All in one mathematics cheatsheet

ALL IN ONE CHEAT SHEET 7 Page 1 of 286 ALL IN ONE MATHEMATICS CHEAT SHEET V2 6 eiπ + 1 = 0 CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL AND UNIVERSITY MATHEMATICS COMPILED FROM MANY SOURCES BY ALEX.

ALL IN ONE MATHEMATICS CHEAT SHEET V2.6 Euler’s Identity: iπ e +1=0 CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL AND UNIVERSITY MATHEMATICS COMPILED FROM MANY SOURCES BY ALEX SPARTALIS 2009-2012 Page of 286 REVISION HISTORY 2.1 08/06/2012 UPDATED: Format NEW: Multivariable Calculus UPDATED: Convergence tests UPDATED: Composite Functions 2.2 10/07/2012 NEW: Three Phase – Delta & Y NEW: Electrical Power 2.3 14/08/2012 NEW: Factorial NEW: Electromagnetics NEW: Linear Algebra NEW: Mathematical Symbols NEW: Algebraic Identities NEW: Graph Theory UPDATED: Linear Algebra UPDATED: Linear Transformations 2.4 31/08/2012 NEW: Graphical Functions NEW: Prime numbers NEW: Power Series Expansion NEW: Inner Products UPDATED: Pi Formulas UPDATED: General Trigonometric Functions Expansion UPDATED: Linear Algebra UPDATED: Matrix Inverse 2.5 10/09/2012 NEW: Machin-Like Formulae NEW: Infinite Summations To Pi NEW: Classical Mechanics NEW: Relativistic Formulae NEW: Statistical Distributions NEW: Logarithm Power Series NEW: Spherical Triangle Identities NEW: Bernoulli Expansion UPDATED: Pi Formulas UPDATED: Logarithm Identities UPDATED: Riemann Zeta Function UPDATED: Eigenvalues and Eigenvectors 2.6 3/10/2012 NEW: QR Factorisation NEW: Jordan Forms NEW: Macroeconomics NEW: Golden Ratio & Fibonacci Sequence NEW: Complex Vectors and Matrices NEW: Numerical Computations for Matrices UPDATED: Prime Numbers UPDATED: Errors within Matrix Formula 2.7 2012 TO DO: USV Decomposition Page of 286 CONTENTS REVISION HISTORY CONTENTS PART 1: PHYSICAL CONSTANTS 20 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20 20 21 22 22 22 23 24 25 SI PREFIXES: SI BASE UNITS: SI DERIVED UNITS: UNIVERSAL CONSTANTS: ELECTROMAGNETIC CONSTANTS: ATOMIC AND NUCLEAR CONSTANTS: PHYSICO-CHEMICAL CONSTANTS: ADOPTED VALUES: NATURAL UNITS: PART 2: MATHEMTAICAL SYMBOLS 26 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 26 26 27 28 28 30 30 31 32 BASIC MATH SYMBOLS GEOMETRY SYMBOLS ALGEBRA SYMBOLS LINEAR ALGEBRA SYMBOLS PROBABILITY AND STATISTICS SYMBOLS COMBINATORICS SYMBOLS SET THEORY SYMBOLS LOGIC SYMBOLS CALCULUS & ANALYSIS SYMBOLS PART 3: AREA, VOLUME AND SURFACE AREA 33 3.1 AREA TRIANGLE: RECTANGLE: SQUARE: PARALLELOGRAM: RHOMBUS: TRAPEZIUM: QUADRILATERAL: RECTANGLE WITH ROUNDED CORNERS: REGULAR HEXAGON: REGULAR OCTAGON: REGULAR POLYGON: 3.2 VOLUME CUBE: CUBOID: PYRAMID: 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 Page of 286 TETRAHEDRON: OCTAHEDRON: DODECAHEDRON: ICOSAHEDRON: 3.3 SURFACE AREA: CUBE: CUBOIDS: TETRAHEDRON: OCTAHEDRON: DODECAHEDRON: ICOSAHEDRON: CYLINDER: 3.4 MISELANIOUS DIAGONAL OF A RECTANGLE DIAGONAL OF A CUBOID LONGEST DIAGONAL (EVEN SIDES) LONGEST DIAGONAL (ODD SIDES) TOTAL LENGTH OF EDGES (CUBE): TOTAL LENGTH OF EDGES (CUBOID): CIRCUMFERENCE PERIMETER OF RECTANGLE SEMI PERIMETER EULER’S FORMULA 3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4) 33 33 33 33 33 33 33 33 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 PART 4: ALGEBRA 36 4.1 POLYNOMIAL FORMULA: QUDARATIC: CUBIC: 4.2 ALGEBRAIC EXPANSION: BABYLONIAN IDENTITY: COMMON PRODUCTS AND FACTORS: BINOMIAL THEOREM: BINOMIAL EXPANSION: DIFFERENCE OF TWO SQUARES: BRAHMAGUPTA–FIBONACCI IDENTITY: DEGEN'S EIGHT-SQUARE IDENTITY: 4.3 LIMIT MANIPULATIONS: 4.4 SUMATION MANIPULATIONS: 4.5 COMMON FUNCTIONS: CONSTANT FUNCTION: LINE/LINEAR FUNCTION: PARABOLA/QUADRATIC FUNCTION: CIRCLE: ELLIPSE: HYPERBOLA: 4.6 LINEAR ALGEBRA: VECTOR SPACE AXIOMS: SUBSPACE: COMMON SPACES: ROWSPACE OF A SPANNING SET IN RN COLUMNSPACE OF A SPANNING SET IN RN NULLSPACE: NULLITY: 36 36 36 38 38 38 38 38 39 39 39 40 40 40 40 41 41 41 41 42 42 42 42 42 43 43 43 43 Page of 286 LINEAR DEPENDENCE: BASIS: STANDARD BASIS: ORTHOGONAL COMPLEMENT: ORTHONORMAL BASIS: GRAM-SCHMIDT PROCESS: COORDINATE VECTOR: DIMENSION: 4.7 COMPLEX VECTOR SPACES: FORM: DOT PRODUCT: INNER PRODUCT: 4.8 LINEAR TRANSITIONS & TRANSFORMATIONS: TRANSITION