APPENDIX Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) Definitions of SI Base Units • Names and Symbols for the SI Base Units • SI Derived Units with Special Names and Symbols • Units in Use Together with the SI Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples • Conversion Factors — Metric to English • Conversion Factors — English to Metric • Conversion Factors — General • Temperature Factors • Conversion of Temperatures Physical Constants General • π Constants • Constants Involving e • Numerical Constants Symbols and Terminology for Physical and Chemical Quantities Elementary Algebra and Geometry Fundamental Properties (Real Numbers) • Exponents • Fractional Exponents • Irrational Exponents • Logarithms • Factorials • Binomial Theorem • Factors and Expansion • Progression • Complex Numbers • Polar Form • Permutations • Combinations • Algebraic Equations • Geometry Determinants, Matrices, and Linear Systems of Equations Determinants • Evaluation by Cofactors • Properties of Determinants • Matrices • Operations • Properties • Transpose • Identity Matrix • Adjoint • Inverse Matrix • Systems of Linear Equations • Matrix Solution Trigonometry Triangles • Trigonometric Functions of an Angle • Inverse Trigonometric Functions Analytic Geometry Rectangular Coordinates • Distance between Two Points; Slope • Equations of Straight Lines • Distance from a Point to a Line • Circle • Parabola • Ellipse • Hyperbola (e > 1) • Change of Axes Series Bernoulli and Euler Numbers • Series of Functions • Error Function • Series Expansion Differential Calculus Notation • Slope of a Curve • Angle of Intersection of Two Curves • Radius of Curvature • Relative Maxima and Minima • Points of Inflection of a Curve • Taylor’s Formula • Indeterminant Forms • Numerical Methods • Functions of Two Variables • Partial Derivatives Integral Calculus Indefinite Integral • Definite Integral • Properties • Common Applications of the Definite Integral • Cylindrical and Spherical Coordinates • Double Integration • Surface Area and Volume by Double Integration • Centroid Vector Analysis Vectors • Vector Differentiation • Divergence Theorem (Gauss) • Stokes’ Theorem • Planar Motion in Polar Coordinates © 2003 by CRC Press LLC Special Functions Hyperbolic Functions • Laplace Transforms • z-Transform • Trigonometric Identities • Fourier Series • Functions with Period Other Than 2π • Bessel Functions • Legendre Polynomials • Laguerre Polynomials • Hermite Polynomials • Orthogonality Statistics Arithmetic Mean • Median • Mode • Geometric Mean • Harmonic Mean • Variance • Standard Deviation • Coefficient of Variation • Probability • Binomial Distribution • Mean of Binomially Distributed Variable • Normal Distribution • Poisson Distribution Tables of Probability and Statistics Areas under the Standard Normal Curve • Poisson Distribution • t-Distribution • χ2 Distribution • Variance Ratio Tables of Derivatives Integrals Elementary Forms • Forms Containing (a + bx) The Fourier Transforms Fourier Transforms • Finite Sine Transforms • Finite Cosine Transforms • Fourier Sine Transforms • Fourier Cosine Transforms • Fourier Transforms Numerical Methods Solution of Equations by Iteration • Finite Differences • Interpolation Probability Definitions • Definition of Probability • Marginal and Conditional Probability • Probability Theorems • Random Variable • Probability Function (Discrete Case) • Cumulative Distribution Function (Discrete Case) • Probability Density (Continuous Case) • Cumulative Distribution Function (Continuous Case) • Mathematical Expectation Positional Notation Change of Base • Examples Credits Associations and Societies Ethics Greek Alphabet Greek Letter Greek Name α β γ δ ε ζ η θ ι κ λ µ Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu Α Β Γ ∆ Ε Ζ Η Θ Ι Κ Λ Μ © 2003 by CRC