ALL IN ONE
MATHEMATICS CHEAT SHEET
V2.10
Euler’s Identity: eiπ + 1 = 0
CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL AND UNIVERSITY MATHEMATICS
COMPILED FROM MANY SOURCES BY ALEX SPARTALIS 2009-2013 4/9/2013 9:44:00 PM
Page 1 of 330 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. 25/10/2012 NEW: USV Decomposition NEW: Ordinary Differential Equations Using Matrices NEW: Exponential Identities UPDATED: Matrix Inverse CORRECTION: Left and Right Matrix Inverse 2.8. 31/12/2012 NEW: Applications of Functions NEW: Higher Order Integration NEW: Root Expansions
Page 2 of 330 NEW: Mathematical Constants UPDATED: Applications of Integration UPDATED: Basic Statistical Operations UPDATED: Pi UPDATED: Identities Between Relationships UPDATED: Vector Space Axioms 2.9. 4/03/2012 UPDATED: Prime Numbers UPDATED: Martricies 2.10. 9/04/2012 NEW: Boolean Algebra NEW: Functions of Random Variables NEW: Transformation of the Joint Density UPDATED: Venn Diagrams: UPDATED: Basic Statistical Operations: UPDATED: Discrete Random Variables: UPDATED: Common DRVs: UPDATED: Undetermined Coefficients UPDATED: Variation of Parameters
Page 3 of 330 CONTENTS
REVISION HISTORY 2
CONTENTS 4
PART 1: PHYSICAL CONSTANTS 26
1.1 SI PREFIXES: 26 1.2 SI BASE UNITS: 26 1.3 SI DERIVED UNITS: 27 1.4 UNIVERSAL CONSTANTS: 28 1.5 ELECTROMAGNETIC CONSTANTS: 28 1.6 ATOMIC AND NUCLEAR CONSTANTS: 28 1.7 PHYSICO-CHEMICAL CONSTANTS: 29 1.8 ADOPTED VALUES: 30 1.9 NATURAL UNITS: 31 1.10 MATHEMATICAL CONSTANTS: 31
PART 2: MATHEMTAICAL SYMBOLS 33
2.1 BASIC MATH SYMBOLS 33 2.2 GEOMETRY SYMBOLS 33 2.3 ALGEBRA SYMBOLS 34 2.4 LINEAR ALGEBRA SYMBOLS 35 2.5 PROBABILITY AND STATISTICS SYMBOLS 35 2.6 COMBINATORICS SYMBOLS 37 2.7 SET THEORY SYMBOLS 37 2.8 LOGIC SYMBOLS 38 2.9 CALCULUS & ANALYSIS SYMBOLS 39
PART 3: AREA, VOLUME AND SURFACE AREA 40
3.1 AREA: 40 TRIANGLE : 40 RECTANGLE : 40 SQUARE : 40 PARALLELOGRAM : 40 RHOMBUS : 40 TRAPEZIUM : 40 QUADRILATERAL : 40 RECTANGLE WITH ROUNDED CORNERS : 40 REGULAR HEXAGON : 40 REGULAR OCTAGON : 40 REGULAR POLYGON : 40 3.2 VOLUME: 40 CUBE : 40 CUBOID : 40 Page 4 of 330 PRISIM : 40 PYRAMID : 40 TETRAHEDRON : 40 OCTAHEDRON : 40 DODECAHEDRON : 40 ICOSAHEDRON : 40 3.3 SURFACE AREA: 40 CUBE : 40 CUBOIDS : 40 TETRAHEDRON : 41 OCTAHEDRON : 41 DODECAHEDRON : 41 ICOSAHEDRON : 41 CYLINDER : 41 3.4 MISCELLANEOUS: 41 DIAGONAL OF A RECTANGLE 41 DIAGONAL OF A CUBOID 41 LONGEST DIAGONAL (E VEN SIDES ) 41 LONGEST DIAGONAL (O DD SIDES ) 41 TOTAL LENGTH OF EDGES (C UBE ): 41 TOTAL LENGTH OF EDGES (C UBOID ): 41 CIRCUMFERENCE 41 PERIMETER OF RECTANGLE 41 SEMI PERIMETER 41 EULER ’S FORMULA 41 3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4) 41
PART 4: ALGEBRA & ARITHMETIC 43
4.1 POLYNOMIAL FORMULA: 43 QUDARATIC : 43 CUBIC : 43 4.2 FUNDAMENTALS OF ARITHMETIC: 45 RATIONAL NUMBERS : 45 IRRATIONAL NUMBERS : 45 4.3 ALGEBRAIC EXPANSION: 45 BABYLONIAN IDENTITY : 45 COMMON PRODUCTS AND FACTORS : 45 BINOMIAL THEOREM : 45 BINOMIAL EXPANSION : 45 DIFFERENCE OF TWO SQUARES : 46 BRAHMAGUPTA –FIBONACCI IDENTITY : 46 DEGEN 'S EIGHT -SQUARE IDENTITY : 46 4.4 ROOT EXPANSIONS: 47 4.5 LIMIT MANIPULATIONS: 47 L’H OPITAL ’S RULE : 47 4.6 SUMMATION MANIPULATIONS: 48 4.7 COMMON FUNCTIONS: 48 CONSTANT FUNCTION : 48 LINE /L INEAR FUNCTION : 48 PARABOLA /Q UADRATIC FUNCTION : 49 CIRCLE : 49 ELLIPSE : 49 HYPERBOLA : 49 4.8 LINEAR ALGEBRA: 50 Page 5 of 330 VECTOR SPACE AXIOMS : 50 SUBSPACE : 50 COMMON SPACES : 50 ROWSPACE OF A SPANNING SET IN RN 51 COLUMNSPACE OF A SPANNING SET IN RN 51 NULLSPACE : 51 NULLITY : 51 LINEAR DEPENDENCE : 51 BASIS : 51 STANDARD BASIS : 52 ORTHOGONAL COMPLEMENT : 52 ORTHONORMAL BASIS : 52 GRAM -SCHMIDT PROCESS : 52 COORDINATE VECTOR : 53 DIMENSION : 53 4.9 COMPLEX VECTOR SPACES: 53 FORM : 53 DOT PRODUCT : 53 INNER PRODUCT : 54 4.10 LINEAR TRANSITIONS & TRANSFORMATIONS: 54 TRANSITION MATRIX : 54 CHANGE OF BASIS TRANSITION MATRIX : 54 TRANSFORMATION MATRIX : 54 4.11 INNER PRODUCTS: 54 DEFINITION : 54 AXIOMS : 54 UNIT VECTOR : 55 CAVCHY -SCHUARZ INEQUALITY : 55 INNER PRODUCT SPACE : 55 ANGLE BETWEEN TWO VECTORS : 55 DISTANCE BETWEEN TWO VECTORS : 55 GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS : 55 4.12 PRIME NUMBERS: 55 DETERMINATE : 55 LIST OF PRIME NUMBERS : 55 FUNDAMENTAL THEORY OF ARITHMETIC : 56 LAPRANGE ’S THEOREM : 56 ADDITIVE PRIMES : 56 ANNIHILATING PRIMES : 56 BELL NUMBER PRIMES : 56 CAROL PRIMES : 56 CENTERED DECAGONAL PRIMES : 56 CENTERED HEPTAGONAL PRIMES : 56 CENTERED SQUARE PRIMES: 57 CENTERED TRIANGULAR PRIMES : 57 CHEN PRIMES : 57 CIRCULAR PRIMES : 57 COUSIN PRIMES : 57 CUBAN PRIMES : 57 CULLEN PRIMES : 57 DIHEDRAL PRIMES : 57 DOUBLE FACTORIAL PRIMES : 57 DOUBLE MERSENNE PRIMES : 58 EISENSTEIN PRIMES WITHOUT IMAGINARY PART : 58 EMIRPS : 58 EUCLID PRIMES : 58 Page 6 of 330 EVEN PRIME : 58 FACTORIAL PRIMES : 58 FERMAT PRIMES : 58 FIBONACCI PRIMES : 58 FORTUNATE PRIMES : 58 GAUSSIAN PRIMES : 58 GENERALIZED FERMAT PRIMES BASE 10: 58 GENOCCHI NUMBER PRIMES: 59 GILDA 'S PRIMES : 59 GOOD PRIMES : 59 HAPPY PRIMES : 59 HARMONIC PRIMES : 59 HIGGS PRIMES FOR SQUARES : 59 HIGHLY COTOTIENT NUMBER PRIMES : 59 IRREGULAR PRIMES : 59 (P, P −5) IRREGULAR PRIMES : 59 (P, P −9) IRREGULAR PRIMES : 59 ISOLATED PRIMES : 59 KYNEA PRIMES : 59 LEFT -TRUNCATABLE PRIMES : 60 LEYLAND PRIMES : 60 LONG PRIMES : 60 LUCAS PRIMES : 60 LUCKY PRIMES : 60 MARKOV PRIMES : 60 MERSENNE PRIMES : 60 MERSENNE PRIME EXPONENTS : 60 MILLS PRIMES : 60 MINIMAL PRIMES : 60 MOTZKIN PRIMES : 60 NEWMAN –SHANKS –WILLIAMS PRIMES : 61 NON -GENEROUS PRIMES : 61 ODD PRIMES : 61 PADOVAN PRIMES : 61 PALINDROMIC PRIMES : 61 PALINDROMIC WING PRIMES : 61 PARTITION PRIMES : 61 PELL PRIMES : 61 PERMUTABLE PRIMES : 61 PERRIN PRIMES : 61 PIERPONT PRIMES : 61 PILLAI PRIMES : 62 PRIMEVAL PRIMES : 62 PRIMORIAL PRIMES : 62 PROTH PRIMES : 62 PYTHAGOREAN PRIMES : 62 PRIME QUADRUPLETS : 62 PRIMES OF BINARY QUADRATIC FORM : 62 QUARTAN PRIMES : 62 RAMANUJAN PRIMES : 62 REGULAR PRIMES : 62 REPUNIT PRIMES : 62 PRIMES IN RESIDUE CLASSES : 62 RIGHT -TRUNCATABLE PRIMES : 63 SAFE PRIMES : 63 SELF PRIMES IN BASE 10: 63 Page 7 of 330 SEXY PRIMES : 63 SMARANDACHE –WELLIN PRIMES : 63 SOLINAS PRIMES : 63 SOPHIE GERMAIN PRIMES : 63 STAR PRIMES : 63 STERN PRIMES : 64 SUPER -PRIMES : 64 SUPERSINGULAR PRIMES : 64 SWINGING PRIMES : 64 THABIT NUMBER PRIMES : 64 PRIME TRIPLETS : 64 TWIN PRIMES : 64 TWO -SIDED PRIMES : 64 ULAM NUMBER PRIMES : 64 UNIQUE PRIMES : 64 WAGSTAFF PRIMES : 64 WALL -SUN -SUN PRIMES : 65 WEAKLY PRIME NUMBERS 65 WIEFERICH PRIMES : 65 WIEFERICH PRIMES : BASE 3 (M IRIMANOFF PRIMES :) 65 WIEFERICH PRIMES : BASE 5 65 WIEFERICH PRIMES : BASE 6 65 WIEFERICH PRIMES : BASE 7 65 WIEFERICH PRIMES : BASE 10 65 WIEFERICH PRIMES : BASE 11 65 WIEFERICH PRIMES : BASE 12 65 WIEFERICH PRIMES : BASE 13 65 WIEFERICH PRIMES : BASE 17 65 WIEFERICH PRIMES : BASE 19 66 WILSON PRIMES : 66 WOLSTENHOLME PRIMES : 66 WOODALL PRIMES : 66 4.13 GENERALISATIONS FROM PRIME NUMBERS: 66 PERFECT NUMBERS : 66 LIST OF PERFECT NUMBERS : 66 AMICABLE NUMBERS : 67 LIST OF AMICABLE NUMBERS : 67 SOCIABLE NUMBERS : 68 LIST OF SOCIABLE NUMBERS : 68 4.14 GOLDEN RATIO & FIBONACCI SEQUENCE: 71 RELATIONSHIP : 71 INFINITE SERIES : 71 CONTINUED FRACTIONS : 71 TRIGONOMETRIC EXPRESSIONS : 72 FIBONACCI SEQUENCE : 72 4.15 FERMAT’S LAST THEOREM: 72 4.16 BOOLEAN ALGEBRA: 72 AXIOMS : 72 THEOREMS OF ONE VARIABLE : 73 THEOREMS OF SEVERAL VARIABLES : 73
PART 5: COUNTING TECHNIQUES & PROBABILITY 74
5.1 2D 74 TRIANGLE NUMBER 74 Page 8 of 330 SQUARE NUMBER 74 PENTAGONAL NUMBER 74 5.2 3D 74 TETRAHEDRAL NUMBER 74 SQUARE PYRAMID NUMBER 74 5.3 PERMUTATIONS 74 PERMUTATIONS : 74 PERMUTATIONS (WITH REPEATS ): 74 5.4 COMBINATIONS 74 ORDERED COMBINATIONS : 74 UNORDERED COMBINATIONS : 74 ORDERED REPEATED COMBINATIONS : 74 UNORDERED REPEATED COMBINATIONS : 74 GROUPING : 74 5.5 MISCELLANEOUS: 74 TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE : 74 NUMBER OF INTERPRETERS : 74 MAX NUMBER OF PIZZA PIECES : 74 MAX PIECES OF A CRESCENT : 74 MAX PIECES OF CHEESE : 74 CARDS IN A CARD HOUSE : 75 DIFFERENT ARRANGEMENT OF DOMINOS : 75 UNIT FRACTIONS : 75 ANGLE BETWEEN TWO HANDS OF A CLOCK : 75 WINNING LINES IN NOUGHTS AND CROSSES : 75 BAD RESTAURANT SPREAD : 75 FIBONACCI SEQUENCE : 75 ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) 75 5.6 FACTORIAL: 75 DEFINITION : 75 TABLE OF FACTORIALS : 75 APPROXIMATION : 76 5.7 THE DAY OF THE WEEK: 76 5.8 BASIC PROBABILITY: 76 AXIOM ’S OF PROBABILITY : 76 COMMUTATIVE LAWS : 76 ASSOCIATIVE LAWS : 76 DISTRIBUTIVE LAWS : 76 INDICATOR FUNCTION : 76 5.9 VENN DIAGRAMS: 76 COMPLEMENTARY EVENTS : 76 NULL SET : 76 TOTALITY : 76 CONDITIONAL PROBABILITY : 77 UNION : 77 INDEPENDENT EVENTS : 77 MUTUALLY EXCLUSIVE : 77 SUBSETS : 77 BAYE ’S THEOREM : 77 5.11 BASIC STATISTICAL OPERATIONS: 77 VARIANCE : 77 ARITHMETIC MEAN : 77 GEOMETRIC MEAN : 77 HARMONIC MEAN : 77 STANDARDIZED SCORE : 77 QUANTILE : 77 Page 9 of 330 5.12 DISCRETE RANDOM VARIABLES: 77 STANDARD DEVIATION : 77 EXPECTED VALUE : 77 VARIANCE : 78 PROBABILITY MASS FUNCTION : 78 CUMULATIVE DISTRIBUTION FUNCTION : 78 5.13 COMMON DRV S: 78 BERNOULLI TRIAL : 78 BINOMIAL TRIAL : 78 POISSON DISTRIBUTION : 78 GEOMETRIC BINOMIAL TRIAL : 79 NEGATIVE BINOMIAL TRIAL : 79 HYPERGEOMETRIC TRIAL : 79 5.14 CONTINUOUS RANDOM VARIABLES: 79 PROBABILITY DENSITY FUNCTION : 79 CUMULATIVE DISTRIBUTION FUNCTION : 79 INTERVAL PROBABILITY : 79 EXPECTED VALUE : 80 VARIANCE : 80 5.15 COMMON CRV S: 80 UNIFORM DISTRIBUTION : 80 EXPONENTIAL DISTRIBUTION : 80 NORMAL DISTRIBUTION : 81 5.16 BIVARIABLE DISCRETE: 81 PROBABILITY : 81 MARGINAL DISTRIBUTION : 82 EXPECTED VALUE : 82 INDEPENDENCE : 82 COVARIANCE : 82 5.17 BIVARIABLE CONTINUOUS: 82 CONDITIONS : 82 PROBABILITY : 82 MARGINAL DISTRIBUTION : 82 MEASURE : 83 EXPECTED VALUE : 83 INDEPENDENCE : 83 CONDITIONAL : 83 COVARIANCE : 83 CORRELATION COEFFICIENT : 83 BIVARIATE UNIFROM DISTRIBUTION : 83 MULTIVARIATE UNIFORM DISTRIBUTION : 83 BIVARIATE NORMAL DISTRIBUTION : 83 5.18 FUNCTIONS OF RANDOM VARIABLES: 84 SUMS (D ISCRETE ): 84 SUMS (C ONTINUOUS ): 84 QUOTIENTS (D ISCRETE ): 84 QUOTIENTS (C ONTINUOUS ): 84 MAXIMUM : 85 MINIMUM : 85 ORDER STATISTICS : 85 5.19 TRANSFORMATION OF THE JOINT DENSITY: 86 BIVARIATE FUNCTIONS : 86 MULTIVARIATE FUNCTIONS : 86 JACOBIAN : 86 JOINT DENSITY : 86 ABBREVIATIONS 86 Page 10 of 330 PART 6: STATISTICAL ANALYSIS 88
