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Handbook of Mathematics, Physics andAstronomy DataSchool of Physical and Geographical SciencesUniversity of KeeleKeeleUniversityc 2013

Contents1 Reference Data1.1 Physical Constants . . . . . . . . . . . .1.2 Astrophysical Quantities . . . . . . . . .1.3 Periodic Table . . . . . . . . . . . . . .1.4 Electron Configurations of the Elements1.5 Greek Alphabet and SI Prefixes . . . . .1234562 Mathematics2.1 Mathematical Constants and Notation2.2 Algebra . . . . . . . . . . . . . . . . .2.3 Trigonometrical Identities . . . . . . .2.4 Hyperbolic Functions . . . . . . . . . .2.5 Differentiation . . . . . . . . . . . . .2.6 Standard Derivatives . . . . . . . . . .2.7 Integration . . . . . . . . . . . . . . .2.8 Standard Indefinite Integrals . . . . .2.9 Definite Integrals . . . . . . . . . . . .2.10 Curvilinear Coordinate Systems . . . .2.11 Vectors and Vector Algebra . . . . . .2.12 Complex Numbers . . . . . . . . . . .2.13 Series . . . . . . . . . . . . . . . . . .2.14 Ordinary Differential Equations . . . .2.15 Partial Differentiation . . . . . . . . .2.16 Partial Differential Equations . . . . .2.17 Determinants and Matrices . . . . . .2.18 Vector Calculus . . . . . . . . . . . . .2.19 Fourier Series . . . . . . . . . . . . . .2.20 Statistics . . . . . . . . . . . . . . . .7891012131415161819222527303335363942453 Selected Physics Formulae3.1 Equations of Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Equations of Relativistic Kinematics and Mechanics . . . . . . . . . . . . . . . .3.3 Thermodynamics and Statistical Physics . . . . . . . . . . . . . . . . . . . . . . .47484950i.

Reference Data1.11.21.31.41.5Physical Constants . . . . . . . . . . . . . .Astrophysical Quantities . . . . . . . . . .Periodic Table . . . . . . . . . . . . . . . .Electron Configurations of the Elements .Greek Alphabet and SI Prefixes . . . . . .1.23456

1.1 Physical ConstantsSymbolchh̄GemempmnuNAkBRµBµNR a0σασTµ0ǫ0eVgatmQuantitySpeed of light in free spacePlanck constanth/2πUniversal gravitation constantElectron chargeElectron rest massProton rest massNeutron rest massAtomic mass unitValue2.998 108 m s 16.626 10 34 J s1.055 10 34 J s6.674 10 11 N m2 kg 21.602 10 19 C9.109 10 31 kg1.673 10 27 kg1.675 10 27 kg1( 12mass of 12 C) 1.661 10 27 kgAvogadro’s constant6.022 1023 mol 1 6.022 1026 (kg-mole) 1Boltzmann constant1.381 10 23 J K 1Gas constant N k8.314 103 J K 1 (kg-mole) 18.314 J K 1 mol 1Bohr magneton9.274 10 24 J T 1 (or A m2 )Nuclear magneton5.051 10 27 J T 1Rydberg constant10973732 m 1Bohr radius5.292 10 11 mStefan-Boltzmann constant5.670 10 8 J K 4 m 2 s 1Fine structure constant1/137.04Thomson cross-section6.652 10 29 m2Permeability of free space4π 10 7 H m 1Permittivity of free space1/(µ0 c2 ) 8.854 10 12 F m 1Electron volt1.602 10 19 JStandard acceleration of gravity 9.807 m s 2Standard atmosphere101325 N m 2 101325 Pa2

1.2 Astrophysical QuantitiesSymbolM R L Mbol MvisT MJRJM R M R AUlypcJyH0QuantityValueMass of Sun1.989 1030 kgRadius of Sun6.955 108 mBolometric luminosity of Sun3.846 1026 WAbsolute bolometric magnitude of Sun 4.75Absolute visual magnitude of Sun 4.83Effective temperature of Sun5778 KSpectral type of SunG2 VMass of Jupiter1.899 1027 kgEquatorial radius of Jupiter71492 kmMass of Earth5.974 1024 kgEquatorial radius of Earth6378 kmMass of Moon7.348 1022 kgEquatorial radius of Moon1738 kmSidereal year3.156 107 sAstronomical Unit1.496 1011 mLight year9.461 1015 mParsec3.086 1016 mJansky10 26 W m 2 Hz 1Hubble constant72 5 km s 1 Mpc 13