Press LLC ϑ Greek Letter EnglishEquivalent a b g d e z e th i k l m Ν Ξ Ο Π P Σ Τ Y Φ X Ψ Ω ν ξ ο π ρ σ τ υ φ χ ψ ω s ϕ Greek Name English Equivalent Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega n x o p r s t u ph ch ps o– International System of Units (SI) The International System of Units (SI) was adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960 It is a coherent system of units built from seven SI base units, one for each of the seven dimensionally independent base quantities: the meter, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively The definitions of the SI base units are given below The SI derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity In the International System there is only one SI unit for each physical quantity This is either the appropriate SI base unit itself or the appropriate SI derived unit However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units It is recommended that only SI units be used in science and technology (with SI prefixes where appropriate) Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units This section is based on information supplied by IUPAC Definitions of SI Base Units Meter — The meter is the length of path traveled by light in vacuum during a time interval of 1/299 792 458 of a second (17th CGPM, 1983) Kilogram — The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (3rd CGPM, 1901) Second — The second is the duration of 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM, 1967) Ampere — The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed meter apart in vacuum, would produce between these conductors a force equal to × 10–7 newton per meter of length (9th CGPM, 1948) Kelvin — The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (13th CGPM, 1967) Mole — The mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12 When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, or other particles, or specified groups of such particles (14th CGPM, 1971) Examples of the use of the mole: mol of H2 contains about 6.022 × 1023 H2 molecules, or 12.044 × 1023 H atoms mol of HgCl has a mass of 236.04 g mol of Hg2Cl2 has a mass of 472.08 g mol of Hg2+2 has a mass of 401.18 g and a charge of 192.97 kC mol of Fe0.91S has a mass of 82.88 g mol of e– has a mass of 548.60 µg and a charge of – 96.49 kC mol of photons whose frequency is 1014 Hz has energy of about 39.90 kJ Candela — The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of (1/683) watt per steradian (16th CGPM, 1979) © 2003 by CRC Press LLC Names and Symbols for the SI Base Units Physical Quantity Name of SI Unit Symbol for SI Unit Meter Kilogram Second Ampere Kelvin Mole Candela m kg s A K mol cd Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity SI Derived Units with Special Names and Symbols Physical Quantity Frequency Force Pressure, stress Energy, work, heat Power, radiant flux Electric charge Electric potential, electromotive force Electric resistance Electric conductance Electric capacitance Magnetic flux density Magnetic flux Inductance Celsius temperature Luminous flux Illuminance Activity (radioactive) Absorbed dose (of radiation) Dose equivalent (dose equivalent index) Plane angle Solid angle Name of SI Unit Symbol for SI Unit Expression in Terms of SI Base Units Hertz Newton Pascal Joule Watt Coulomb Volt Hz N Pa J W C V s–1 m kg s–2 N m–2 = m–1 kg s–2 N m = m2 kg s–2 J s–1 = m2 kg s–3 As J C –1 = m2 kg s–3A –1 Ohm Siemens Farad Tesla Weber Henry Degree Celsius Lumen Lux Becquerel Gray Sievert Ω S F T Wb H °C lm lx Bq Gy Sv V A –1 = m2 kg s–3A –2 Ω–1 = m–2 kg–1 s3A C V –1 = m–2 kg–1 s4A V s m–2 = kg s–2A –1 V s = m2 kg s–2A –1 V A –1 s = m2 kg s–2A –2 K cd sr cd sr m–2 s–1 J kg –1 = m2 s–2 J kg –1 = m2 s–2 Radian Steradian rad sr I = m m–1 I = m2 m–2 For radial (circular) frequency and for angular