MATRIX: CHANGE OF BASIS TRANSITION MATRIX: TRANSFORMATION MATRIX: 4.9 INNER PRODUCTS: DEFINITION: AXIOMS: UNIT VECTOR: CAVCHY-SCHUARZ INEQUALITY: INNER PRODUCT SPACE: ANGLE BETWEEN TWO VECTORS: DISTANCE BETWEEN TWO VECTORS: GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS: 4.10 PRIME NUMBERS: DETERMINATE: LIST OF PRIME NUMBERS: PERFECT NUMBERS: LIST OF PERFECT NUMBERS: AMICABLE NUMBERS: LIST OF AMICABLE NUMBERS: SOCIABLE NUMBERS: LIST OF SOCIABLE NUMBERS: 4.11 GOLDEN RATIO & FIBONACCI SEQUENCE: RELATIONSHIP: INFINITE SERIES: CONTINUED FRACTIONS: TRIGONOMETRIC EXPRESSIONS: FIBONACCI SEQUENCE: 4.12 FERMAT’S LAST THEOREM: 43 43 44 44 44 45 45 45 45 45 45 46 46 46 46 46 46 46 46 47 47 47 47 47 47 47 47 47 48 48 49 49 50 50 53 53 53 53 54 54 54 PART 5: COUNTING TECHNIQUES & PROBABILITY 55 5.1 2D TRIANGLE NUMBER SQUARE NUMBER PENTAGONAL NUMBER 5.2 3D TETRAHEDRAL NUMBER SQUARE PYRAMID NUMBER 5.3 PERMUTATIONS PERMUTATIONS: PERMUTATIONS (WITH REPEATS): 5.4 COMBINATIONS 55 55 55 55 55 55 55 55 55 55 55 Page of 286 ORDERED COMBINATIONS: UNORDERED COMBINATIONS: ORDERED REPEATED COMBINATIONS: UNORDERED REPEATED COMBINATIONS: GROUPING: 5.5 MISCELLANEOUS: TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE: NUMBER OF INTERPRETERS: MAX NUMBER OF PIZZA PIECES: MAX PIECES OF A CRESCENT: MAX PIECES OF CHEESE: CARDS IN A CARD HOUSE: DIFFERENT ARRANGEMENT OF DOMINOS: UNIT FRACTIONS: ANGLE BETWEEN TWO HANDS OF A CLOCK: WINNING LINES IN NOUGHTS AND CROSSES: BAD RESTAURANT SPREAD: FIBONACCI SEQUENCE: ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) 5.6 FACTORIAL: DEFINITION: TABLE OF FACTORIALS: APPROXIMATION: 5.7 THE DAY OF THE WEEK: 5.8 BASIC PROBABILITY: 5.9 VENN DIAGRAMS: COMPLEMENTARY EVENTS: TOTALITY: CONDITIONAL PROBABILITY: UNION : INDEPENDENT EVENTS: MUTUALLY EXCLUSIVE: BAYE’S THEOREM: 5.11 BASIC STATISTICAL OPERATIONS: VARIANCE: MEAN: STANDARDIZED SCORE: 5.12 DISCRETE RANDOM VARIABLES: STANDARD DEVIATION: EXPECTED VALUE: VARIANCE: PROBABILITY MASS FUNCTION: CUMULATIVE DISTRIBUTION FUNCTION: 5.13 COMMON DRVS: BERNOULLI TRIAL: BINOMIAL TRIAL: GEOMETRIC TRIAL: NEGATIVE BINOMIAL TRIAL: 5.14 CONTINUOUS RANDOM VARIABLES: PROBABILITY DENSITY FUNCTION: CUMULATIVE DISTRIBUTION FUNCTION: INTERVAL PROBABILITY: EXPECTED VALUE: VARIANCE: 5.15 COMMON CRVS: UNIFORM DISTRIBUTION: Page of 286 55 55 55 55 55 55 55 55 55 55 55 56 56 56 56 56 56 56 56 56 56 56 57 57 57 57 57 57 57 57 57 57 57 58 58 58 58 58 58 58 58 58 58 58 58 58 59 59 59 59 59 59 59 59 59 59 EXPONENTIAL DISTRIBUTION: NORMAL DISTRIBUTION: 5.16 MULTIVARIABLE DISCRETE: PROBABILITY: MARGINAL DISTRIBUTION: EXPECTED VALUE: INDEPENDENCE: COVARIANCE: 5.17 MULTIVARIABLE CONTINUOUS: PROBABILITY: MARGINAL DISTRIBUTION: EXPECTED VALUE: INDEPENDENCE: COVARIANCE: CORRELATION COEFFICIENT: ABBREVIATIONS 60 61 61 61 61 61 61 62 62 62 62 62 62 62 62 62 PART 6: FINANCIAL 64 6.1 GENERAL FORMUALS: PROFIT: PROFIT MARGIN: SIMPLE INTEREST: COMPOUND INTEREST: CONTINUOUS INTEREST: ABBREVIATIONS (6.1): 6.2 MACROECONOMICS: GDP: RGDP: NGDP: GROWTH: NET EXPORTS: WORKING AGE POPULATION: LABOR FORCE: UNEMPLOYMENT: NATURAL UNEMPLOYMENT: UNEMPLOYMENT RATE: EMPLOYMENT RATE: PARTICIPATION RATE: CPI: INFLATION RATE: ABBREVIATIONS (6.2) 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 PART 7: PI 66 7.1 AREA: CIRCLE: CYCLIC QUADRILATERAL: AREA OF A SECTOR (DEGREES) AREA OF A SECTOR (RADIANS) AREA OF A SEGMENT (DEGREES) AREA OF AN ANNULUS: ELLIPSE : 7.2 VOLUME: 66 66 66 66 66 66 66 66 66 Page of 286 SPHERE: CAP OF A SPHERE: CONE: ICE-CREAM & CONE: DOUGHNUT: SAUSAGE: ELLIPSOID: 7.3 SURFACE AREA: SPHERE: HEMISPHERE: DOUGHNUT: SAUSAGE: CONE: 7.4 MISELANIOUS: LENGTH OF ARC (DEGREES) LENGTH OF CHORD (DEGREES) PERIMETER OF AN ELLIPSE 7.6 PI: JOHN WALLIS: ISAAC NEWTON: JAMES GREGORY: LEONARD EULER: JOZEF HOENE-WRONSKI: FRANCISCUS VIETA: INTEGRALS: INFINITE SERIES: CONTINUED FRACTIONS: 7.7 CIRCLE GEOMETRY: RADIUS OF CIRCUMSCRIBED CIRCLE FOR RECTANGLES: RADIUS OF CIRCUMSCRIBED CIRCLE FOR SQUARES: RADIUS OF CIRCUMSCRIBED CIRCLE FOR TRIANGLES: RADIUS OF CIRCUMSCRIBED CIRCLE FOR QUADRILATERALS: RADIUS OF INSCRIBED CIRCLE FOR SQUARES: RADIUS OF INSCRIBED CIRCLE FOR TRIANGLES: RADIUS OF CIRCUMSCRIBED CIRCLE: RADIUS OF INSCRIBED CIRCLE: 7.8 ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7): 7.9 CRESCENT GEOMETRY: AREA OF A LUNAR CRESCENT: AREA OF AN ECLIPSE CRESCENT: 7.10 ABBREVIATIONS (7.9): 66 66 66 66 66 66 66 66 66 66 66 66 66 67 67 67 67 67 67 67 67 67 67 67 67 68 68 69 69 69 69 69 69 69 69 69 69 70 70 70 70 PART 8: PHYSICS 71 8.1 MOVEMENT: STOPPING DISTANCE: CENTRIPETAL ACCELERATION: CENTRIPETAL FORCE: DROPPING TIME : FORCE: KINETIC ENERGY: MAXIMUM HEIGHT OF A CANNON: PENDULUM SWING TIME: POTENTIAL ENERGY: RANGE OF A CANNON: 71 71 71 71 71 71 71 71 71 71 71 Page of 286 TIME IN FLIGHT OF A CANNON: UNIVERSAL GRAVITATION: ABBREVIATIONS (8.1): 8.2 CLASSICAL MECHANICS: NEWTON’S LAWS: INERTIA: MOMENTS OF INERTIA: VELOCITY AND SPEED: ACCELERATION: TRAJECTORY (DISPLACEMENT): KINETIC ENERGY: CENTRIPETAL FORCE: CIRCULAR MOTION: ANGULAR MOMENTUM: TORQUE: WORK: LAWS OF CONSERVATION: ABBREVIATIONS (8.2) 8.3 RELATIVISTIC EQUATIONS: KINETIC ENERGY: MOMENTUM: TIME DILATION: LENGTH CONTRACTION: RELATIVISTIC MASS: 71 71 71 72 72 72 73 76 76 76 77 77 77 77 78 78 78 78 78 78 79 79 79 79 PART 9: TRIGONOMETRY 80 9.1 CONVERSIONS: 9.2 BASIC RULES: SIN RULE: COS RULE: TAN RULE: AUXILIARY ANGLE: PYTHAGORAS THEOREM: 9.3 RECIPROCAL FUNCTIONS 9.4 BASIC IDENTITES: 9.5 IDENTITIES (SINΘ): 9.6 IDENTITIES (COSΘ): 9.7 IDENTITIES (TANΘ): 9.8 IDENTITIES (CSCΘ): 9.9 IDENTITIES (COTΘ): 9.10 ADDITION FORMULAE: 9.11 DOUBLE ANGLE FORMULAE: 9.12 TRIPLE ANGLE FORMULAE: 9.13 HALF ANGLE FORMULAE: 9.14 POWER REDUCTION: 9.15 PRODUCT TO SUM: 9.16 SUM TO PRODUCT: 9.17 HYPERBOLIC EXPRESSIONS: 9.18 HYPERBOLIC RELATIONS: 9.19 MACHIN-LIKE FORMULAE: FORM: FORMULAE: IDENTITIES: 9.20 SPHERICAL TRIANGLE IDENTITIES: 80 80 80 80 80 80 80 81 81 81 82 82 82 82 83 83 84 84 85 86 86 86 87 87 87 87 88 88 Page of 286 9.21 ABBREVIATIONS (9.1-9.19) 88 PART 10: EXPONENTIALS & LOGARITHIMS 90 10.1 10.2 10.3 10.4 10.5 10.6 90 90 90 90 91 91 FUNDAMENTAL THEORY: IDENTITIES: CHANGE OF BASE: LAWS FOR LOG TABLES: COMPLEX NUMBERS: LIMITS INVOLVING LOGARITHMIC TERMS PART 11: COMPLEX NUMBERS 92 11.1 GENERAL: FUNDAMENTAL: STANDARD FORM: POLAR FORM: ARGUMENT: MODULUS: CONJUGATE: EXPONENTIAL: DE MOIVRE’S FORMULA: EULER’S IDENTITY: 11.2 OPERATIONS: ADDITION: SUBTRACTION: MULTIPLICATION: DIVISION: SUM OF SQUARES: 11.3 IDENTITIES: EXPONENTIAL: LOGARITHMIC: TRIGONOMETRIC: HYPERBOLIC: 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 93 PART 12: DIFFERENTIATION 94 12.1 GENERAL RULES: PLUS OR MINUS: PRODUCT RULE: QUOTIENT RULE: POWER RULE: CHAIN RULE: BLOB RULE: BASE A LOG: NATURAL LOG: EXPONENTIAL (X): FIRST PRINCIPLES: 12.2 EXPONETIAL FUNCTIONS: 12.3 LOGARITHMIC FUNCTIONS: 12.4 TRIGONOMETRIC FUNCTIONS: 12.5 HYPERBOLIC FUNCTIONS: 12.5 PARTIAL DIFFERENTIATION: 94 94 94 94 94 94 94 94 94 94 94 94 95 95 95 96 Page 10 of 286 ≡ 1/4 long cwt = st = 28 lb av ≡ ¼ short tn ≡ ¼ long tn ≡ 100 kg ≡ 20 gr ≡ 1/700 lb av ≡ gee × lb av × s2/ft ≡ 14 lb av ≡ mg × long tn ÷ oz t ≡ mg × sh tn ÷ oz t quarter (Imperial) quarter (informal) quarter, long (informal) quintal (metric) scruple (apothecary) sheet q s ap slug; geepound slug stone st ton, assay (long) AT ton, assay (short) AT ton, long ton, short tonne (mts unit) long tn or ton sh tn t Zentner 99.7 Ztr = 226.796 185 kg = 254.011 7272 kg = 100 kg = 1.295 9782 g = 647.9891 mg ≈ 14.593 903 kg = 6.350 293 18 kg ≈ 32.666 667 g ≈ 29.166 667 g ≡ 240 lb = 016.046 9088 kg ≡ 000 lb ≡ 000 kg = 907.184 74 kg = 000 kg = 114.305 277 24 kg (variants exist) ≡ 252 lb = 18 st wey = 12.700 586 36 kg Definitions vary; see [19] and.