Handbook of Mathematics, Physics and Astronomy Data School of Chemical and Physical Sciences c 2017 Contents 1 Reference Data 1 1.1 PhysicalConstants ............................... .... 2 1.2 AstrophysicalQuantities. ....... 3 1.3 PeriodicTable ................................... 4 1.4 ElectronConfigurationsoftheElements . ......... 5 1.5 GreekAlphabetandSIPrefixes. ..... 6 2 Mathematics 7 2.1 MathematicalConstantsandNotation . ........ 8 2.2 Algebra ......................................... 9 2.3 TrigonometricalIdentities . ........ 10 2.4 HyperbolicFunctions. ..... 12 2.5 Differentiation .................................. 13 2.6 StandardDerivatives. ..... 14 2.7 Integration ..................................... 15 2.8 StandardIndefiniteIntegrals . ....... 16 2.9 DefiniteIntegrals ................................ 18 2.10 CurvilinearCoordinateSystems. ......... 19 2.11 VectorsandVectorAlgebra . ...... 22 2.12ComplexNumbers ................................. 25 2.13Series ......................................... 27 2.14 OrdinaryDifferentialEquations . ......... 30 2.15 PartialDifferentiation . ....... 33 2.16 PartialDifferentialEquations . ......... 35 2.17 DeterminantsandMatrices . ...... 36 2.18VectorCalculus................................. 39 2.19FourierSeries .................................. 42 2.20Statistics ..................................... 45 3 Selected Physics Formulae 47 3.1 EquationsofElectromagnetism . ....... 48 3.2 Equations of Relativistic Kinematics and Mechanics . ............. 49 3.3 Thermodynamics and Statistical Physics . .......... 50 i Reference Data 1.1 PhysicalConstants................................. 2 1.2 AstrophysicalQuantities . 3 1.3 PeriodicTable ................................... 4 1.4 Electron Configurations of the Elements . 5 1.5 GreekAlphabetandSIPrefixes. 6 1 1.1 Physical Constants Symbol Quantity Value c Speed of light in free space 2.998 108 ms−1 h Planck constant 6.626 × 10−34 Js h¯ h/2π 1.055 × 10−34 Js × − − G Universal gravitation constant 6.674 10 11 Nm2 kg 2 e Electron charge 1.602 × 10−19 C × −31 me Electron rest mass 9.109 10 kg × −27 mp Proton rest mass 1.673 10 kg × −27 mn Neutron rest mass 1.675 10 kg 1 × 12 u Atomicmassunit ( 12 mass of C) = 1.661 10−27 kg × 23 −1 NA Avogadro constant 6.022 10 mol = 6.022× 1026 (kg-mole)−1 × −23 −1 k or kB Boltzmann constant 1.381 10 JK R Molar gas constant 8.314 × 103 J K−1 (kg-mole)−1 × − − 8.314 J K 1 mol 1 −24 −1 2 µB Bohr magneton 9.274 10 J T (or Am ) × −27 −1 µN Nuclear magneton 5.051 10 J T × −1 R∞ Rydberg constant 10973732 m Ry Rydberg energy 13.606 eV −11 a0 Bohr radius 5.292 10 m σ Stefan-Boltzmann constant 5.670 × 10−8 JK−4 m−2 s−1 b Wien displacement constant 2.898 × 10−3 mK α Fine-structure constant 1/137.04× −29 2 σe or σT Thomson cross section 6.652 10 m ×−7 −1 µ0 Permeability of free space 4π 10 Hm × − − = 1.257 10 6 Hm 1 2 × ǫ0 Permittivity of free space 1/(µ0c ) = 8.854 10−12 Fm−1 eV Electronvolt 1.602 10× −19 J g Standard acceleration of gravity 9.807× ms−2 atm Standard atmosphere 101325 