral form: /integraldisplay SB≤dS= 0 /integraldisplay SD≤dS=/integraldisplay τρdτ /contintegraldisplay E≤dl=−/integraldisplay∂B ∂t≤dS /contintegraldisplay H≤dl=/integraldisplay S/parenleftigg J+∂D ∂t/parenrightigg ≤dS Energy density in an electric field =ǫrǫ0E2 2 Energy density in a magnetic field =θrθ0H2 2 Velocity of plane waves in a linear, homogeneous and isotropic medium u= (θrǫrθ0ǫ0)−1/2 48 3.2 Equations of Relativistic Kinematics and Mechanics Definitions E= energy; m0= rest mass p= linear momentum v= relative velocity of reference frames in x,x′direction γ= 1//radicalig (1−v2/c2) Lorentz transformations Two inertial frames, SandS′, are such that S′moves relative to Salong the positive x direction, with velocity vas measured in S; the origins coincide at time t=t′= 0. The Lorentz transformations are: x=γ(x′+vt′) x′=γ(x−vt) t=γ/parenleftigg t′+vx′ c2/parenrightigg t′=γ/parenleftbigg t−vx c2/parenrightbigg E2= (pc)2+ (m0c2)2 49 3.3 Thermodynamics and Statistical Physics Maxwell speed distribution For a gas, molecular weight m, in thermodynamic equilibrium at temperature T, the fraction f(v)dvof molecules with speed in the range v→v+dvis f(v)dv= 4πv2/parenleftbiggm 2πkBT/parenrightbigg3/2 exp/bracketleftigg −mv2 2kBT/bracketrightigg dv Thermodynamic variables Helmholtz free energy: Gibbs function: Enthalpy:F=U−TS G=U−TS+PV H=U+PV Maxwell’s thermodynamic relations /parenleftigg∂T ∂V/parenrightigg S=−/parenleftigg∂P ∂S/parenrightigg V/parenleftigg∂T ∂P/parenrightigg S=/parenleftigg∂V ∂S/parenrightigg P /parenleftigg∂V ∂T/parenrightigg P=−/parenleftigg∂S ∂P/parenrightigg T/parenleftigg∂S ∂V/parenrightigg T=/parenleftigg∂P ∂T/parenrightigg V Statistical physics Partition function: Z=/summationdisplay ie−βEi=/summationdisplay ie−Ei/kT Helmholtz free energy: F=−kTlnZ Entropy: S=kln Ω = −k/summationdisplay ipilnpi Blackbody radiation The energy emitted per unit area, per unit time, into unit soli d angle, in the frequency range ν→ν+dνis B(T,ν) =2hν3 c21 (exp[hν/kT ]−1) 50 Quantum statistics Distribution function: f(Es) =/bracketleftigg exp/parenleftigg(Es−θ) kT/parenrightigg ±1/bracketrightigg−1 Fermi-Dirac: + sign; θ=EF Bose-Einstein: −sign N.B. for photons θ= 0. For high energies E≫kTboth distributions reduce to the classical Maxwell-Boltzmann distribution. 51 Bibliography Tables of Physical and Chemical constants , compiled by G. W. C. Kaye & T. H. Laby, Longmans. Allen’s Astrophysical Quantities , ed A. C. Cox, AIP Press, Springer Handbook of Space Astronomy and Astrophysics , M. V. Zombeck, Cambridge University Press. Electricity and Magnetism , Chapter 13, W. J. Duffin, McGraw Hill. Special Relativity, A. P. French, Van Nostrand. Mathematical methods for Science Students , 2nd. edition, G. Stephenson, Longmans. The Elements , J. Emsley, Clarendon Press. Table of Isotopes , C. M. Lederer, J. M. Hollander & I. Perlman, John Wiley & Sons. Handbook of mathematical functions , Eds M. Abramowitz & I. E. Stegun, Dover Publica- tions. Contains many useful mathematical formulae and tables of special functions. An introduction to applied mathematics , Jaeger, Oxford University Press. Mathematics of Physics and Chemistry , Margenau & Murphy, van Nostrand. Tables of integrals, series and products , I. S. Gradshteyn & I. M. Ryzhik, Academic Press. Contains virtually every integral ever solved. 52