[14] DENSITY: Name of unit gram per millilitre kilogram per cubic metre (SI unit) kilogram per litre ounce (avoirdupois) per cubic foot Symbol g/mL kg/m3 kg/L oz/ft3 Definition ≡ g/mL ≡ kg/m3 ≡ kg/L ≡ oz/ft3 ounce (avoirdupois) per cubic inch oz/in3 ≡ oz/in3 ounce (avoirdupois) per gallon (Imperial) oz/gal ounce (avoirdupois) per gallon (U.S oz/gal fluid) pound (avoirdupois) per cubic foot lb/ft3 ≡ oz/gal Relation to SI units = 1,000 kg/m3 = kg/m3 = 1,000 kg/m3 ≈ 1.001153961 kg/m3 ≈ 1.729994044×103 kg/m3 ≈ 6.236023291 kg/m3 ≡ oz/gal ≈ 7.489151707 kg/m3 ≡ lb/ft3 lb/in3 ≡ lb/in3 ≈ 16.01846337 kg/m3 ≈ 2.767990471×104 kg/m3 ≈ 99.77637266 kg/m3 pound (avoirdupois) per cubic inch pound (avoirdupois) per gallon (Imperial) lb/gal ≡ lb/gal pound (avoirdupois) per gallon (U.S lb/gal ≡ lb/gal ≈ 119.8264273 kg/m3 fluid) slug per cubic foot slug/ft3 ≡ slug/ft3 ≈ 515.3788184 kg/m3 Page 272 of 286 99.8 TIME: Name of unit atomic unit of time Symbol au Definition ≡ a0/(α·c) fortnight helek ≡ 441 mo (hollow) + 499 mo (full) = 76 a of 365.25 d ≡ 100 a (see below for definition of year length) = 24 h ≡ Time needed for the Earth to rotate once around its axis, determined from successive transits of a very distant astronomical object across an observer's meridian (International Celestial Reference Frame) ≡ 10 a (see below for definition of year length) ≡ wk ≡ 1/1 080 h Hipparchic cycle ≡ Callippic cycles - d Callippic cycle century day d day (sidereal) d decade hour jiffy jiffy (alternate) h ≡ 60 ≡ 1/60 s ≡ 1/100 s ke (quarter of an hour) ≡ ¼ h = 1/96 d ke (traditional) ≡ 1/100 d lustre; lustrum Metonic cycle; enneadecaeteris ≡ a of 365 d ≡ 110 mo (hollow) + 125 mo (full) = 6940 d ≈ 19 a ≡ 000 a (see below for definition of year length) millennium milliday md ≡ 1/1 000 d minute moment month (full) ≡ 60 s ≡ 90 s ≡ 30 d[20] mo Page 273 of 286 Relation to SI units ≈ 2.418 884 254 × 10−17 s = 2.398 3776 × 109 s = 100 × year = 86400 s ≈ 86 164.1 s = 10 × year = 209 600 s = 3.3 s = 9.593 424 × 109 s = 600 s = 016 s = 10 ms = 60 × 60 / s = 900 s = 60 / = 15 = 24 × 60 × 60 / 100 s = 864 s = 24 * 60 / 100 = 14.4 = 1.5768 × 108 s = 5.996 16 × 108 s = 1000 × year = 24 × 60 × 60 / 000 s = 86.4 s = 60 s = 90 s = 592 000 s Average Gregorian month = 365.2425/12 d = 30.436875 d ≡ 29 d[20] Cycle time of moon phases ≈ 29.530589 days (Average) = 48 mo (full) + 48 mo (hollow) + mo (full)[21][22] = a of 365.25 d = 2922 d month (Greg av.) mo month (hollow) mo month (synodic) mo octaeteris ≡ (Gℏ /c5)½ Planck time second s shake sigma Sothic cycle svedberg week S wk year (Gregorian) a, y, or yr year (Julian) a, y, or yr year (sidereal) a, y, or yr year (tropical) a, y, or yr 99.9 ≈ 2.6297 × 106 s = 505 600 s ≈ 2.551 × 106 s = 2.524 608 × 108 s ≈ 1.351 211 868 × 10−43 s time of 192 631 770 periods of the radiation corresponding to the transition between the hyperfine levels of the (SI base unit) ground state of the caesium 133 atom at [8] K (but other seconds are sometimes used in astronomy) ≡ 10−8 s = 10 ns −6 ≡ 10 s = µs = 4.607 ≡ 461 a of 365 d 4096 × 1010 s ≡ 10−13 s = 100 fs ≡7d = 604 800 s = 365.2425 d average, calculated from common years (365 d) plus leap years = 31 556 952 s (366 d) on most years divisible by See leap year for details = 365.25 d average, calculated from common years (365 d) plus one leap year = 31 557 600 s (366 d) every four years ≡ time taken for Sun to return to the ≈ 365.256 363 d ≈ same position with respect to the stars of 31 558 149.7632 s the celestial sphere ≡ Length of time it takes for the Sun to ≈ 365.242 190 d ≈ return to the same position in the cycle of 31 556 925 s seasons FREQUENCY: Name of unit Symbol hertz (SI unit) Hz revolutions per rpm minute Definition ≡ Number of cycles per second ≡ One unit rpm equals one rotation completed around a fixed axis in one minute of time 99.10 SPEED OR VELOCITY: Page 274 of 286 Relation to SI units = Hz = 1/s ≈ 0.104719755 rad/s Name of unit foot per hour foot per minute foot per second furlong per fortnight inch per minute inch per second kilometre per hour knot knot (Admiralty) Symbol fph ≡ ft/h Relation to SI units ≈ 8.466 667 × 10−5 m/s fpm ≡ ft/min = 5.08 × 10−3 m/s fps ≡ ft/s = 3.048 × 10−1 m/s ≡ furlong/fortnight ≈ 1.663 095 × 10−4 m/s ipm ≡ in/min ≈ 4.23 333 × 10−4 m/s ips ≡ in/s = 2.54 × 10−2 m/s km/h ≡ km/h ≈ 2.777 778 × 10−1 m/s kn ≡ NM/h = 1.852 km/h ≡ NM (Adm)/h = 1.853 184 km/h[citation needed] The ratio of the speed of an object moving through a fluid to the speed of sound in the same medium; typically used as a measure of aircraft speed ≈ 0.514 444 m/s m/s ≡ m/s = m/s mph ≡ mi/h = 0.447 04 m/s mpm ≡ mi/min = 26.8224 m/s mps ≡ mi/s = 609.344 m/s c ≡ 299 792 458 m/s = 299 792 458 m/s kn mach number M metre per