6.1 GENERAL PRINCIPLES: 88 MEAN SQUARE VALUE OF X : 88 F-STATISTIC OF X : 88 F-STATISTIC OF THE NULL HYPOTHESIS : 88 P-VALUE : 88 RELATIVE EFFICIENCY : 88 6.2 CONTINUOUS REPLICATE DESIGN (CRD): 88 TREATMENTS : 88 FACTORS : 88 REPLICATIONS PER TREATMENT : 88 TOTAL TREATMENTS : 88 MATHEMATICAL MODEL : 88 TEST FOR TREATMENT EFFECT : 89 ANOVA: 89 6.3 RANDOMISED BLOCK DESIGN (RBD): 89 TREATMENTS : 89 FACTORS : 89 REPLICATIONS PER TREATMENT : 89 TOTAL TREATMENTS : 89 MATHEMATICAL MODEL : 89 TEST FOR TREATMENT EFFECT : 90 TEST FOR BLOCK EFFECT : 90 RELATIVE EFFICIENCY : 90 ANOVA: 90 6.4 LATIN SQUARE DESIGN (LSD): 90 TREATMENTS : 90 FACTORS : 90 REPLICATIONS PER TREATMENT : 90 TOTAL TREATMENTS : 90 MATHEMATICAL MODEL : 90 TEST FOR TREATMENT EFFECT : 91 RELATIVE EFFICIENCY : 91 ANOVA: 91 6.5 ANALYSIS OF COVARIANCE: 91 MATHEMATICAL MODEL : 91 ASSUMPTIONS : 91 6.6 RESPONSE SURFACE METHODOLOGY: 92 DEFINITION : 92 1ST ORDER : 92 2ND ORDER : 92 COMMON DESIGNS 92 CRITERION FOR DETERMINING THE OPTIMATILITY OF A DESIGN : 92 6.7 FACTORIAL OF THE FORM 2N: 92 GENERAL DEFINITION : 92 CONTRASTS FOR A 22 DESIGN : 92 SUM OF SQUARES FOR A 22 DESIGN : 92 HYPOTHESIS FOR A CRD 22 DESIGN : 92 HYPOTHESIS FOR A RBD 22 DESIGN : 93 6.8 GENERAL FACTORIAL: 93 GENERAL DEFINITION : 93 ORDER : 93 DEGREES OF FREEDOM FOR MAIN EFFECTS : 93 DEGREES OF FREEDOM FOR HIGHER ORDER EFFECTS : 93 Page 11 of 330 6.9 ANOVA ASSUMPTIONS: 93 ASSUMPTIONS : 93 LEVENE ’S TEST : 94 6.10 CONTRASTS: 94 LINEAR CONTRAST : 94 ESTIMATED MEAN OF CONTRAST : 94 ESTIMATED VARIANCE OF CONTRAST : 94 F OF CONTRAST : 94 ORTHOGONAL CONTRASTS : 94 6.11 POST ANOVA MULTIPLE COMPARISONS: 94 BONDERRONI METHOD : 94 FISHER ’S LEAST SIGNIFICANT DIFFERENCE : 94 TUKEY ’S W PROCEDURE : 94 SCHEFFE ’S METHOD : 94
PART 7: PI 96
7.1 AREA: 96 CIRCLE : 96 CYCLIC QUADRILATERAL : 96 AREA OF A SECTOR (DEGREES ) 96 AREA OF A SECTOR (RADIANS ) 96 AREA OF A SEGMENT (DEGREES ) 96 AREA OF AN ANNULUS : 96 ELLIPSE : 96 7.2 VOLUME: 96 SPHERE : 96 CAP OF A SPHERE : 96 CONE : 96 ICE -CREAM & CONE : 96 DOUGHNUT : 96 SAUSAGE : 96 ELLIPSOID : 96 7.3 SURFACE AREA: 96 SPHERE : 96 HEMISPHERE : 96 DOUGHNUT : 96 SAUSAGE : 96 CONE : 96 7.4 MISELANIOUS: 97 LENGTH OF ARC (DEGREES ) 97 LENGTH OF CHORD (DEGREES ) 97 PERIMETER OF AN ELLIPSE 97 7.6 PI: 97 ARCHIMEDES ’ BOUNDS : 97 JOHN WALLIS : 97 ISAAC NEWTON : 97 JAMES GREGORY : 97 SCHULZ VON STRASSNITZKY : 97 JOHN MACHIN : 97 LEONARD EULER : 97 JOZEF HOENE -WRONSKI : 97 FRANCISCUS VIETA : 97 INTEGRALS : 98 INFINITE SERIES : 98 Page 12 of 330 CONTINUED FRACTIONS : 99 7.7 CIRCLE GEOMETRY: 99 RADIUS OF CIRCUMSCRIBED CIRCLE FOR RECTANGLES : 99 RADIUS OF CIRCUMSCRIBED CIRCLE FOR SQUARES : 99 RADIUS OF CIRCUMSCRIBED CIRCLE FOR TRIANGLES : 99 RADIUS OF CIRCUMSCRIBED CIRCLE FOR QUADRILATERALS : 99 RADIUS OF INSCRIBED CIRCLE FOR SQUARES : 99 RADIUS OF INSCRIBED CIRCLE FOR TRIANGLES : 99 RADIUS OF CIRCUMSCRIBED CIRCLE : 99 RADIUS OF INSCRIBED CIRCLE : 100 7.8 ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7): 100 7.9 CRESCENT GEOMETRY: 100 AREA OF A LUNAR CRESCENT : 100 AREA OF AN ECLIPSE CRESCENT : 100 7.10 ABBREVIATIONS (7.9): 101
PART 8: APPLIED FIELDS: 102
8.1 MOVEMENT: 102 STOPPING DISTANCE : 102 CENTRIPETAL ACCELERATION : 102 CENTRIPETAL FORCE : 102 DROPPING TIME : 102 FORCE : 102 KINETIC ENERGY : 102 MAXIMUM HEIGHT OF A CANNON : 102 PENDULUM SWING TIME : 102 POTENTIAL ENERGY : 102 RANGE OF A CANNON : 102 TIME IN FLIGHT OF A CANNON : 102 UNIVERSAL GRAVITATION : 102 ABBREVIATIONS (8.1): 102 8.2 CLASSICAL MECHANICS: 103 NEWTON ’S LAWS : 103 INERTIA : 103 MOMENTS OF INERTIA : 104 VELOCITY AND SPEED : 107 ACCELERATION : 107 TRAJECTORY (D ISPLACEMENT ): 107 KINETIC ENERGY : 108 CENTRIPETAL FORCE : 108 CIRCULAR MOTION : 108 ANGULAR MOMENTUM : 108 TORQUE : 109 WORK : 109 LAWS OF CONSERVATION : 109 ABBREVIATIONS (8.2) 109 8.3 RELATIVISTIC EQUATIONS: 109 KINETIC ENERGY : 109 MOMENTUM : 110 TIME DILATION : 110 LENGTH CONTRACTION : 110 RELATIVISTIC MASS : 110 8.4 ACCOUNTING: 110 PROFIT : 110 Page 13 of 330 PROFIT MARGIN : 110 SIMPLE INTEREST : 110 COMPOUND INTEREST : 110 CONTINUOUS INTEREST : 110 ABBREVIATIONS (8.4): 110 8.5 MACROECONOMICS: 110 GDP: 110 RGDP: 110 NGDP: 110 GROWTH : 110 NET EXPORTS : 110 WORKING AGE POPULATION : 110 LABOR FORCE : 111 UNEMPLOYMENT : 111 NATURAL UNEMPLOYMENT : 111 UNEMPLOYMENT RATE : 111 EMPLOYMENT RATE : 111 PARTICIPATION RATE : 111 CPI: 111 INFLATION RATE : 111 ABBREVIATIONS (8.5) 111
PART 9: TRIGONOMETRY 112
9.1 CONVERSIONS: 112 9.2 BASIC RULES: 112 SIN RULE : 112 COS RULE : 112 TAN RULE : 112 AUXILIARY ANGLE : 112 PYTHAGORAS THEOREM : 112 PERIODICY : 112 9.3 RECIPROCAL FUNCTIONS 113 9.4 BASIC IDENTITES: 113 9.5 IDENTITIES BETWEEN RELATIONSHIPS: 113 9.6 ADDITION FORMULAE: 114 9.7 DOUBLE ANGLE FORMULAE: 114 9.8 TRIPLE ANGLE FORMULAE: 115 9.9 HALF ANGLE FORMULAE: 115 9.10 POWER REDUCTION: 116 9.11 PRODUCT TO SUM: 117 9.12 SUM TO PRODUCT: 117 9.13 HYPERBOLIC EXPRESSIONS: 117 9.14 HYPERBOLIC RELATIONS: 118 9.15 MACHIN-LIKE FORMULAE: 118 FORM : 118 FORMULAE : 118 IDENTITIES : 119 9.16 SPHERICAL TRIANGLE IDENTITIES: 119 9.17 ABBREVIATIONS (9.1-9.16) 119
PART 10: EXPONENTIALS & LOGARITHIMS 121