velocity, the unit rad s –1, or simply s–1, should be used, and this may not be simplified to Hz The unit Hz should be used only for frequency in the sense of cycles per second The Celsius temperature θ is defined by the equation: θ ⁄ °C = T ⁄ K – 273.15 The SI unit of Celsius temperature interval is the degree Celsius, °C, which is equal to the kelvin, K °C should be treated as a single symbol, with no space between the ° sign and the letter C (The symbol °K, and the symbol °, should no longer be used.) © 2003 by CRC Press LLC Units in Use Together with the SI These units are not part of the SI, but it is recognized that they will continue to be used in appropriate contexts SI prefixes may be attached to some of these units, such as milliliter, ml; millibar, mbar; megaelectronvolt, MeV; and kilotonne, ktonne Physical Quantity Time Time Time Planeangle Planeangle Planeangle Length Area Volume Mass Pressure Energy Mass Name of Unit Symbol for Unit Value in SI Units Minute Hour Day Degree Minute Second Ångstrom Barn Liter Tonne Bar Electronvolt Unified atomic mass unit2,3 h d ° ′ ″ Å b l, L t bar eV (= e × V) u (= ma( 12C)/12) 60 s 3600 s 86 400 s (π /180) rad (π /10 800) rad (π /648 000) rad 10 –10 m 10 –28 m2 dm3 = 10–3 m3 Mg = 103 kg 10 Pa = 10 N m–2 ≈ 1.60218 × 10–19 J ≈ 1.66054 × 10–27 kg The ångstrom and the bar are approved by CIPM for “temporary use with SI units,” until CIPM makes a further recommendation However, they should not be introduced where they are not used at present The values of these units in terms of the corresponding SI units are not exact, since they depend on the values of the physical constants e (for the electronvolt) and NA (for the unified atomic mass unit), which are determined by experiment The unified atomic mass unit is also sometimes called the dalton, with symbol Da, although the name and symbol have not been approved by CGPM Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples Multiples and Submultiples 18 10 10 15 10 12 10 10 10 10 10 © 2003 by CRC Press LLC Prefixes exa peta tera giga mega kilo hecto deca Symbols E P T G M k h da Multiples and Submultiples –1 10 10 –2 10 –3 10 –6 10 –9 10 –12 10 –15 10 –18 Prefixes Symbols deci centi milli micro nano pico femto atto d c m µ (Greek mu) n p f a Conversion Factors — Metric to English To obtain Inches Feet Yards Miles Ounces Pounds Gallons (U.S liquid) Fluid ounces Square inches Square feet Square yards Cubic inches Cubic feet Cubic yards Multiply By Centimeters Meters Meters Kilometers Grams Kilograms Liters Milliliters (cc) Square centimeters Square meters Square meters Milliliters (cc) Cubic meters Cubic meters 0.3937007874 3.280839895 1.093613298 0.6213711922 3.527396195 × 10–2 2.204622622 0.2641720524 3.381402270 × 10–2 0.1550003100 10.76391042 1.195990046 6.102374409 × 10–2 35.31466672 1.307950619 Conversion Factors — English to Metric* To obtain Microns Centimeters Meters Meters Kilometers Grams Kilograms Liters Millimeters (cc) Square centimeters Square meters Square meters Milliliters (cc) Cubic meters Cubic meters Multiply By Mils Inches Feet Yards Miles Ounces Pounds Gallons (U.S liquid) Fluid ounces Square inches Square feet Square yards Cubic inches Cubic feet Cubic yards 25.4 2.54 0.3048 0.9144 1.609344 28.34952313 0.45359237 3.785411784 29.57352956 6.4516 0.09290304 0.83612736 16.387064 2.831684659 × 10–2 0.764554858 * Boldface numbers are exact; others are given to ten significant figures where so indicated by the multiplier factor Conversion Factors — General* To obtain Atmospheres Atmospheres Atmospheres BTU BTU Cubic feet Degree (angle) Ergs Feet Feet of water @ 4°C Foot-pounds Foot-pounds Foot-pounds per Horsepower Inches of mercury @ 0°C © 2003 by CRC Press LLC Multiply By Feet of water @ 4°C Inches of mercury @ 0°C Pounds per square inch Foot-pounds