second (SI unit) mile per hour mile per minute mile per second speed of light in vacuum speed of sound in air Definition = 0.514 773 m/s Unitless Actual speed of sound varies depending on atmospheric conditions See "speed of sound" below for one specific condition ≈ 344 m/s at 20 °C, 60% relative humidity [23] s 99.11 FLOW (VOLUME): Name of unit cubic foot per minute cubic foot per second cubic inch per minute cubic inch per second cubic metre per second (SI unit) gallon (U.S fluid) per day Symbol CFM ft3/s in3/min in3/s m3/s GPD Definition ≡ ft3/min ≡ ft3/s ≡ in3/min ≡ in3/s ≡ m3/s ≡ gal/d Page 275 of 286 Relation to SI units = 4.719474432×10−4 m3/s = 0.028316846592 m3/s = 2.7311773 × 10−7 m3/s = 1.6387064×10−5 m3/s = m3/s = 4.381263638 × 10−8 m3/s gallon (U.S fluid) per hour gallon (U.S fluid) per minute litre per minute GPH GPM LPM ≡ gal/h = 1.051503273 × 10−6 m3/s ≡ gal/min = 6.30901964×10−5 m3/s ≡ L/min = 1.6 × 10−5 m3/s 99.12 ACCELERATION: Name of unit foot per hour per second foot per minute per second foot per second squared gal; galileo inch per minute per second inch per second squared knot per second metre per second squared (SI unit) mile per hour per second mile per minute per second mile per second squared standard gravity Symbol fph/s fpm/s fps2 Gal ipm/s ips2 kn/s m/s2 mph/s mpm/s mps2 g Definition ≡ ft/(h·s) ≡ ft/(min·s) ≡ ft/s2 ≡ cm/s2 ≡ in/(min·s) ≡ in/s2 ≡ kn/s ≡ m/s2 ≡ mi/(h·s) ≡ mi/(min·s) ≡ mi/s2 ≡ 9.806 65 m/s2 Relation to SI units ≈ 8.466 667 × 10−5 m/s2 = 5.08 × 10−3 m/s2 = 3.048 × 10−1 m/s2 = 10−2 m/s2 ≈ 4.233 333 × 10−4 m/s2 = 2.54 × 10−2 m/s2 ≈ 5.144 444 × 10−1 m/s2 = m/s2 = 4.4704 × 10−1 m/s2 = 26.8224 m/s2 = 1.609 344 × 103 m/s2 = 9.806 65 m/s2 99.13 FORCE: Name of unit dyn ≡ g·cm/s2 Relation to SI units ≈ 8.238 722 06 × 10−8 N [24] = 10−5 N kgf; kp; Gf ≡ g × kg = 9.806 65 N Symbol ≡ me· α2·c2/a0 atomic unit of force dyne (cgs unit) kilogram-force; kilopond; graveforce Definition kip; kip-force kip; kipf; ≡ g × 000 lb klbf milligrave-force, gravet-force mGf; gf newton (SI unit) N ounce-force ozf pound lb pound-force lbf ≡g×1g = 4.448 221 615 2605 × 103 N = 9.806 65 mN A force capable of giving a mass of one =1N= kg an acceleration of one meter per kg·m/s2 [25] second, per second = 0.278 013 850 ≡ g × oz 953 7812 N = 4.448 230 531 ≡ slug·ft/s2 N = 4.448 221 615 ≡ g × lb 2605 N Page 276 of 286 poundal pdl ≡ lb·ft/s2 sthene (mts unit) sn ≡ t·m/s2 ton-force tnf ≡ g × sh tn = 0.138 254 954 376 N = × 103 N = 8.896 443 230 521 × 103 N 99.14 PRESSURE OR MECHANICAL STRESS: Name of unit Symbol atmosphere (standard) atm atmosphere (technical) at bar barye (cgs unit) bar centimetre of mercury cmHg Definition ≡ kgf/cm2 ≡ dyn/cm2 ≡ 13 595.1 kg/m3 × cm × g centimetre of water (4 °C) cmH2O ≈ 999.972 kg/m3 × cm × g foot of mercury ≡ 13 595.1 kg/m3 × ft × g ftHg (conventional) foot of water (39.2 °F) ftH2O ≈ 999.972 kg/m3 × ft × g inch of mercury inHg ≡ 13 595.1 kg/m3 × in × g (conventional) inch of water (39.2 °F) inH2O ≈ 999.972 kg/m3 × in × g kilogram-force per square kgf/mm2 ≡ kgf/mm2 millimetre ≡ kipf/sq in Relation to SI units ≡ 101 325 Pa [26] = 9.806 65 × 104 Pa [26] ≡ 105 Pa = 0.1 Pa ≈ 1.333 22 × 103 Pa [26] ≈ 98.0638 Pa [26] ≈ 40.636 66 × 103 Pa [26] ≈ 2.988 98 × 103 Pa [26] ≈ 3.386 389 × 103 Pa [26] ≈ 249.082 Pa [26] = 9.806 65 × 106 Pa [26] ≈ 6.894 757 × 106 Pa [26] ≈ 0.133 3224 Pa kip per square inch ksi micron (micrometre) of mercury pound per square foot ≡ 13 595.1 kg/m3 × µm × g ≈ [26] 0.001 torr ≡ 13 595.1 kg/m3 × mm × g ≈ mmHg ≈ 133.3224 Pa [26] torr ≈ 999.972 kg/m3 × mm × g = mmH2O = 9.806 38 Pa 0.999 972 kgf/m2 Pa ≡ N/m2 = kg/(m·s2) = Pa [27] = × 103 Pa = pz ≡ 000 kg/m·s2 kPa ≈ 47.880 25 Pa psf ≡ lbf/ft2 [26] pound per square inch psi poundal per square foot pdl/sq ft ≡ pdl/sq ft millimetre of mercury millimetre of water (3.98 °C) pascal (SI unit) pièze (mts unit) µmHg ≡ lbf/in2 Page 277 of 286 ≈ 6.894 757 × 103 Pa [26] ≈ 1.488 164 Pa [26] ≡ sh tn × g / sq ft short ton per square foot torr ≡ 101 325/760 Pa torr ≈ 95.760 518 × 103 Pa ≈ 133.3224 Pa [26] 99.15 TORQUE OR MOMENT OF FORCE: Name of unit foot-pound force foot-poundal inch-pound force metre