10.1 FUNDAMENTAL THEORY: 121 Page 14 of 330 10.2 EXPONENTIAL IDENTITIES: 121 10.3 LOG IDENTITIES: 121 10.4 LAWS FOR LOG TABLES: 122 10.5 COMPLEX NUMBERS: 122 10.6 LIMITS INVOLVING LOGARITHMIC TERMS 122
PART 11: COMPLEX NUMBERS 123
11.1 GENERAL: 123 FUNDAMENTAL : 123 STANDARD FORM : 123 POLAR FORM : 123 ARGUMENT : 123 MODULUS : 123 CONJUGATE : 123 EXPONENTIAL : 123 DE MOIVRE ’S FORMULA : 123 EULER ’S IDENTITY : 123 11.2 OPERATIONS: 123 ADDITION : 123 SUBTRACTION : 123 MULTIPLICATION : 123 DIVISION : 123 SUM OF SQUARES : 123 11.3 IDENTITIES: 123 EXPONENTIAL : 123 LOGARITHMIC : 123 TRIGONOMETRIC : 123 HYPERBOLIC : 124
PART 12: DIFFERENTIATION 125
12.1 GENERAL RULES: 125 PLUS OR MINUS : 125 PRODUCT RULE : 125 QUOTIENT RULE : 125 POWER RULE : 125 CHAIN RULE : 125 BLOB RULE : 125 BASE A LOG : 125 NATURAL LOG : 125 EXPONENTIAL (X): 125 FIRST PRINCIPLES : 125 ANGLE OF INTERSECTION BETWEEN TWO CURVES : 126 12.2 EXPONETIAL FUNCTIONS: 126 12.3 LOGARITHMIC FUNCTIONS: 126 12.4 TRIGONOMETRIC FUNCTIONS: 126 12.5 HYPERBOLIC FUNCTIONS: 127 12.5 PARTIAL DIFFERENTIATION: 127 FIRST PRINCIPLES : 127 GRADIENT : 128 TOTAL DIFFERENTIAL : 128 CHAIN RULE : 128 IMPLICIT DIFFERENTIATION : 129 Page 15 of 330 HIGHER ORDER DERIVATIVES : 129
PART 13: INTEGRATION 130
13.1 GENERAL RULES: 130 POWER RULE : 130 BY PARTS : 130 CONSTANTS : 130 13.2 RATIONAL FUNCTIONS: 130 13.3 TRIGONOMETRIC FUNCTIONS (SINE): 131 13.4 TRIGONOMETRIC FUNCTIONS (COSINE): 132 13.5 TRIGONOMETRIC FUNCTIONS (TANGENT): 133 13.6 TRIGONOMETRIC FUNCTIONS (SECANT): 133 13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT): 134 13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE): 134 13.9 TRIGONOMETRIC FUNCTIONS (SINE & TANGENT): 136 13.10 TRIGONOMETRIC FUNCTIONS (COSINE & TANGENT): 136 13.11 TRIGONOMETRIC FUNCTIONS (SINE & COTANGENT): 136 13.12 TRIGONOMETRIC FUNCTIONS (COSINE & COTANGENT): 136 13.13 TRIGONOMETRIC FUNCTIONS (ARCSINE): 136 13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE): 137 13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT): 137 13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT): 137 13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT): 138 13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT): 138 13.19 EXPONETIAL FUNCTIONS 138 13.20 LOGARITHMIC FUNCTIONS 140 13.21 HYPERBOLIC FUNCTIONS 141 13.22 INVERSE HYPERBOLIC FUNCTIONS 143 13.23 ABSOLUTE VALUE FUNCTIONS 144 13.24 SUMMARY TABLE 144 13.25 SQUARE ROOT PROOFS 145 13.26 CARTESIAN APPLICATIONS 148 AREA UNDER THE CURVE : 148 VOLUME : 148 VOLUME ABOUT X AXIS : 148 VOLUME ABOUT Y AXIS : 149 SURFACE AREA ABOUT X AXIS : 149 LENGTH WRT X -ORDINATES : 149 LENGTH WRT Y -ORDINATES : 149 LENGTH PARAMETRICALLY : 149 LINE INTEGRAL OF A SCALAR FIELD : 149 LINE INTEGRAL OF A VECTOR FIELD : 149 AREA OF A SURFACE : 149 13.27 HIGHER ORDER INTEGRATION 149 PROPERTIES OF DOUBLE INTEGRALS : 150 VOLUME USING DOUBLE INTEGRALS : 150 VOLUME USING TRIPLE INTEGRALS : 151 CENTRE OF MASS : 153 13.28 WORKING IN DIFFERENT COORDINATE SYSTEMS: 153 CARTESIAN : 153 POLAR : 153 CYLINDRICAL : 153 SPHERICAL : 154 CARTESIAN TO POLAR : 154 Page 16 of 330 POLAR TO CARTESIAN : 154 CARTESIAN TO CYLINDRICAL : 154 CYLINDRICAL TO CARTESIAN : 154 SPHERICAL TO CARTESIAN : 154
PART 14: FUNCTIONS 155
14.1 ODD & EVEN FUNCTIONS: 155 DEFINITIONS : 155 COMPOSITE FUNCTIONS : 155 BASIC INTEGRATION : 155 14.2 MULTIVARIABLE FUNCTIONS: 155 LIMIT : 155 DISCRIMINANT : 155 CRITICAL POINTS : 155 14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS: 156 SEPARABLE 156 LINEAR 156 HOMOGENEOUS 156 EXACT 157 BERNOULLI FORM : 157 14.4 SECOND ORDE, FIRST DEGREE, DIFFERENTIAL EQUATIONS: 158 GREG ’S LEMMA : 158 HOMOGENEOUS 158 UNDETERMINED COEFFICIENTS 158 VARIATION OF PARAMETERS 159 EULER TYPE 160 REDUCTION OF ORDER 160 POWER SERIES SOLUTIONS : 161 14.5 ORDINARY DIFFERENTIAL EQUATIONS USING MATRICES: 163 DERIVATION OF METHODS : 163 FUNDAMENTAL MATRIX : 163 HOMOGENEOUS SOLUTION : 163 INHOMOGENEOUS SOLUTION : 164 NTH ORDER LINEAR , CONSTANT COEFFICIENT ODE: 164 14.6 APPLICATIONS OF FUNCTIONS 166 TERMINOLOGY : 166 GRADIENT VECTOR OF A SCALAR FIELD : 166 DIRECTIONAL DERIVATIVES : 166 OPTIMISING THE DIRECTIONAL DERIVATIVE : 166 14.7 ANALYTIC FUNCTIONS 166
PART 15: MATRICIES 167
15.1 BASIC PRINICPLES: 167 SIZE 167 15.2 BASIC OPERTAIONS: 167 ADDITION : 167 SUBTRACTION : 167 SCALAR MULTIPLE : 167 TRANSPOSE : 167 SCALAR PRODUCT : 167 SYMMETRY : 167 CRAMER ’S RULE : 167 Page 17 of 330 LEAST SQUARES SOLUTION 167 15.3 SQUARE MATRIX: 167 DIAGONAL : 168 LOWER TRIANGLE MATRIX : 168 UPPER TRIANGLE MATRIX : 168 15.4 DETERMINATE: 168 2X2 168 3X3 168 NXN 168 RULES 168 15.5 INVERSE 170 2X2: 170 3X3: 170 MINOR : 170 COFACTOR : 171 ADJOINT METHOD FOR INVERSE : 171 LEFT INVERSE : 171 RIGHT INVERSE : 171 PSEUDO INVERSE : 171 15.6 LINEAR TRANSFORMATION 171 AXIOMS FOR A LINEAR TRANSFORMATION : 172 TRANSITION MATRIX : 172 ZERO TRANSFORMATION : 172 IDENTITY TRANSFORMATION : 172 15.7 COMMON TRANSITION MATRICIES 172 ROTATION (C LOCKWISE ): 172 ROTATION (A NTICLOCKWISE ): 172 SCALING : 172 SHEARING (PARALLEL TO X -AXIS ): 172 SHEARING (PARALLEL TO Y -AXIS ): 172 15.8 EIGENVALUES AND EIGENVECTORS 172 DEFINITIONS : 172 EIGENVALUES : 172 EIGENVECTORS : 172 