Joules Cords Radians Foot-pounds Miles Atmospheres Horsepower-hours Kilowatt-hours Horsepower Foot-pounds per sec Pounds per square inch 2.950 × 10–2 3.342 × 10–2 6.804 × 10–2 1.285 × 10–3 9.480 × 10–4 128 57.2958 1.356 × 107 5280 33.90 1.98 × 106 2.655 × 106 3.3 × 104 1.818 × 10–3 2.036 To obtain Multiply Joules Joules Kilowatts Kilowatts Kilowatts Knots Miles Nautical miles Radians Square feet Watts BTU Foot-pounds BTU per Foot-pounds per Horsepower Miles per hour Feet Miles Degrees Acres BTU per By 1054.8 1.35582 1.758 × 10–2 2.26 × 10–5 0.745712 0.86897624 1.894 × 10–4 0.86897624 1.745 × 10–2 43560 17.5796 * Boldface numbers are exact; others are given to ten significant figures where so indicated by the multiplier factor Temperature Factors °F = ⁄ ( °C ) + 32 Fahrenheit temperature = 1.8 (temperature in kelvins) – 459.67 °C = ⁄ [ ( °F ) – 32 ] Celsius temperature = temperature in kelvins – 273.15 Fahrenheit temperature = 1.8 (Celsius temperature) + 32 Conversion of Temperatures From °Celsius °Fahrenheit Kelvin °Rankine To °Fahrenheit t F = ( t C × 1.8 ) + 32 Kelvin T K = t C + 273.15 °Rankine T R = ( t C + 273.15 ) × 18 °Celsius t F – 32 t C = 1.8 Kelvin t F – 32 T K = + 273.15 1.8 °Rankine T R = t F + 459.67 °Celsius t C = T K – 273.15 °Rankine T R = T K × 1.8 °Fahrenheit t F = T R – 459.67 Kelvin T T K = R1.8 Physical Constants General Equatorial radius of the earth = 6378.388 km = 3963.34 miles (statute) Polar radius of the earth = 6356.912 km = 3949.99 miles (statute) degree of latitude at 40° = 69 miles © 2003 by CRC Press LLC international nautical mile = 1.15078 miles (statute) = 1852 m = 6076.115 ft Mean density of the earth = 5.522 g/cm3 = 344.7 lb/ft3 Constant of gravitation (6.673 ± 0.003) × 10–8 cm3 gm–1s–2 Acceleration due to gravity at sea level, latitude 45° = 980.6194 cm/s2 = 32.1726 ft/s2 Length of seconds pendulum at sea level, latitude 45° = 99.3575 cm = 39.1171 in knot (international) = 101.269 ft/min = 1.6878 ft/s = 1.1508 miles (statute)/h micron = 10 –4 cm ångstrom = 10 –8 cm Mass of hydrogen atom = (1.67339 ± 0.0031) × 10–24 g Density of mercury at 0° C = 13.5955 g/ml Density of water at 3.98° C = 1.000000 g/ml Density, maximum, of water, at 3.98° C = 0.999973 g/cm3 Density of dry air at 0° C, 760 mm = 1.2929 g/l Velocity of sound in dry air at 0° C = 331.36 m/s = 1087.1 ft/s Velocity of light in vacuum = (2.997925 ± 0.000002) × 1010 cm/s Heat of fusion of water 0° C = 79.71 cal/g Heat of vaporization of water 100° C = 539.55 cal/g Electrochemical equivalent of silver = 0.001118 g/s international amp Absolute wavelength of red cadmium light in air at 15° C, 760 mm pressure = 6438.4696 Å Wavelength of orange-red line of krypton 86 = 6057.802 Å ␲ Constants π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511 ⁄ π = 0.31830 98861 83790 67153 77675 26745 02872 40689 19291 48091 π = 9.8690 44010 89358 61883 44909 99876 15113 53136 99407 24079 log e π = 1.14472 98858 49400 17414 34273 51353 05871 16472 94812 91531 log 10 π = 0.49714 98726 94133 85435 12682 88290 89887 36516 78324 38044 og 10 π = 0.39908 99341 79057 52478 25035 91507 69595 02099 34102 92128 Constants Involving e e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996 ⁄ e = 0.36787 94411 71442 32159 55237 70161 46086 74458 11131 03177 e = 7.38905 60989 30650 22723 04274 60575 00781 31803 15570 55185 M = log 10 e = 0.43429 44819 03251 82765 11289 18916 60508 22943 97005 80367 ⁄ M = log e 10 = 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 62877 log 10 M = 9.63778 43113 00536 78912 29674 98645 – 10 Numerical Constants = 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37695 = 1.25992 10498 94873 16476 72106 07278 22835 05702 51464 70151 log e = 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 36026 og 10 = 0.30102 99956 63981 19521 37388 94724 49302 