1.3 Periodic Table4

1.4 Electron Configurations of the 6666666666666Electron configuration3d 4s 4p 4d s5p122211111122222221234565d5f

1.5 Greek Alphabet and SI PrefixesThe Greek alphabetABΓ EZHΘIKΛMαβγδǫ, εζηθ, πpiρ,̺ rhoσ, ς sigmaτtauυupsilonφ, ϕ phiχchiψpsiωomegaSI PrefixesName Prefixyotta Yzetta ZexaEpetaPteraTgigaGmega Mkilokhecto hdecadadecidcenti cmillimmicro µnano npicopfemto fattoazepto zyocto y6Factor1024102110181015101210910610310210110 110 210 310 610 910 1210 1510 1810 2110 24

2.132.142.152.162.172.182.192.20Mathematical Constants and NotationAlgebra . . . . . . . . . . . . . . . . . .Trigonometrical Identities . . . . . . .Hyperbolic Functions . . . . . . . . . .Differentiation . . . . . . . . . . . . . .Standard Derivatives . . . . . . . . . .Integration . . . . . . . . . . . . . . . .Standard Indefinite Integrals . . . . .Definite Integrals . . . . . . . . . . . . .Curvilinear Coordinate Systems . . . .Vectors and Vector Algebra . . . . . .Complex Numbers . . . . . . . . . . . .Series . . . . . . . . . . . . . . . . . . . .Ordinary Differential Equations . . . .Partial Differentiation . . . . . . . . . .Partial Differential Equations . . . . .Determinants and Matrices . . . . . .Vector Calculus . . . . . . . . . . . . . .Fourier Series . . . . . . . . . . . . . . .Statistics . . . . . . . . . . . . . . . . . .7.89101213141516181922252730333536394245

2.1 Mathematical Constants and NotationConstantsπeln 10log10 eln x1 radian1 degree 3.141592654 . . .(N.B. π 2 10)2.718281828 . . .2.302585093 . . .0.434294481 . . .2.302585093 log10 x180/π 57.2958 degreesπ/180 0.0174533 radiansNotationFactorial n n! n (n 1) (n 2) . . . 2 1(N.B. 0! 1)µ ¶nnStirling’s approximationn! (2πn)1/2 (n 1)e 4% for n 15)ln n! n ln n n(Error n (n 2) · · · 5 3 1Double Factorial n!! n (n 2) · · · 6 4 2 1exp(x)ln xarcsin xarccos xarctan xnXi 1nYi 1 for n 0 oddfor n 0 evenfor n 1, 0exloge xsin 1 xcos 1 xtan 1 xAi A1 A2 A3 · · · An sum of n termsAi A1 A2 A3 · · · An product of n terms 1Sign function: sgn x 0 18if x 0if x 0if x 0

2.2 AlgebraPolynomial expansions(a b)2 a2 2ab b2(ax b)2 a2 x2 2abx b2(a b)3 a3 3a2 b 3ab2 b3(ax b)3 a3 x3 3a2 bx2 3ab2 x b3Quadratic equationsax2 bx c 0 b b2 4acx 2aLogarithms and ExponentialsIf y axtheny ex ln aa0 a x a x ay ax /ay (ax )y ln 1 ln(1/x) ln(xy) ln(x/y) ln xy andloga y x11/axax yax y(ay )x axy0 ln xln x ln yln x ln yy ln xChange of base: loga y and in particular ln y 9logb ylogb alog10 y 2.303 log10 ylog10 e

2.3 Trigonometrical IdentitiesTrigonometric functions θ c absin θ csc θ 1sin θaccos θ sec θ bctan θ 1cos θabcot θ 1tan θBasic relations(sin θ)2 (cos θ)2 sin2 θ cos2 θ1 tan2 θ1 cot2 θsin θcos θ 1 sec2 θ csc2 θ tan θSine and Cosine Rules½½T½½ βT½Ta½T c½T½½T½½γαTbSine RuleCosine Rulebca sin αsin βsin γa2 b2 c2 2bc cos α10T