kilogram Newton metre (SI unit) Symbol ft lbf ft pdl in lbf m kg N·m Definition ≡ g × lb × ft ≡ lb·ft2/s2 ≡ g × lb × in ≡N×m/g ≡ N × m = kg·m2/s2 Relation to SI units = 1.355 817 948 331 4004 N·m = 4.214 011 009 380 48 × 10−2 N·m = 0.112 984 829 027 6167 N·m ≈ 0.101 971 621 N·m = N·m 99.16 ENERGY, WORK, OR AMOUNT OF HEAT: Name of unit barrel of oil equivalent British thermal unit (ISO) British thermal unit (International Table) British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (39 °F) British thermal unit (59 °F) British thermal unit (60 °F) British thermal unit (63 °F) calorie (International Table) calorie (mean) calorie (thermochemical) calorie (3.98 °C) calorie (15 °C) calorie (20 °C) Celsius heat unit (International Table) bboe ≈ 5.8 × 106 BTU59 °F Relation to SI units ≈ 6.12 × 109 J BTUISO ≡ 1.0545 × 103 J = 1.0545 × 103 J Symbol Definition BTUIT = 1.055 055 852 62 × 103 J BTUmean ≈ 1.055 87 × 103 J BTUth ≈ 1.054 350 × 103 J BTU39 °F ≈ 1.059 67 × 103 J BTU59 °F ≡ 1.054 804 × 103 J = 1.054 804 × 103 J BTU60 °F ≈ 1.054 68 × 103 J BTU63 °F ≈ 1.0546 × 103 J calIT ≡ 4.1868 J ≈ 4.190 02 J calmean calth = 4.1868 J ≡ 4.184 J cal3.98 °C cal15 °C cal20 °C ≡ 4.1855 J CHUIT ≡ BTUIT × K/°R Page 278 of 286 = 4.184 J ≈ 4.2045 J = 4.1855 J ≈ 4.1819 J = 1.899 100 534 716 × 103 J cubic centimetre of atmosphere; standard cubic centimetre cubic foot of atmosphere; standard cubic foot cc atm; scc = 0.101 325 J cu ft atm; ≡ atm × ft3 scf = 2.869 204 480 9344 × 103 J ≡ 000 BTUIT = 1.055 055 852 62 × 106 J ≡ atm × yd3 = 77.468 520 985 2288 × 103 J cubic foot of natural gas cubic yard of atmosphere; standard cubic yard ≡ atm × cm3 cu yd atm; scy electronvolt eV ≡e×1V erg (cgs unit) erg ≡ g·cm2/s2 foot-pound force ft lbf ≡ g × lb × ft foot-poundal ft pdl ≡ lb·ft2/s2 gallon-atmosphere (imperial) imp gal atm US gal gallon-atmosphere (US) atm hartree, atomic unit of Eh energy ≡ atm × gal (imp) horsepower-hour hp·h ≡ hp × h inch-pound force in lbf ≡ g × lb × in ≡ atm × gal (US) ≡ me· α2·c2 (= Ry) ≈ 1.602 177 33 × 10−19 ± 4.9 × 10−26 J = 10−7 J = 1.355 817 948 331 4004 J = 4.214 011 009 380 48 × 10−2 J = 460.632 569 25 J = 383.556 849 0138 J ≈ 4.359 744 × 10−18 J = 2.684 519 537 696 172 792 × 106 J = 0.112 984 829 027 6167 J The work done when a force of one newton moves the point of = J = m·N = joule (SI unit) J its application a distance of one kg·m2/s2 meter in the direction of the force.[25] kilocalorie; large calorie kcal; Cal ≡ 000 calIT = 4.1868 × 103 J kilowatt-hour; Board of kW·h; ≡ kW × h = 3.6 × 106 J Trade Unit B.O.T.U litre-atmosphere l atm; sl ≡ atm × L = 101.325 J = 1.055 055 852 quad ≡ 1015 BTUIT 62 × 1018 J ≈ 2.179 rydberg Ry ≡ R∞· ℎ ·c 872 × 10−18 J = 105.505 585 therm (E.C.) ≡ 100 000 BTUIT 262 × 106 J Page 279 of 286 therm (U.S.) thermie ton of coal equivalent ton of oil equivalent ton of TNT th TCE TOE tTNT ≡ 100 000 BTU59 °F ≡ McalIT ≡ Gcalth ≡ 10 Gcalth ≡ Gcalth = 105.4804 × 106 J = 4.1868 × 106 J = 29.3076 × 109 J = 41.868 × 109 J = 4.184 × 109 J 99.17 POWER OR HEAT FLOW RATE: Name of unit atmosphere-cubic centimetre per minute atmosphere-cubic centimetre per second atmosphere-cubic foot per hour atmosphere-cubic foot per minute atmosphere-cubic foot per second BTU (International Table) per hour BTU (International Table) per minute BTU (International Table) per second calorie (International Table) per second foot-pound-force per hour foot-pound-force per minute foot-pound-force per second horsepower (boiler) horsepower (European electrical) horsepower (Imperial electrical) horsepower (Imperial mechanical) horsepower (metric) litre-atmosphere per minute Symbol Definition Relation to SI units atm ccm ≡ atm × cm3/min = 1.688 75 × 10−3 W atm ccs ≡ atm × cm3/s = 0.101 325 W atm cfh ≡ atm × cu ft/h atm·cfm ≡ atm × cu ft/min atm cfs ≡ atm × cu ft/s = 0.797 001 244 704 W = 47.820 074 682 24 W = 2.869 204 480 9344 × 103 W BTUIT/h ≡ BTUIT/h BTUIT/min ≡ BTUIT/min ≈ 0.293 071 W ≈ 17.584 264 W BTUIT/s ≡ BTUIT/s = 1.055 055 852 62 × 103 W calIT/s ≡ calIT/s = 4.1868 W ft lbf/h ≡ ft lbf/h ft lbf/min ≡ ft lbf/min ≈ 3.766 161 × 10−4 W = 2.259 696 580 552 334 × 10−2 W = 1.355 817 948 331 4004 W ≈ 