CHARACTERISTIC POLYNOMIAL : 172 ALGEBRAIC MULTIPLICITY : 173 GEOMETRIC MULTIPLICITY : 173 TRANSFORMATION : 173 LINEARLY INDEPENDENCE : 173 DIGITALIZATION : 173 CAYLEY -HAMILTON THEOREM : 173 ORTHONORMAL SET : 173 QR FACTORISATION : 173 15.9 JORDAN FORMS 174 GENERALISED DIAGONLISATION : 174 JORDAN BLOCK : 174 JORDAN FORM : 174 ALGEBRAIC MULTIPLICITY : 174 GEOMETRIC MULTIPLICITY : 174 GENERALISED CHAIN : 174 POWERS : 175 15.12 SINGULAR VALUE DECOMPOSITION 175 FUNDAMENTALLY : 175 SIZE : 175 PSEUDO INVERSE : 175 PROCEDURE : 175 Page 18 of 330 15.11 COMPLEX MATRICIS: 176 CONJUGATE TRANSPOSE : 176 HERMITIAN MATRIX : 176 SKEW -HERMITIAN : 176 UNITARY MATRIX : 177 NORMAL MATRIX : 177 DIAGONALISATION : 177 SPECTRAL THEOREM : 177 15.12 NUMERICAL COMPUTATIONS: 177 RAYLEIGH QUOTIENT : 177 POWER METHOD : 179 15.13 POWER SERIES: 179
PART 16: VECTORS 180
16.1 BASIC OPERATIONS: 180 ADDITION : 180 SUBTRACTION : 180 EQUALITY : 180 SCALAR MULTIPLICATION : 180 PARALLEL : 180 MAGNITUDE : 180 UNIT VECTOR : 180 ZERO VECTOR : 180 DOT PRODUCT : 180 ANGLE BETWEEN TWO VECTORS : 180 ANGLE OF A VECTOR IN 3D: 180 PERPENDICULAR TEST : 180 SCALAR PROJECTION : 181 VECTOR PROJECTION : 181 CROSS PRODUCT : 181 16.2 LINES 181 16.3 PLANES 181 GENERALLY : 181 TANGENT PLANE : 181 NORMAL LINE : 182 16.4 CLOSEST APPROACH 182 TWO POINTS : 182 POINT AND LINE : 182 POINT AND PLANE : 182 TWO SKEW LINES : 182 16.5 GEOMETRY 182 AREA OF A TRIANGLE : 182 AREA OF A PARALLELOGRAM : 182 AREA OF A PARALLELEPIPED : 182 16.6 SPACE CURVES 182 WHERE : 182 VELOCITY : 183 ACCELERATION : 183 DEFINITION OF “S”: 183 UNIT TANGENT : 183 CHAIN RULE : 183 NORMAL : 183 CURVATURE : 184 UNIT BINOMIAL : 184 Page 19 of 330 TORTION : 184 ABBREVIATIONS 184
PART 17: SERIES 185
17.1 MISCELLANEOUS 185 GENERAL FORM : 185 INFINITE FORM : 185 PARTIAL SUM OF A SERIES : 185 0.99…=1: 185 17.2 TEST FOR CONVERGENCE AND DIVERGENCE 185 TEST FOR CONVERGENCE : 185 TEST FOR DIVERGENCE : 185 GEOMETRIC SERIES 185 P SERIES 185 THE SANDWICH THEOREM 185 THE INTEGRAL TEST 185 THE DIRECT COMPARISON TEST 186 THE LIMIT COMPARISON TEST 186 D’ ALMBERT ’S RATIO COMPARISON TEST 186 THE N TH ROOT TEST 186 ABEL ’S TEST : 186 NEGATIVE TERMS 186 ALTERNATING SERIES TEST 186 ALTERNATING SERIES ERROR 187 17.3 ARITHMETIC PROGRESSION: 187 DEFINITION : 187 NTH TERM : 187 SUM OF THE FIRST N TERMS : 187 17.4 GEOMETRIC PROGRESSION: 187 DEFINITION : 187 NTH TERM : 187 SUM OF THE FIRST N TERMS : 187 SUM TO INFINITY : 187 GEOMETRIC MEAN : 187 17.5 SUMMATION SERIES 187 LINEAR : 187 QUADRATIC : 187 CUBIC : 187 17.6 APPROXIMATION SERIES 187 TAYLOR SERIES 187 MACLAURUN SERIES 188 LINEAR APPROXIMATION : 188 QUADRATIC APPROXIMATION : 188 CUBIC APPROXIMATION : 188 17.7 MONOTONE SERIES 188 STRICTLY INCREASING : 188 NON -DECREASING : 188 STRICTLY DECREASING : 188 NON -INCREASING : 188 CONVERGENCE : 188 17.8 RIEMANN ZETA FUNCTION 188 FORM : 188 EULER ’S TABLE : 188 ALTERNATING SERIES : 189 Page 20 of 330 PROOF FOR N =2: 189 17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS 190 17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS 190 17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS 191 17.12 INFINITE SUMMATIONS TO PI 193 17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS 193 ABBREVIATIONS 193 17.14 POWER SERIES EXPANSION 193 EXPONENTIAL : 193 TRIGONOMETRIC : 194 EXPONENTIAL AND LOGARITHM SERIES : 196 FOURIER SERIES : 197 17.15 BERNOULLI EXPANSION : 197 FUNDAMENTALLY : 197 EXPANSIONS : 198 LIST OF BERNOULLI NUMBERS : 198
PART 18: ELECTRICAL 200
18.1 FUNDAMENTAL THEORY 200 CHARGE : 200 CURRENT : 200 RESISTANCE : 200 OHM ’S LAW : 200 POWER : 200 CONSERVATION OF POWER : 200 ELECTRICAL ENERGY : 200 KIRCHOFF ’S VOLTAGE LAW : 200 KIRCHOFF ’S CURRENT LAW : 200 AVERAGE CURRENT : 200 RMS CURRENT : 200 TO Y CONVERSION : 200 18.2 COMPONENTS 201 RESISTANCE IN SERIES : 201 RESISTANCE IN PARALLEL : 201 INDUCTIVE IMPEDANCE : 201 CAPACITOR IMPEDANCE : 201 CAPACITANCE IN SERIES : 201 CAPACITANCE IN PARALLEL : 201 VOLTAGE , CURRENT & POWER SUMMARY : 201 18.3 THEVENIN’S THEOREM 201 THEVENIN ’S THEOREM : 201 MAXIMUM POWER TRANSFER THEOREM : 202 18.4 FIRST ORDER RC CIRCUIT 202 18.5 FIRST ORDER RL CIRCUIT 202 18.6 SECOND ORDER RLC SERIES CIRCUIT 202 CALCULATION USING KVL: 202 IMPORTANT VARIABLES 202 SOLVING : 203 MODE 1: 203 MODE 2: 203 MODE 3: 204 MODE 4: 204 CURRENT THROUGH INDUCTOR : 205 PLOTTING MODES : 205 Page 21 of 330 18.7 SECOND ORDER RLC PARALLEL CIRCUIT 206 CALCULATION USING KCL: 206 IMPORTANT VARIABLES 206 SOLVING : 207 18.8 LAPLANCE TRANSFORMATIONS 207 IDENTITIES : 207 PROPERTIES : 208 18.9 THREE PHASE – Y 209 LINE VOLTAGE : 209 PHASE VOLTAGE : 209 LINE CURRENT : 209 PHASE CURRENT : 209 POWER : 209 18.10 THREE PHASE – DELTA 209 LINE VOLTAGE : 209 PHASE VOLTAGE : 209 LINE CURRENT : 209 PHASE CURRENT : 209 POWER : 209 18.11 POWER 209 INSTANTANEOUS : 209 AVERAGE : 210 MAXIMUM POWER : 210 TOTAL POWER : 210 COMPLEX POWER : 210 18.12 ELECTROMAGNETICS 210 DEFINITIONS : 210 PERMEABILITY OF FREE SPACE : 210 MAGNETIC FIELD INTENSITY : 210 RELUCTANCE : 210 OHM ’S LAW : 210 MAGNETIC FORCE ON A CONDUCTOR : 210 ELECTROMAGNETIC INDUCTION : 210 MAGNETIC FLUX : 210 ELECTRIC FIELD : 210 MAGNETIC FORCE ON A PARTICLE : 210