Expansions for compound anglessin(A B)sin(A B)cos(A B)cos(A B) sin A cos B cos A sin Bsin A cos B cos A sin Bcos A cos B sin A sin Bcos A cos B sin A sin Btan A tan Btan(A B) 1 tan A tan Btan A tan Btan(A B) 1 tan A tan B¶µ¶µππ cos θsin θ cos θsin θ 2¶2µµ¶ππ sin θcoscos θ θ sin θ22sin(π θ) sin θsin(π θ) sin θcos(π θ) cos θcos(π θ) cos θ1cos A cos B [cos(A B) cos(A B)]21sin A sin B [cos(A B) cos(A B)]21[sin(A B) sin(A B)]sin A cos B 21cos A sin B [sin(A B) sin(A B)]2sin 2A 2 sin A cos A2cos 2A cos A sin2 A 2 cos2 A 1 1 2 sin2 A2 tan Atan 2A 1 tan2 AFactor formulaesin A sin B sin A sin B cos A cos B cos A cos B µ¶µ¶A BA B 2 sincos22¶µ¶µA BA Bsin 2 cos22¶µ¶µA BA Bcos 2 cos22¶µ¶µA BA Bsin 2 sin2211

2.4 Hyperbolic FunctionsDefinitions and basic relationsex e x2ex e xcosh x 2sinh xe2x 1tanh x 2xcosh xe 1sinh x sech x 1/ cosh xcosech x 1/ sinh xcoth x 1/ tanh xcosh2 x sinh2 x 11 tanh2 x sech2 xcoth2 x 1 cosech2 x sinh 1 x loge [x x2 1] cosh 1 x loge [x x2 1]¶µ1 x1 1(x2 1)logetanh x 21 xExpansions for compound argumentssinh(A B) sinh A cosh B cosh A sinh Bcosh(A B) cosh A cosh B sinh A sinh Btanh A tanh Btanh(A B) (1 tanh A tanh B)sinh 2A 2 sinh A cosh Acosh 2A cosh2 A sinh2 A 2 cosh2 A 1 1 2 sinh2 A2 tanh Atanh 2A (1 tanh2 A)Factor formulaesinh A sinh B sinh A sinh B cosh A cosh B cosh A cosh B µ¶µ¶A BA Bcosh2 sinh22¶µ¶µA BA Bsinh2 cosh22µ¶µ¶A BA B2 coshcosh22µ¶µ¶A BA B2 sinhsinh2212

2.5 DifferentiationDefinition"df (x δx) f (x)f (x) limf (x) δx 0dxδx′f ′′ (x) #d ′d2f (x) f (x)2dxdxdnf (x) the nth order differential,dxnobtained by taking n successive differentiations of f (x).f n (x) The overdot notation is often used to indicate a derivative taken with respect to time:ẏ d2 ydy, ÿ 2 ,dtdtetc.Rules of differentiationIf u u(x) and v v(x) then:sum ruleddu dv(u v) dxdx dxfactor ruleproduct rulequotient rulechain ruledud(ku) kdxdxwhere k is any constantddvdu(uv) u vdxdxdxÃ!µ ¶dudv . 2d u v uvdx vdxdxdy dudy dxdu dxLeibnitz’ formulaLeibnitz’ formula for the nth derivative of a product of two functions u(x) and v(x):[uv]n un v nun 1 v1 n(n 1)n(n 1)(n 2)un 2 v2 un 3 v3 · · · uvn ,2!3!where un dn u/dxn etc.13

2.6 Standard Derivativesd n(x ) nxn 1dxd(exp[ax]) a exp[ax]dxd x(a ) ax ln adxd(ln x) x 1dxda(ln(ax b)) dx(ax b)d(loga x) x 1 loga edxd(sin(ax b)) a cos(ax b)dxd(cos(ax b)) a sin(ax b)dxd(tan(ax b)) a sec2 (ax b)dxd(sinh(ax b)) a cosh(ax b)dxd(cosh(ax b)) a sinh(ax b)dxd(tanh(ax b)) a sech2 (ax b)dxd(arcsin(ax b)) a [1 (ax b)2 ] 1/2dxd(arccos(ax b)) a [1 (ax b)2 ] 1/2dxd(arctan(ax b)) a[1 (ax b)2 ] 1dxd(exp [axn ]) anx(n 1) exp [axn ]dxd(sin2 x) 2 sin x cos xdxd(cos2 x) 2 sin x cos xdx14