Nm−2 = 101325 Pa 2 1.2 Astrophysical Quantities Symbol Quantity Value 30 M⊙ Mass of Sun 1.989 10 kg × 8 R⊙ Radius of Sun 6.957 10 m × 26 L⊙ Bolometric luminosity of Sun 3.828 10 W ⊙ × Mbol Absolute bolometric magnitude of Sun +4.74 ⊙ Mvis Absolute visual magnitude of Sun +4.83 ⊙ Teff Effective temperature of Sun 5770 K Spectral type of Sun G2 V M Mass of Jupiter 1.898 1027 kg J × RJ Equatorial radius of Jupiter 71492 km 24 M⊕ Mass of Earth 5.972 10 kg × R⊕ Equatorial radius of Earth 6378 km M Mass of Moon 7.348 1022 kg × R $ Equatorial radius of Moon 1738 km Sidereal year 3.156 107 s $ × au Astronomical Unit 1.496 1011 m ly Lightyear 9.461 × 1015 m pc Parsec 3.086 × 1016 m Jy Jansky 10−26 ×Wm−2 Hz−1 H Hubble constant 72 5 kms−1 Mpc−1 0 ± 3 1.3 Periodic Table 4 1.4 Electron Configurations of the Elements Z Element Electron configuration 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 1 H 1 2 He 2 3 Li 2 1 4 Be 2 2 5 B 2 2 1 6 C 2 2 2 7 N 2 2 3 8 O 2 2 4 9 F 2 2 5 10 Ne 2 2 6 11 Na 2 2 6 1 12 Mg 2 2 6 2 13 Al 2 2 6 2 1 14 Si 2 2 6 2 2 15 P 2 2 6 2 3 16 S 2 2 6 2 4 17 Cl 2 2 6 2 5 18 Ar 2 2 6 2 6 19 K 2 2 6 2 6 - 1 20 Ca 2 2 6 2 6 - 2 21 Sc 2 2 6 2 6 1 2 22 Ti 2 2 6 2 6 2 2 23 V 2 2 6 2 6 3 2 24 Cr 2 2 6 2 6 5 1 25 Mn 2 2 6 2 6 5 2 26 Fe 2 2 6 2 6 6 2 27 Co 2 2 6 2 6 7 2 28 Ni 2 2 6 2 6 8 2 29 Cu 2 2 6 2 6 10 1 30 Zn 2 2 6 2 6 10 2 31 Ga 2 2 6 2 6 10 2 1 32 Ge 2 2 6 2 6 10 2 2 33 As 2 2 6 2 6 10 2 3 34 Se 2 2 6 2 6 10 2 4 35 Br 2 2 6 2 6 10 2 5 36 Kr 2 2 6 2 6 10 2 6 37 Rb 2 2 6 2 6 10 2 6 - - 1 38 Sr 2 2 6 2 6 10 2 6 - - 2 39 Y 2 2 6 2 6 10 2 6 1 - 2 40 Zr 2 2 6 2 6 10 2 6 2 - 2 41 Nb 2 2 6 2 6 10 2 6 4 - 1 42 Mo 2 2 6 2 6 10 2 6 5 - 1 43 Tc 2 2 6 2 6 10 2 6 6 - 1 44 Ru 2 2 6 2 6 10 2 6 7 - 1 45 Rh 2 2 6 2 6 10 2 6 8 - 1 46 Pd 2 2 6 2 6 10 2 6 10 - - 47 Ag 2 2 6 2 6 10 2 6 10 - 1 48 Cd 2 2 6 2 6 10 2 6 10 - 2 49 In 2 2 6 2 6 10 2 6 10 - 2 1 50 Sn 2 2 6 2 6 10 2 6 10 - 2 2 51 Sb 2 2 6 2 6 10 2 6 10 - 2 3 52 Te 2 2 6 2 6 10 2 6 10 - 2 4 53 I 2 2 6 2 6 10 2 6 10 - 2 5 54 Xe 2 2 6 2 6 10 2 6 10 - 2 6 5 1.5 Greek Alphabet and SI Prefixes The Greek alphabet A α alpha N ν nu B β beta Ξ ξ xi Γ γ gamma O o omicron ∆ δ delta Π π pi E ǫ,ε epsilon P ρ,̺ rho Z ζ zeta Σ σ, ς sigma H η eta T τ tau Θ θ,ϑ theta Y υ upsilon I ι iota Φ φ,ϕ phi K κ kappa X χ chi Λ λ lambda Ψ ψ psi M µ 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 µ 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 MathematicalConstantsandNotation. 8 2.2 Algebra ....................................... 9 2.3 TrigonometricalIdentities . 10 2.4 HyperbolicFunctions ............................... 12 2.5 Differentiation ................................... 13 2.6 StandardDerivatives .............................. 14 2.7 Integration ..................................... 15 2.8 StandardIndefiniteIntegrals . 16 2.9 DefiniteIntegrals.................................. 18 2.10 CurvilinearCoordinateSystems. ... 19 2.11 VectorsandVectorAlgebra . 22 2.12 ComplexNumbers................................. 25 2.13Series......................................... 