PART 19: GRAPH THEORY 211
19.1 FUNDAMENTAL EXPLANATIONS: 211 LIST OF VERTICES : 211 LIST OF EDGES : 211 SUBGAPHS : 211 TREE : 211 DEGREE OF VERTEX : 211 DISTANCE : 211 DIAMETER : 211 TOTAL EDGES IN A SIMPLE BIPARTITE GRAPH : 211 TOTAL EDGES IN K-REGULAR GRAPH : 211 19.2 FACTORISATION: 211 1 FACTORISATION : 211
1 FACTORS OF A K ,nn BIPARTITE GRAPH : 211
1 FACTORS OF A K 2n GRAPH : 211 19.3 VERTEX COLOURING: 211 Page 22 of 330 CHROMATIC NUMBER : 212 UNION /I NTERSECTION : 212 EDGE CONTRACTION : 212 COMMON CHROMATIC POLYNOMIALS : 212 19.4 EDGE COLOURING: 212 COMMON CHROMATIC POLYNOMIALS : 212
PART 98: LIST OF DISTRIBUTION FUNCTIONS 213
5.18 FINITE DISCRETE DISTRIBUTIONS 213 BERNOULLI DISTRIBUTION 213 RADEMACHER DISTRIBUTION 213 BINOMIAL DISTRIBUTION 214 BETA -BINOMIAL DISTRIBUTION 215 DEGENERATE DISTRIBUTION 216 DISCRETE UNIFORM DISTRIBUTION 217 HYPERGEOMETRIC DISTRIBUTION 219 POISSON BINOMIAL DISTRIBUTION 220 FISHER 'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (U NIVARIATE ) 220 FISHER 'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (M ULTIVARIATE ) 221 WALLENIUS ' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (U NIVARIATE ) 221 WALLENIUS ' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (M ULTIVARIATE ) 222 5.19 INFINITE DISCRETE DISTRIBUTIONS 222 BETA NEGATIVE BINOMIAL DISTRIBUTION 222 MAXWELL –BOLTZMANN DISTRIBUTION 223 GEOMETRIC DISTRIBUTION 224 LOGARITHMIC (S ERIES ) DISTRIBUTION 226 NEGATIVE BINOMIAL DISTRIBUTION 227 POISSON DISTRIBUTION 228 CONWAY –MAXWELL –POISSON DISTRIBUTION 229 SKELLAM DISTRIBUTION 230 YULE –SIMON DISTRIBUTION 230 ZETA DISTRIBUTION 232 ZIPF 'S LAW 233 ZIPF –MANDELBROT LAW 234 5.20 BOUNDED INFINITE DISTRIBUTIONS 234 ARCSINE DISTRIBUTION 234 BETA DISTRIBUTION 236 LOGITNORMAL DISTRIBUTION 238 CONTINUOUS UNIFORM DISTRIBUTION 239 IRWIN -HALL DISTRIBUTION 240 KUMARASWAMY DISTRIBUTION 241 RAISED COSINE DISTRIBUTION 242 TRIANGULAR DISTRIBUTION 243 TRUNCATED NORMAL DISTRIBUTION 245 U-QUADRATIC DISTRIBUTION 246 VON MISES DISTRIBUTION 247 WIGNER SEMICIRCLE DISTRIBUTION 248 5.21 SEMI-BOUNDED CUMULATIVE DISTRIBUTIONS 250 BETA PRIME DISTRIBUTION 250 CHI DISTRIBUTION 251 NONCENTRAL CHI DISTRIBUTION 252 CHI -SQUARED DISTRIBUTION 252 INVERSE -CHI -SQUARED DISTRIBUTION 253 NONCENTRAL CHI -SQUARED DISTRIBUTION 255 Page 23 of 330 SCALED -INVERSE -CHI -SQUARED DISTRIBUTION 256 DAGUM DISTRIBUTION 257 EXPONENTIAL DISTRIBUTION 258 FISHER 'S Z-DISTRIBUTION 261 FOLDED NORMAL DISTRIBUTION 261 FRÉCHET DISTRIBUTION 262 GAMMA DISTRIBUTION 263 ERLANG DISTRIBUTION 264 INVERSE -GAMMA DISTRIBUTION 265 INVERSE GAUSSIAN /W ALD DISTRIBUTION 266 LÉVY DISTRIBUTION 267 LOG -CAUCHY DISTRIBUTION 269 LOG -LOGISTIC DISTRIBUTION 270 LOG -NORMAL DISTRIBUTION 271 MITTAG –LEFFLER DISTRIBUTION 272 PARETO DISTRIBUTION 273 RAYLEIGH DISTRIBUTION 274 RICE DISTRIBUTION 275 TYPE -2 GUMBEL DISTRIBUTION 276 WEIBULL DISTRIBUTION 277 5.22 UNBOUNDED CUMULATIVE DISTRIBUTIONS 278 CAUCHY DISTRIBUTION 278 EXPONENTIALLY MODIFIED GAUSSIAN DISTRIBUTION 279 FISHER –TIPPETT / GENERALIZED EXTREME VALUE DISTRIBUTION 281 GUMBEL DISTRIBUTION 282 FISHER 'S Z-DISTRIBUTION 283 GENERALIZED NORMAL DISTRIBUTION 283 GEOMETRIC STABLE DISTRIBUTION 285 HOLTSMARK DISTRIBUTION 285 HYPERBOLIC DISTRIBUTION 286 HYPERBOLIC SECANT DISTRIBUTION 287 LAPLACE DISTRIBUTION 288 LÉVY SKEW ALPHA -STABLE DISTRIBUTION 289 LINNIK DISTRIBUTION 291 LOGISTIC DISTRIBUTION 291 NORMAL DISTRIBUTION 293 NORMAL -EXPONENTIAL -GAMMA DISTRIBUTION 294 SKEW NORMAL DISTRIBUTION 294 STUDENT 'S T-DISTRIBUTION 295 NONCENTRAL T-DISTRIBUTION 297 VOIGT DISTRIBUTION 297 GENERALIZED PARETO DISTRIBUTION 298 TUKEY LAMBDA DISTRIBUTION 299 5.23 JOINT DISTRIBUTIONS 299 DIRICHLET DISTRIBUTION 299 BALDING –NICHOLS MODEL 300 MULTINOMIAL DISTRIBUTION 301 MULTIVARIATE NORMAL DISTRIBUTION 301 NEGATIVE MULTINOMIAL DISTRIBUTION 302 WISHART DISTRIBUTION 303 INVERSE -WISHART DISTRIBUTION 303 MATRIX NORMAL DISTRIBUTION 303 MATRIX T-DISTRIBUTION 304 5.24 OTHER DISTRIBUTIONS 304 CATEGORICAL DISTRIBUTION 304 CANTOR DISTRIBUTION 305 Page 24 of 330 PHASE -TYPE DISTRIBUTION 306 TRUNCATED DISTRIBUTION 306
PART 99: CONVERSIONS 308
99.1 LENGTH: 308 99.2 AREA: 310 99.3 VOLUME: 311 99.4 PLANE ANGLE: 314 99.5 SOLID ANGLE: 315 99.6 MASS: 315 99.7 DENSITY: 317 99.8 TIME: 317 99.9 FREQUENCY: 319 99.10 SPEED OR VELOCITY: 319 99.11 FLOW (VOLUME): 320 99.12 ACCELERATION: 320 99.13 FORCE: 321 99.14 PRESSURE OR MECHANICAL STRESS: 321 99.15 TORQUE OR MOMENT OF FORCE: 322 99.16 ENERGY, WORK, OR AMOUNT OF HEAT: 322 99.17 POWER OR HEAT FLOW RATE: 324 99.18 ACTION: 325 99.19 DYNAMIC VISCOSITY: 325 99.20 KINEMATIC VISCOSITY: 326 99.21 ELECTRIC CURRENT: 326 99.22 ELECTRIC CHARGE: 326 99.23 ELECTRIC DIPOLE: 327 99.24 ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE: 327 99.25 ELECTRICAL RESISTANCE: 327 99.26 CAPACITANCE: 327 99.27 MAGNETIC FLUX: 327 99.28 MAGNETIC FLUX DENSITY: 328 99.29 INDUCTANCE: 328 99.30 TEMPERATURE: 328 99.31 INFORMATION ENTROPY: 328 99.32 LUMINOUS INTENSITY: 329 99.33 LUMINANCE: 329 99.34 LUMINOUS FLUX: 329 99.35 ILLUMINANCE: 329 99.36 RADIATION - SOURCE ACTIVITY: 329 99.37 RADIATION – EXPOSURE: 330 99.38 RADIATION - ABSORBED DOSE: 330 99.39 RADIATION - EQUIVALENT DOSE: 330