2.7 IntegrationDefinitionsThe area (A) under a curve is given byA limdxi 0The Indefinite Integral isZXf (xi )dxi Zf (x) dxf (x) dx F (x) Cwhere F (x) is a function such that F ′ (x) f (x) and C is the constant of integration.The Definite Integral isZba· bf (x) dx F (b) F (a) F (x)awhere a is the lower limit of integration and b the upper limit of integration.Rules of integrationsum rulefactor ruleZ(f (x) g(x)) dx ZZkf (x) dx kZZZf (x) dx Zg(x) dxf (x) dx where k is any constantdxdu where u g(x) is any function of xduN.B. for definite integrals you must also substitute the values of u into the limits ofthe integral.substitutionf (x) dx f (x)Integration by partsAn integral of the formZu(x)q(x) dx can sometimes be solved if q(x) can be integratedZdv, so v q(x) dx, then the Integration byand u(x) differentiated. So if we let q(x) dxParts formula isZZdvduu dx uv v dxdxdxdvNote that if you pick u andthe wrong way round you will end up with an integraldxdvsuch thatthat is even more complex than the initial one. The aim is to pick u anddxduis simplified.dx15

2.8 Standard Indefinite IntegralsIn the following table C is the constant of integration.ZZZZZZZln x dx x ln x x Csin x dx cos x Ccos x dx sin x Ctan x dx ln cos x Ccot x dx ln sin x Ccos2 x dx Z(n 6 1)x 1 dx ln x CZZZxn 1 Cn 1sec2 x dx tan x CZZxn dx ZZZZcsc2 x dx cot x Csin2 x dx sinn x dx cosn x dx sin x cos x dx cos mx cos nx dx sin mx sin nx dx sin mx cos nx dx 11x sin x cos x C2211x sin x cos x C22sinn 1 x cos x (n 1) Z sinn 2 x dx Cnncosn 1 x sin x (n 1) Z cosn 2 x dx Cnn1 2sin x C2sin(m n)x sin(m n)x C (m2 6 n2 )2(m n)2(m n)sin(m n)x sin(m n)x C (m2 6 n2 )2(m n)2(m n)cos(m n)x cos(m n)x C (m2 6 n2 )2(m n)2(m n)Zx2 cos x dx (x2 2) sin x 2x cos x CZx cos nx dx Zx2 sin x dx (2 x2 ) cos x 2x sin x Cx sin nx cos nx C2nnZx cos nx sin nx Cx sin nx dx 2nnZeax Ceax dx a16

ZZxeax dx eax (x 1/a)/a Cµ¶1 1 ix e inx Cxedx n nZeax (a sin kx k cos kx)eax sin kx dx C(a2 k 2 )Zeax (a cos kx k sin kx)axe cos kx dx C(a2 k 2 )ZZZZZZ inxsinh x dx cosh x Ccosh x dx sinh x Ctanh x dx ln cosh x Csech2 x dx tanh x Ccsch2 x dx coth x Cµ ¶11x Cdx arctan22a xaaµ ¶Z11 1 xdx tanha2 x 2aa¶µ1a x Cln 2aa xµ ¶Z1x Cdx arcsin221/2(a x )aµ ¶x C arccosaµ ¶Z1 1 x Cdx cosh(x2 a2 )1/2aµ ¶Zx2x Cdx x a arctan22(a x )aZ1dx ln[x (x2 a2 )1/2 ] C2(a x2 )1/2µ ¶ 1 x C sinha·µ ¶ Z1x/a2 Cdx sin arctan223/2(a x )a1x 2 2 Ca (a x2 )1/2Zq x1/2dx aarcsin((x/a)) ax/a (x/a)2 C(a x)1/2Zxdx (a2 x2 )1/2 C(a2 x2 )1/2Z x11dx arctan[x (b/a)] C222(a bx )2a(a bx ) 2a (ab)17