67881 89881 46211 = 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81039 3 = 1.44224 95703 07408 38232 16383 10780 10958 83918 69253 49935 log e = 1.09861 22886 68109 69139 52452 36922 52570 46474 90557 82275 og 10 = 0.47712 12547 19662 43729 50279 03255 11530 92001 28864 19070 © 2003 by CRC Press LLC Symbols and Terminology for Physical and Chemical Quantities Name Symbol Mass Reduced mass Density, mass density Relative density Surface density Specific volume Momentum Angular momentum, action Moment of inertia Force Torque, moment of a force Energy Potential energy Kinetic energy Work Hamilton function Lagrange function Definition Classical Mechanics m µ µ = m1m2/(m1 + m2) ρ ρ = m/V d d = ρ/ρθ ρA, ρS ρA = m/A v v = V/m = 1/ρ p p = mv L L =r×p I, J l = Σ miri2 F F = dp/dt = ma T, (M) T =r×F E Ep, V, Φ Ep = – ∫ F ⋅ ds Ek, T, K Ek = (1/2)mv2 W, w W = ∫ F ⋅ ds H H (q, p) = T(q, p) + V(q) · L L (q, q) Pressure Surface tension Weight Gravitational constant Normal stress Shear stress Linear strain, relative elongation Modulus of elasticity, Young’s modulus Shear strain Shear modulus Volume strain, bulk strain Bulk modulus Compression modulus Viscosity, dynamic viscosity, fluidity Kinematic viscosity Friction coefficient Power Sound energy flux Acoustic factors Reflection factor Acoustic absorption factor Transmission factor Dissipation factor kg kg kg m–3 kg m–2 m3 kg –1 kg m s–1 Js kg m2 N Nm J J J J J J p, P γ, σ G, (W, P) G σ τ ε, e E · – V(q) = T(q, q) p = F/A γ = dW/dA G = mg F = Gm1m2/r2 σ = F/A τ = F/A ε = ∆l/l E = σ/ε Pa, N m –2 N m–1, J m–2 N N m2 kg–2 Pa Pa l Pa γ G θ K η, µ φ ν µ, ( f ) P P, Pa γ = ∆x/d G = τ/γ θ = ∆V/V0 K = –V0 (dp/dV) τx,z = η(dvx/dz) φ = 1/η ν = η/ρ Ffrict = µFnorm P = dW/dt P = dE/dt l Pa Pa Pa s m kg–1 s m2 s–1 l W W ρ αa, (α) τ δ ρ = Pr /P0 αa = – ρ τ = Ptr/P0 δ = αa – τ 1 1 Elementary Algebra and Geometry Fundamental Properties (Real Numbers) a+b = b+a Commutative Law for Addition (a + b) + c = a + (b + c) Associative Law for Addition © 2003 by CRC Press LLC SI unit a+0 = 0+a Identity Law for Addition a + ( –a ) = ( –a ) + a = Inverse Law for Addition a ( bc ) = ( ab )c Associative Law for Multiplication 1 a   =   a = 1, a ≠  a  a Inverse Law for Multiplication (a)(1) = (1)(a) = a Identity Law for Multiplication ab = ba Commutative Law for Multiplication a ( b + c ) = ab + ac Distributive Law DIVISION BY ZERO IS NOT DEFINED Exponents For integers m and n n m a a a ⁄a n m n m (a ) = a n+m = a n–m = a nm ( ab ) m = a b (a ⁄ b) m = a ⁄b m m m m Fractional Exponents a p⁄q = (a 1⁄q p ) where a1/q is the positive qth root of a if a > and the negative qth root of a if a is negative and q is odd Accordingly, the five rules of exponents given above (for integers) are also valid if m and n are fractions, provided a and b are positive Irrational Exponents If an exponent is irrational, e.g., , the quantity, such as a , is the limit of the sequence, a1.4, a1.41, a1.414, K Operations with Zero m = 0; a = Logarithms If x, y, and b are positive and b ≠ log b ( xy ) = log b x + log b y log b ( x ⁄ y ) = log b x – log b y p log b x = p log b x log b ( ⁄ x ) = – log b x log b b = log b = © 2003 by CRC Press LLC Note: b log x b = x The probability that the value of X is some real number x is given by f (x) = P [X = x], where f is called the probability function of the random variable X Cumulative Distribution Function (Discrete Case) The probability that the value of a random variable X is less than or equal to some real number x is defined as F(x) = P(X ≤ x) = Σ f ( xi ), –∞ < x < ∞ where the summation extends over those values of i such that xi ≤ x Probability Density (Continuous Case) The random variable X will be called a continuous random variable if there exists a function f such that ∞ f (x) ≥ and ∫ f ( x ) dx = for all x in interval −∞