Page 25 of 330 PART 1: PHYSICAL CONSTANTS
1.1 SI PREFIXES:
m n Prefix Symbol 1000 10 Decimal Scale 8 24 yotta Y 1000 10 1000000000000000000000000 Septillion 7 21 zetta Z 1000 10 1000000000000000000000 Sextillion 6 18 exa E 1000 10 1000000000000000000 Quintillion 5 15 peta P 1000 10 1000000000000000 Quadrillion 4 12 tera T 1000 10 1000000000000 Trillion 3 9 giga G 1000 10 1000000000 Billion 2 6 mega M 1000 10 1000000 Million 1 3 kilo k 1000 10 1000 Thousand 2⁄ 2 hecto h 1000 3 10 100 Hundred 1⁄ 1 deca da 1000 3 10 10 Ten 1000 0 10 0 1 One −1 ⁄ −1 deci d 1000 3 10 0.1 Tenth −2 ⁄ −2 centi c 1000 3 10 0.01 Hundredth −1 −3 milli m 1000 10 0.001 Thousandth −2 −6 micro µ 1000 10 0.000001 Millionth −3 −9 nano n 1000 10 0.000000001 Billionth −4 −12 pico p 1000 10 0.000000000001 Trillionth −5 −15 femto f 1000 10 0.000000000000001 Quadrillionth −6 −18 atto a 1000 10 0.000000000000000001 Quintillionth −7 −21 zepto z 1000 10 0.000000000000000000001 Sextillionth −8 −24 yocto y 1000 10 0.000000000000000000000001 Septillionth
1.2 SI BASE UNITS:
Quantity Unit Symbol length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd
Page 26 of 330 1.3 SI DERIVED UNITS:
Expression in terms of other SI Quantity Unit Symbol units angle, plane radian* rad m/m = 1 angle, solid steradian* sr m2/m 2 = 1 degree K Celsius temperature Celsius °C electric capacitance farad F C/V electric charge, quantity A·s of electricity coulomb C electric conductance siemens S A/V electric inductance henry H Wb/A electric potential difference, electromotive W/A force volt V electric resistance ohm V/A energy, work, quantity of N·m heat joule J force newton N kg·m/s 2 frequency (of a periodic 1/s phenomenon) hertz Hz illuminance lux lx lm/m 2 luminous flux lumen lm cd·sr magnetic flux weber Wb V·s magnetic flux density tesla T Wb/m 2 power, radiant flux watt W J/s pressure, stress pascal Pa N/m 2 activity (referred to a 1/s radionuclide) becquerel Bq
absorbed dose, specific J/kg energy imparted, kerma gray Gy dose equivalent, ambient dose equivalent, directional dose J/kg equivalent, personal dose equivalent, organ dose equivalent sievert Sv catalytic activity katal kat mol/s
Page 27 of 330 1.4 UNIVERSAL CONSTANTS:
Relative Standard Quantity Symbol Value Uncertainty speed of light in 299 792 458 m·s −1 defined vacuum Newtonian constant of 6.67428(67)×10 −11 m3·kg −1 ·s −2 1.0 × 10 −4 gravitation −34 −8 Planck constant 6.626 068 96(33) × 10 J·s 5.0 × 10 reduced Planck 1.054 571 628(53) × 10 −34 J·s 5.0 × 10 −8 constant
1.5 ELECTROMAGNETIC CONSTANTS:
Relative Standard Quantity Symbol Value (SI units) Uncertainty magnetic constant 4π × 10 −7 N·A −2 = 1.256 defined (vacuum permeability) 637 061... × 10 −6 N·A −2 electric constant (vacuum 8.854 187 817... × 10 −12 defined permittivity) F·m −1 characteristic impedance 376.730 313 461... defined of vacuum 8.987 551 787... × 10 9 Coulomb's constant defined N·m²·C −2 1.602 176 487(40) × 10 −19 elementary charge 2.5 × 10 −8 C 927.400 915(23) × 10 −26 Bohr magneton 2.5 × 10 −8 J·T −1 7.748 091 7004(53) × 10 −5 conductance quantum 6.8 × 10 −10 S inverse conductance 12 906.403 7787(88) 6.8 × 10 −10 quantum 4.835 978 91(12) × 10 14 Josephson constant 2.5 × 10 −8 Hz·V −1 2.067 833 667(52) × 10 −15 magnetic flux quantum 2.5 × 10 −8 Wb 5.050 783 43(43) × 10 −27 nuclear magneton 8.6 × 10 −8 J·T −1 −10 von Klitzing constant 25 812.807 557(18) 6.8 × 10
1.6 ATOMIC AND NUCLEAR CONSTANTS:
Page 28 of 330 Relative Standard Quantity Symbol Value (SI units) Uncertainty 5.291 772 −11 −9 Bohr radius 108(18) × 10 3.3 × 10 m 2.817 940 classical 2894(58) × 10 −15 2.1 × 10 −9 electron radius m 9.109 3 82 15(45) electron mass 5.0 × 10 −8 × 10 −31 kg Fermi coupling 1.166 39(1) × 8.6 × 10 −6 constant 10 −5 GeV −2 fine-structure 7.297 352 537 6.8 × 10 −10 constant 6(50) × 10 −3 4.359 744 17(75) Hartree energy 1.7 × 10 −7 × 10 −18 J 1.672 621 −27 −8 proton mass 637(83) × 10 5.0 × 10 kg 3.636 947 quantum of 550(24) × 10 −4 6.7 × 10 −9 circulation m² s −1 Rydberg 10 973 731.568 6.6 × 10 −12 constant 525(73) m −1 Thomson cross 6.652 458 73(13) 2.0 × 10 −8 section × 10 −29 m² weak mixing 0.222 15(76) 3.4 × 10 −3 angle
1.7 PHYSICO-CHEMICAL CONSTANTS:
Relative Quantity Symbol Value (SI units) Standard Uncertainty atomic mass unit 1.660 538 86(28) × (unified atomic mass 1.7 × 10 −7 10 −27 kg unit) 6.022 141 5(10) × Avogadro's number 1.7 × 10 −7 10 23 mol −1 1.380 6504(24) × Boltzmann constant 1.8 × 10 −6 10 −23 J·K −1 96 Faraday constant 8.6 × 10 −8 485.3383(83)C·mol −1 first 3.741 771 18(19) × 5.0 × 10 −8 radiation 10 −16 W·m²
Page 29 of 330 constant for 1.191 042 82(20) × spectral 1.7 × 10 −7 10 −16 W·m² sr −1 radiance at T=273.15 Loschmidt 2.686 777 3(47) × K and 1.8 × 10 −6 constant 10 25 m −3 p=101.325 kPa 8.314 472(15) gas constant 1.7 × 10 −6 J·K −1 ·mol −1 3.990 312 716(27) × molar Planck constant 6.7 × 10 −9 10 −10 J·s·mol −1 at T=273.15 2.2710 981(40) × 1.7 × 10 −6 K and 10 −2 m³·mol −1 molar p=100 kPa volume of an ideal at T=273.15 gas 2.2413 996(39) × K and 1.7 × 10 −6 10 −2 m³·mol −1 p=101.325 kPa at T=1 K and p=100 −1.151 704 7(44) 3.8 × 10 −6 Sackur- kPa Tetrode at T=1 K constant and −1.164 867 7(44) 3.8 × 10 −6 p=101.325 kPa second radiation 1.438 775 2(25) × 1.7 × 10 −6 constant 10 −2 m·K Stefan–Boltzmann 5.670 400(40) × 10 −8 7.0 × 10 −6 constant W·m −2 ·K −4 Wien displacement 2.897 768 5(51) × 1.7 × 10 −6 law constant 4.965 114 231... 10 −3 m·K