2.9 Definite IntegralsZ x0Z 01/2 xen xx e Zπdx 2dx Z0µ1lnx10 ¶n 2.61 πx1/2dx (ex 1)2 Γ(n 1)the Gamma functionNote: for n an integer greater than 0, Γ(n 1) n!, Γ(1) 0! 12 Z u x2 e dx erf(u)π 0the Error function2 Z ax21Note that erf( ) 1, so that edx .π 0aZ 2e ax dx Z0 2n ax2x erπa1 3 5 · · · (2n 1)dx 2n 1 anZ rπ Sa(n 1, 2, 3 . . .)2x2n e ax dx 2S Z0 2x2n 1 e ax dx Z n!2an 1(a 0; n 0, 1, 2 . . .)2x2n 1 e ax dx 0 Z 0Z 0Z0x2 ln(1 e x )dx a k2π/2ndx sin(π/2n)e ax cos(kx)dx 11 x2n π 44518a2

2.10 Curvilinear Coordinate SystemsDefinitionsSpherical Coordinatesx r sin θ cos φy r sin θ sin φz r cos θCylindrical Coordinatesx r cos φy r sin φz zElements of area and volumeElements of AreaCartesian (x, y)Plane polar (r, θ)dS dx dydS r dr dθElements of VolumeCartesian (x, y, z)Spherical polar (r, θ, φ)Cylindrical polar (r, φ, z)dV dx dy dzdV r2 sin θ dr dθ dφdV r dr dφ dzmiscellaneousArea of elementary circular annulus, width dr, centred on the origin: dS 2πrdrVolume of elementary cylindrical annulus of height dz and thickness dr: dV 2πrdrdzVolume of elementary spherical shell of thickness dr, centred on the origin: dV 4πr2 dr19

ZZx r cos φy r sin φz zz direction110000 P(r, φ,z)11YYφ directiondzdrrdirectionXXCylindrical polar co-ordinatesx r sin θ cos φy r sin θ sin φz r cos φZrr dφVolume elementZdrP(r, θ,φ)θφYYr sin θ d φr dθXSpherical Polar co-ordinatesXVolume elementFigure 2.1: Coordinate Systems and Elements of volume.20

Solid angleNormal toelementary area daθrElementarysolid angle d Ωdad Ω da cos θr2steradiansFigure 2.2: Solid angle.1. The solid angle subtended by any closed surface at any point inside the surface is4π;2. The solid angle subtended by any closed surface at a point outside the surface iszero.21

2.11 Vectors and Vector AlgebraVectors are quantities with both magnitude and direction; they are combined by thetriangle rule (see Fig. 2.3).A B B A CCBθAFigure 2.3: Vector addition. orVectors may be denoted by bold type A, by putting a little arrow over the symbol A,by underlining the symbol A. Unit vectors are usually denoted by a circumflex accent(e.g. î).Magnitude etc. A (A · A) (A2x A2y A2z )The angle θ between two vectors A and B is given bycos θ A·BAx Bx Ay By Az Bz q A B (A2x A2y A2z )(Bx2 By2 Bz2 )Unit vectorsUnit vector in the direction of A A/ A Cartesian co-ordinates: î, ĵ, k̂ are unit vectors in the directions of the x, y, z cartesian axesrespectively.If Ax , Ay , Az are the cartesian components of A thenA îAx ĵAy k̂AzAddition and subtractionA B B A(Commutative law)(A B) C A (B C)22(Associative law)

BθpAFigure 2.4: Vector (or Cross) Product; the vector p is directed out of the page.ProductsScalar productA · B A B cos θ B · A(a scalar)î · î ĵ · ĵ k̂ · k̂ 1î · ĵ ĵ · k̂ k̂ · î 0A · B Ax Bx Ay By Az BzA · (B C) A · B A · CVector (or Cross) productSee Fig. 2.4A B B A ( A B sin θ)p̂ (a vector)where p̂ is a unit vector perpendicular to both A and B. Note that the vector product isnon-commutative.î ĵ k̂ĵ k̂ îk̂ î ĵî î ĵ ĵ k̂ k̂ 0Also, in cartesian co-ordinates,A B (Ay Bz Az By )î (Az Bx Ax Bz )ĵ (Ax By Ay Bx )k̂ îĵAx AyBx By k̂ Az Bz 23

Scalar triple product(A B) · C (B C) · A (C A) · B (a scalar)(Note the cyclic order: A B C) Ax Ay AzBx B y B zCx Cy Cz Ax (By Cz Bz Cy ) Ay (Bz Cx Bx Cz ) Az (Bx Cy By Cx )Vector triple productA (B C) (A · C)B (A · B)CA (B C) B (C A) C (A B) 024(a vector)(Note the cyclic order: A B C)