27 2.14 OrdinaryDifferentialEquations . ... 30 2.15 PartialDifferentiation. 33 2.16 PartialDifferentialEquations . ... 35 2.17 DeterminantsandMatrices . 36 2.18 VectorCalculus................................... 39 2.19 FourierSeries.................................... 42 2.20 Statistics....................................... 45 7 2.1 Mathematical Constants and Notation Constants π = 3.141592654 ... (N.B. π2 10) ≃ e = 2.718281828 ... ln10 = 2.302585093 ... log10 e = 0.434294481 ... ln x = 2.302585093 log10 x 1radian = 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) n n Stirling’s approximation n! (2πn)1/2 (n 1) ≃ e ≫ ln n! n ln n n (Error < 4% for n 15) ≃ − ∼ ≥ n (n 2) 5 3 1 for n> 0 odd × − ×···× × × Double Factorial n!! = n (n 2) 6 4 2 for n> 0 even × − ×···× × × 1 for n = 1, 0 − exp(x) = ex ln x = loge x arcsin x = sin−1 x arccos x = cos−1 x arctan x = tan−1 x n Ai = A1 + A2 + A3 + + An = sum of n terms i=1 ··· Xn A = A A A A = product of n terms i 1 × 2 × 3 ×···× n iY=1 +1 if x> 0 Sign function: sgn x = 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 b √b2 4ac x = − ± − 2a Logarithms and Exponentials x x ln a If y = a then y = e and loga y = x a0 = 1 a−x = 1/ax ax ay = ax+y × ax/ay = ax−y (ax)y = (ay)x = axy ln1 = 0 ln(1/x) = ln x − ln(xy) = ln x + ln y ln(x/y) = ln x ln y − ln xy = y ln x logb y Change of base: loga y = logb a log10 y and in particular ln y = 2.303 log10 y log10 e ≃ 9 2.3 Trigonometrical Identities Trigonometric functions ✑ ✑ ✑ ✑ ✑ ✑ c✑ ✑ ✑ a ✑ ✑ ✑ ✑ ✑ ✑ θ b a b a sin θ = cos θ = tan θ = c c b 1 1 1 csc θ = sec θ = cot θ = sin θ cos θ tan θ Basic relations (sin θ)2 + (cos θ)2 sin2 θ + cos2 θ = 1 ≡ 1+tan2 θ = sec2 θ 1+cot2 θ = csc2 θ sin θ = tan θ cos θ Sine and Cosine Rules ✚✚❚ ✚ β ❚ ✚ ❚ ✚ a✚ ❚ c ✚ ❚ ✚ ✚ ❚ ✚ ❚ ✚ γ α ❚ b a b c Sine Rule = = sin α sin β sin γ Cosine Rule a2 = b2 + c2 2bc cos α − 10 Expansions for compound angles sin(A + B) = sin A cos B + cos A sin B sin(A B) = sin A cos B cos A sin B − − cos(A + B) = cos A cos B sin A sin B − cos(A B) = cos A cos B + sin A sin B − tan A + tan B tan(A + B) = 1 tan A tan B tan− A tan B tan(A B) = − − 1+tan A tan B π π sin θ + = +cos θ sin θ = +cos θ 2 2 − π π cos θ + = sin θ cos θ = +sin θ 2 − 2 − sin(π + θ) = sin θ sin(π θ) = +sin θ − − cos(π + θ) = cos θ cos(π θ) = cos θ − − − 1 cos A cos B = [cos(A + B) + cos(A B)] 2 − 1 sin A sin B = [cos(A B) cos(A + B)] 2 − − 1 sin A cos B = [sin(A + B) + sin(A B)] 2 − 1 cos A sin B = [sin(A + B) sin(A B)] 2 − − sin2A = 2sin A cos A cos2A = cos2 A sin2 A = 2cos2 A 1 − − = 1 2sin2 A −2tan A tan2A = 1 tan2 A − Factor formulae A + B A B sin A + sin B = +2sin cos − 2 2 A + B A B sin A sin B = +2cos sin − − 2 2 A + B A B cos A + cos B = +2cos cos − 2 2 A + B A B cos A cos B = 2sin sin − − − 2 2 11 2.4 Hyperbolic Functions Definitions and basic relations ex e−x sinh x = − 2 ex + e−x cosh x = 2 sinh x e2x 1 tanh x = = − cosh x e2x +1 sech x = 1/ cosh x cosh2 x sinh2 x = 1 − cosech x = 1/ sinh x 1 tanh2 x = sech2x − coth x = 1/ tanh x coth2 x 1 = cosech2x − −1 2 sinh x = loge[x + √x