9.810 657 × 103 W ft lbf/s ≡ ft lbf/s bhp ≈ 34.5 lb/h × 970.3 BTUIT/lb hp ≡ 75 kp·m/s = 736 W hp ≡ 746 W = 746 W hp ≡ 550 ft lbf/s hp ≡ 75 m kgf/s = 745.699 871 582 270 22 W = 735.498 75 W L·atm/min ≡ atm × L/min Page 280 of 286 = 1.688 75 W litre-atmosphere per second lusec poncelet square foot equivalent direct radiation ton of air conditioning ton of refrigeration (Imperial) ton of refrigeration (IT) watt (SI unit) L·atm/s ≡ atm × L/s = 101.325 W lusec p ≡ LÃàmHg/s [14] 100 m kgf/s 1.333 ì 10−4 W = 980.665 W sq ft EDR ≡ 240 BTUIT/h ≈ 70.337 057 W ≡ t ice melted / 24 h ≡ BTUIT × lng tn/lb ÷ 10 min/s ≡ BTUIT × sh tn/lb ÷ 10 min/s The power which in one second of time gives rise to one joule of energy.[25] W ≈ 504 W ≈ 3.938 875 × 103 W ≈ 3.516 853 × 103 W = W = J/s = N·m/s = kg·m2/s3 99.18 ACTION: Name of unit Symbol Definition atomic unit of action au Relation to SI units ≡ ℏ = ℎ /2π ≈ 1.054 571 68 × 10−34 J·s[28] 99.19 DYNAMIC VISCOSITY: Name of unit pascal second (SI unit) poise (cgs unit) pound per foot hour pound per foot second pound-force second per square foot pound-force second per square inch Symbol Pa·s P lb/(ft·h) lb/(ft·s) lbf·s/ft2 lbf·s/in2 Definition ≡ N·s/m2 , kg/(m·s) ≡ 10−1 Pa·s ≡ lb/(ft·h) ≡ lb/(ft·s) ≡ lbf·s/ft2 ≡ lbf·s/in2 Relation to SI units = Pa·s = 0.1 Pa·s ≈ 4.133 789 × 10−4 Pa·s ≈ 1.488164 Pa·s ≈ 47.88026 Pa·s ≈ 6,894.757 Pa·s 99.20 KINEMATIC VISCOSITY: Name of unit square foot per second square metre per second (SI unit) stokes (cgs unit) Symbol ft2/s m2/s St Definition ≡ ft2/s ≡ m2/s ≡ 10−4 m2/s Relation to SI units = 0.09290304 m2/s = m2/s = 10−4 m2/s 99.21 ELECTRIC CURRENT: Name of unit ampere (SI base unit) Symbol A Definition Relation to SI units ≡ The constant current needed to produce a force of × 10−7 newton per = A metre between two straight parallel Page 281 of 286 conductors of infinite length and negligible circular cross-section placed one metre apart in a vacuum.[8] electromagnetic unit; abampere (cgs abamp ≡ 10 A unit) esu per second; statampere (cgs esu/s ≡ (0.1 A·m/s) / c unit) = 10 A ≈ 3.335641×10−10 A 99.22 ELECTRIC CHARGE: Name of unit Symbol Definition abcoulomb; electromagnetic unit (cgs abC; emu ≡ 10 C unit) Relation to SI units = 10 C atomic unit of charge au ≡e ≈ 1.602 176 462 × 10−19 C coulomb (SI unit) C ≡ The amount of electricity carried in one second of time by one ampere of current.[25] = C = A·s faraday F ≡ mol × NA·e ≈ 96 485.3383 C statcoulomb; franklin; electrostatic unit (cgs unit) statC; Fr; ≡ (0.1 A·m) / c esu ≈ 3.335 641 × 10−10 C 99.23 ELECTRIC DIPOLE: Name of unit Symbol Definition Relation to SI units atomic unit of electric dipole moment ea0 ≈ 8.478 352 81 × 10−30 C·m 99.24 ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE: Name of Symbol Definition unit abvolt abV ≡ × 10−8 V (cgs unit) statvolt statV ≡ c· (1 µJ/A·m) (cgs unit) The difference in electric potential across two volt (SI points along a conducting wire carrying one ampere V unit) of constant current when the power dissipated between the points equals one watt Page 282 of 286 Relation to SI units = × 10−8 V = 299.792 458 V = V = W/A = kg·m2/(A·s3) 99.25 ELECTRICAL RESISTANCE: Name Symbol of unit Definition The resistance between two points in a conductor when one volt of electric potential difference, applied to these points, produces one ampere of current in the conductor ohm (SI Ω unit) Relation to SI units = Ω = V/A = kg·m2/(A2·s3) 99.26 CAPACITANCE: Name Symbol of unit Definition The capacitance between two parallel plates that results in one volt of potential difference when charged by one coulomb of electricity farad F (SI unit) Relation to SI units = F = C/V = A2·s4/(kg·m2) 99.27 MAGNETIC FLUX: Name of Symbol Definition unit maxwell Mx ≡ 10−8 Wb (CGS unit) Magnetic flux which, linking a circuit of one weber (SI turn, would produce in it an electromotive force Wb unit) of volt if it were reduced to zero at a uniform rate in second Relation to SI units = × 10−8 Wb = Wb = V·s = kg·m2/(A·s2) 99.28 MAGNETIC FLUX DENSITY: Name of unit Symbol Definition Relation to SI units −4 gauss (CGS unit) G ≡ Mx/cm = 10 T = × 10−4 T tesla (SI unit) T ≡ Wb/m2 = T = Wb/m2 = kg/(A·s2) 99.29 INDUCTANCE: Name Symbol of unit henry H (SI unit) Definition Relation to SI units The inductance of a closed circuit that produces one volt of electromotive force when the current in the = H = Wb/A = circuit varies at a uniform rate of one ampere per kg·m2/(A·s)2 second 99.30 TEMPERATURE: Name of Symbol Definition Page 283 of 286 Conversion to unit degree Celsius degree Delisle degree Fahrenheit degree Newton degree Rankine degree Réaumur kelvin °C = K − 273.15 A unit of °C is the same size as a unit of K; however, their numerical values differ as the zero point of Celsius is set at 273.15 K (the ice point) °C [K] = 373.15 − [°De] × 2/3 °F ≡ freezing pt of H2O+NaCl, 180°F [K] = ([°F] + between freezing and boiling pt of H2O @ 1atm 459.67) × 5/9 [K] = [°N] × 100/33 + 273.15 °De °F °N °R; °Ra °R ≡ absolute zero [K] = [°R] × 5/9 [K] = [°Ré] × 5/4 + 273.15 [K] = ([°Rø] − 7.5) × 40/21 + 273.15 °Ré degree Rømer °Rø kelvin (SI base unit) K [K] = [°C] + 273.15 ≡ 1/273.16 of the thermodynamic temperature of the triple point of water 1K 99.31 INFORMATION ENTROPY: Name of unit Symbol Definition SI unit J/K ≡ J/K nat; nip; nepit nat ≡ kB bit; shannon bit; b; Sh ≡ ln(2) × kB ban; ≡ ln(10) × Hart kB ban; hartley ≡ bits nibble byte B ≡ bits kilobyte (decimal) kB ≡ 000 B kilobyte KB; KiB ≡ 024 B (kibibyte) Relation to SI units Relation to bits = J/K = 1.380 650 5(23) × 10−23 J/K = 9.569 940 (16) × 10−24 J/K = bit = 3.179 065 3(53) × 10−23 J/K = 3.827 976 0(64) × 10−23 = 22 bit J/K = 7.655 952 (13) × 10−23 J/K = 23 bit = 7.655 952 (13) × 10−20 J/K = 7.839 695 (13) × 10−20 J/K = 210 bit 99.32 LUMINOUS INTENSITY: Name of unit Symbol candela (SI base unit); candle cd Definition The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant Page 284 of 286 Relation to SI units = cd intensity in that direction of 1/683 watt per steradian ≡ cd The use of candlepower as a unit is discouraged due to its ambiguity Varies and is poorly reproducible Approximately 0.981 cd candlepower cp (new) candlepower cp (old, pre-1948) = cd ≈ 0.981 cd 99.33 LUMINANCE: Name of unit candela per square foot candela per square inch candela per square metre (SI unit); footlambert lambert stilb (CGS unit) Symbol cd/ft2 cd/in2 cd/m2 fL L sb Definition ≡ cd/ft2 ≡ cd/in2 ≡ cd/m2 ≡ (1/π) cd/ft2 ≡ (104/π) cd/m2 ≡ 104 cd/m2 Relation to SI units ≈ 10.763910417 cd/m2 ≈ 1,550.0031 cd/m2 = cd/m2 ≈ 3.4262590996 cd/m2 ≈ 3,183.0988618 cd/m2 ≈ × 104 cd/m2 99.34 LUMINOUS FLUX: Name of unit Symbol Definition Relation to SI units lumen (SI unit) lm ≡ cd·sr = lm = cd·sr 99.35 ILLUMINANCE: Name of unit footcandle; lumen per square foot lumen per square inch lux (SI unit) phot (CGS unit) Symbol fc lm/in2 lx ph Definition ≡ lm/ft2 ≡ lm/in2 ≡ lm/m2 ≡ lm/cm2 Relation to SI units = 10.763910417 lx ≈ 1,550.0031 lx = lx = lm/m2 = × 104 lx 99.36 RADIATION - SOURCE ACTIVITY: Name of unit becquerel (SI unit) curie rutherford (H) Symbol Definition Bq ≡ Number of disintegrations per second Ci ≡ 3.7 × 1010 Bq rd ≡ MBq 99.37 RADIATION – EXPOSURE: Name of unit Symbol Definition Relation to SI units −4 roentgen R R ≡ 2.58 × 10 C/kg = 2.58 × 10−4 C/kg 99.38 RADIATION - ABSORBED DOSE: Page 285 of 286 Relation to SI units = Bq = 1/s = 3.7 × 1010 Bq = × 106 Bq Name of unit Symbol Definition Relation to SI units 2 gray (SI unit) Gy ≡ J/kg = m /s = Gy rad rad ≡ 0.01 Gy = 0.01 Gy 99.39 RADIATION - EQUIVALENT DOSE: Name of unit Symbol Definition Relation to SI units Röntgen equivalent man rem ≡ 0.01 Sv = 0.01 Sv sievert (SI unit) Sv ≡ J/kg = Sv Page 286 of 286 ... Graph is a horizontal line passing through the point (0,a) x=a Graph is a vertical line passing through the point (a,0) Line/Linear Function: y = mx + c Graph is a line with point (0,c) and slope... COMPONENTS RESISTANCE IN SERIES: RESISTANCE IN PARALLEL: INDUCTIVE IMPEDANCE: CAPACITOR IMPEDANCE: CAPACITANCE IN SERIES: CAPACITANCE IN PARALLEL: VOLTAGE, CURRENT & POWER SUMMARY: 18.3 THEVENIN’S... arcsecond line 1´ = 60´´ line from point A to point B ray line that start from point A | perpendicular perpendicular lines (90º angle) AC | BC || parallel parallel lines AB || CD ≅ congruent