2.12 Complex Numbers z a ib is a complex number where a, b are real and i 1 (N.B. sometimes j isused instead of i).a is the real part of z and b is the imaginary part. Sometimes the real part of a complexquantity z is denoted by ℜ(z), the imaginary part by ℑ(z).If a1 ib1 a2 ib2 then a1 a2 and b1 b2 .Modulus and argument The modulus of z z a2 b2 .The argument of z θ arctan(b/a)Complex conjugateTo form the complex conjugate of any complex number simply replace i by i whereverit occurs in the number. Thus if z a ib then the complex conjugate is z a ib.If z Ae ix then z q A e ix . Note: z zz (a ib)(a ib) a2 b2RationalizationIf z A/B, where A and B are both complex numbers, then the quotient can be ‘rationalized’ as follows:AAB AB z BBB B 2and the denominator is now real.Polar form See Fig. 2.5. If z a ib then z a2 b2 and θ arctan(b/a). Note when evaluatingarctan(b/a), θ must be put in the correct quadrant (see Fig 2.6).eiθ cos θ i sin θsin θ eiθ e iθ2i(Euler’s identity)cos θ eiθ e iθ2z a ib z cos θ i z sin θ z exp[iθ]Ifz1 z1 exp[iθ1 ] and z2 z2 exp[iθ2 ]then z1 z2 z1 z2 exp i[θ1 θ2 ] z1 z1exp i[θ1 θ2 ] andz2 z2 25

Imaginary partof zb z #########a z cos θ#b z sin θ## θ#aReal partof zFigure 2.5: Argand diagram.If z nthen z z n z m z n z1 z2 w where w w exp[iθ] w 1/n exp[i(θ 2kπ)/n] where k 0, 1, 2 . . . (n 1) z n z m n z1 z2 DeMoivre’s theoremeinθ (cos θ i sin θ)n cos nθ i sin nθwhere n is an integerTrigonometric and hyperbolic functionssinh(iθ) i sin θcosh(iθ) cos θtanh(iθ) i tan θsin(iθ) i sinh θcos(iθ) cosh θtan(iθ) i tanh θFigure 2.6: Selecting the correct quadrant for θ arctan(b/a)26

2.13 SeriesArithmetic progression (A.P.)S a (a d) (a 2d) (a 3d) · · · (a [n 1]d)Sum over n terms isSn n[2a (n 1)d]2Geometric progression (G.P.)S a ar ar2 ar3 · · · arn 1Sum over n terms isSn If r 1 the sum to infinity isa(1 rn )(1 r)S a(1 r)Binomial theorem(a b)n an nan 1 b n(n 1) n 2 2 n(n 1)(n 2) n 3 3a b a b ···2!3!If n is a positive integer the series contains (n 1) terms. If n is a negative integer or apositive or negative fraction the series is infinite. The series converges if b/a 1.Special cases:n(n 1) 2 n(n 1)(n 2) 3x x · · · Valid for all n.2!3!1 x x2 x3 x 4 · · ·1 2x 3x2 4x3 5x4 · · ·x x2 x3 5x41 ···2816 128x 3x2 5x3 35x41 ···2816128(1 x)n 1 nx (1 x) 1 (1 x) 2 1(1 x) 2 1(1 x) 2 Maclaurin’s theorem df x2 d2 f x3 d3 f xn dn f f (x) f (0) x ··· ··· dx x 0 2! dx2 x 0 3! dx3 x 0n! dxn x 0where, for example d2 f /dx2 x 0 means the result of forming the second derivative of f (x)with respect to x and then setting x 0.27

Taylor’s theorem (x a)3 d3 f (x a)2 d2 f df ··· f (x) f (a) (x a) dx x a2!dx2 x a3!dx3 x a (x a)n dn f ··· ··· n!dxn x awhere, for example d2 f /dx2 x a again means the result of forming the second derivativeof f (x) with respect to x and then setting x a.Series expansions of trigonometric functionssin θ θ θ3 θ5 θ7 ···3!5!7!cos θ 1 θ2 θ4 θ6 ···2!4!6!For θ in radians and small (i.e. θ 1):sin θ θ 4 % for θ 30 0.52 radiansError tan θ θcos θ 1 4 % for θ 30 0.52 radiansError 4 % for θ 16 0.28 radiansError Series expansions of exponential functionse x 1 x ln(1 x) x For x small (i.e. x 1):x2 x3 x4 ···2!3!4!x2 x3 x4 ···234Convergent for all values of xConvergent for 1 x 1e x exp[ x] 1 x · · ·ln(1 x) x · · ·Series expansions of hyperbolic functions1 xx3 x5(e e x ) x ···23!5!x2 x41 x(e e x ) 1 ···cosh x 22!4!sinh x 28

L’Hôpital’s ruleIf two functions f (x) and g(x) are both zero or infinite at x a the ratio f (a)/g(a) isundefined. However the limit of f (x)/g(x) as x approaches a may exist. This may befound fromf (x)f ′ (a)lim x a g(x)g ′ (a)where f ′ (a) means the result of differentiating f (x) with respect to x and then puttingx a.Convergence TestsD’Alembert’s ratio testIn a series, Xn 1an , let the ratio R n limµ¶an 1.an If R 1 the series is convergent If R 1 the series is divergent If R 1 the test fails.The Integral TestZ A sum to infinity of an converges ifan dn is finite. This can only be applied to series1where an is positive and decreasing as n gets larger.29

2.14 Ordinary Differential EquationsGeneral points1. In general, finding a function which is ‘a solution of’ (i.e. satisfies) any particulardifferential equation is a trial and error process. It involves inductive not deductivereasoning, comparable with integration as opposed to differentiation.2. If the highest differential coefficient in the equation is the nth then the generalsolution must contain n arbitrary constants.3. The known physical conditions–the boundary conditions–may enable one particularsolution or a set of solutions to be selected from the infinite family of possiblemathematical solutions; that is boundary conditions may allow specific values to beassigned to the arbitrary constants in the general solution.4. Virtually all the ordinary differential equations met in basic physics are linear, thatis the differential coefficients occur to the first power only.Definitionsorder of a differential equationThe order of a differential equation is the order of the highest differential coefficient itcontains.degree of a differential equationThe degree of a differential equation is the power to which the highest order differentialcoefficient is raised.dependent and independent variablesOrdinary differential equations involve only two variables, one of which is referred to asthe dependent variable and the other as the independent variable. It is usually clear fromthe nature of the physical problem which is the independent and which is the dependentvariable.30

First Order Differential EquationsDirect IntegrationZdy f (x) has the solution y f (x)dx. Thus it can be solved (indxprinciple) by direct integration.The equationSeparable Variablesdy f (x)g(y), where f (x) is a function ofdxx only and g(y) is a function of y only. Dividing both sides by g(y) and integrating givesZZdy f (x) dx C, which can be used to obtain the solution of y(x).g(y)First order differential equations of the formThe linear equationdy P (x)y Q(x)dxThiscan be solved by multiplying through by an ‘integrating factor’ eI , where I RP (x)dx, so that the original equation can be rewritten asA general first order linear equation of the formd ³ I ye eI Q(x)dxSince Q and I are only functions of x we can integrate both sides to obtainIye ZQ(x)eI dxSecond Order Differential EquationsDirect IntegrationEquations of the formy d2 y f (x), can be solved by integrating twice:dx2Z ·Z f (x)dx C dx Z ·Zf (x)dx dx Cx DNote that there are two arbitrary constants, C and D.31

Homogeneous Second Order Differential Equationsadyd2 y b cy 02dxdxwhere a, b and c are constants. Letting y Aeαx , gives the auxiliary equationaα2 bα c 0This is solved for α using the quadratic equation, which gives two values for α, α1 andα2 . The general solution is the combination of the two, y Aeα1 x Beα2 x .The auxiliary equation has real roots When b2 4ac, both α1 and α2 are real. Thegeneral solution is y Aeα1 x Beα2 x .The auxiliary equation has complex roots When b2 4ac, both α1 and α2 are complex. Using Euler’s Equation, substituting C A B and D i(A B), the generalsolution can be written asy eαx (C cos(βx) D sin(βx))where α b/(2a) and β q(4ac b2 )/(2a).The auxiliary equation has equal roots When b2 4ac, there is only one α. Thegeneral solution is given by y (A Bx)eαxNon-homogeneous Second Order Differential EquationsNon-homogeneous second order differential equations are of the formadyd2 y b cy f (x)2dxdxTo solve, first solve the homogeneous equation (i.e. for right-hand side 0),d2 ydya 2 b cy 0dxdxusing the method given above to get the solutiony Aeα1 x Beα2 xwhich 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 function f (x).f (x) kf (x) kxf (x) kx2f (x) k sin x or k cos xf (x) ekxassumeassumeassumeassumeassume32yyyyy C Cx D Cx2 Dx E C cos x D sin x Cekx

2.15 Partial DifferentiationDefinitionIf f f (x, y) with x and y independent, thenÃ f x! yf (x δx, y) f (x, y)δx 0δxlim derivative with respect to x with y kept constantÃ f y!f (x, y δy) f (x, y)δy derivative with respect to y with x kept constant xlimδy 0The rules of partial differentiation are the same as differentiation, alwaysbearing in mind which term is varying and which are constant.Convenient notationfx 2f 2f f 2f 2f f, fxx ,f ,f ,f ,f xyyyyyx x x2 x y y y 2 y xNote that for functions with continuous derivatives fxy 2f 2f fyx x y y xTotal DerivativesTotal change in f due to infinitesimal changes in both x and y:df Ã!ÃÃ f x!Ã fdx yy!dyx!df fdy f is the total derivative of f with respect to x. dx x y y x dxÃ!Ã!df f fdx is the total derivative of f with respect to y. dy y x x y dyFor a function where each variable depends upon a third parameter, such as f (x, y) wherex and y depend on time (t):df dtÃ f x!yÃ fdx dt y33!xdydt

Maxima and Minima with two or more variablesIf f is a function of two or more variables we can still find the maximum and minimumpoints of the function. Consider a 3-d surface given by f f (x, y). We can identify thefollowing types of stationary points where gradients are zero:peak – a local maximumpit – a local minimumpass or saddle point – minimum in one direction, maximum in the 1010-10Figure 2.7: Surface plots showing a peak (left) and a saddle Point (right)At each peak, pit or pass, the function f is stationary, i.e. f f 0 x yLet f (x0 , y0 ) be a stationary point and define the second derivative test discriminant asD Ã 2f x2!Ã 2f y 2!Ã 2f x y!22 fxx fyy fxywhich is evaluated at (x0 , y0 ) and,if D 0 and fxx 0 we have a pit (minimum)if D 0 and fxx 0 we have a peak (maximum)if D 0 we have a pass (saddle point)if D 0 we do not know, have to test further comparing f (x0 , y0 ), f (x0 dx, y0 ),f (x0 , y0 dy), i.e. compare with values close to f (x0 , y0 ).34

2.16 Partial Differential EquationsThe following partial differential equations are basic to physics:Three dimensionsOne dimension 2φ x2 2φ x2 φ t1 2φc2 t2D 2 φ 0Laplace’s equation2 φ constantPoisson’s equation φ 2 φ DDiffusion equation t1 2φ2 φ 2 2Wave equationc tIn general a partial differential equation can be satisfied by a wide variety of differentfunctions, i.e. if φ f (x, t) or φ f (x, y, z), f may take many different forms which arenot equivalent ways of representing the same set of surfaces. For example, any continuous,differentiable function of (x ct) will fit the one-dimensional wave equation.‘Solving’ these partial differential equations in a particular physical context thereforeinvolves choosing not just constants but also the functions which fit the boundary conditions. Equations involving three or four independent variables, e.g. (x, y, t) or (x, y, z, t)can be solved only when the ‘boundaries’ are surfaces of some simple co-ordinate system, such as rectangular, polar, cylindrical polar, spherical polar. The partial differentialequations can then be separated into a number of ordinary differential equations in theseparate co-ordinates, and solutions can be expressed as expansions of various class

1.2 Astrophysical Quantities Symbol Quantity Value M Mass of Sun 1.989 1030 kg R Radius of Sun 6.955 108 m L Bolometric luminosity of Sun 3.846 1026 W M bol Absolute bolometric magnitude of Sun 4.75 M vis Absolute visual magnitude of Sun 4.83 T Eﬀective temperature of Sun 5778K Spectral type of Sun G2V M J Mass of Jupiter 1.899 1027 kg R J Equatorial radius of Jupiter .

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