1.8 ADOPTED VALUES:
Value (SI Relative Standard Quantity Symbol units) Uncertainty conventional value of 4.835 979 × defined Josephson constant 10 14 Hz·V −1 conventional value of von 25 812.807 defined Klitzing constant 1 × 10 −3 constant −1 defined molar mass kg·mol −2 of carbon-12 1.2 × 10 defined Page 30 of 330 kg·mol −1 standard acceleration of 9.806 65 gravity (gee, free-fall on defined m·s −2 Earth) standard atmosphere 101 325 Pa defined
1.9 NATURAL UNITS:
Name Dimension Expression Value (SI units)
Planck length Length (L) 1.616 252(81) × 10 −35 m
Planck mass Mass (M) 2.176 44(11) × 10 −8 kg
Planck time Time (T) 5.391 24(27) × 10 −44 s
−18 Planck charge Electric charge (Q) 1.875 545 870(47) × 10 C
Planck temperature Temperature ( Θ) 1.416 785(71) × 10 32 K
1.10 MATHEMATICAL CONSTANTS:
(each to 1000 decimal places)
π ≈ 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482 5342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559 6446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104 5432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001 1330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074 4623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907 0217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778 9609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034 4181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083 0264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349 0428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199 e ≈ 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217 8525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233 8298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537 4234544243710753907774499206955170276183860626133138458300075204493382656029760673711320070 9328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770 7854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680 3318252886939849646510582093923982948879332036250944311730123819706841614039701983767932068 3282376464804295311802328782509819455815301756717361332069811250996181881593041690351598888 5193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270 4197862320900216099023530436994184914631409343173814364054625315209618369088870701676839642 4378140592714563549061303107208510383750510115747704171898610687396965521267154688957035035
φ ≈
Page 31 of 330 1.61803398874989484820458683436563811772030917980576286213544862270526046281890244970720720 4189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638 3226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203 4080588795445474924618569536486444924104432077134494704956584678850987433944221254487706647 8091588460749988712400765217057517978834166256249407589069704000281210427621771117778053153 1714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782 9778347845878228911097625003026961561700250464338243776486102838312683303724292675263116533 9247316711121158818638513316203840052221657912866752946549068113171599343235973494985090409 4762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561 9432082750513121815628551222480939471234145170223735805772786160086883829523045926478780178 8992199027077690389532196819861514378031499741106926088674296226757560523172777520353613936
Page 32 of 330
PART 2: MATHEMTAICAL SYMBOLS
2.1 BASIC MATH SYMBOLS
Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+3 ≠ not equal sign inequality 5 ≠ 4 > strict inequality greater than 5 > 4 < strict inequality less than 4 < 5 ≥ inequality greater than or equal to 5 ≥ 4 ≤ inequality less than or equal to 4 ≤ 5 ( ) parentheses calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 + plus sign addition 1 + 1 = 2 − minus sign subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 ∓ minus - plus both minus and plus operations 3 ∓ 5 = -2 and 8 * asterisk multiplication 2 * 3 = 6 × times sign multiplication 2 × 3 = 6 multiplication dot multiplication 2 3 = 6 ÷ division sign / obelus division 6 ÷ 2 = 3 / division slash division 6 / 2 = 3
– horizontal line division / fraction
mod modulo remainder calculation 7 mod 2 = 1 . period decimal point, decimal separator 2.56 = 2+56/100 a b power exponent 23 = 8 a^b caret exponent 2 ^ 3 = 8 √a square root √a · √a = a √9 = ±3 3√a cube root 3√8 = 2 4√a forth root 4√16 = ±2 n√a n-th root (radical) for n=3, n√8 = 2 % percent 1% = 1/100 10% × 30 = 3 ‰ per-mille 1‰ = 1/1000 = 0.1% 10‰ × 30 = 0.3 ppm per-million 1ppm = 1/1000000 10ppm × 30 = 0.0003 ppb per-billion 1ppb = 1/1000000000 10ppb × 30 = 3×10 -7 ppt per-trillion 1ppb = 10 -12 10ppb × 30 = 3×10 -10
2.2 GEOMETRY SYMBOLS
Symbol Symbol Name Meaning / definition Example
∠ angle formed by two rays ∠ABC = 30º
∡ measured angle ∡ABC = 30º ∢ spherical angle ∢AOB = 30º ∟ right angle = 90º α = 90º º degree 1 turn = 360º α = 60º ´ arcminute 1º = 60´ α = 60º59' Page 33 of 330 ´´ arcsecond 1´ = 60´´ α = 60º59'59'' AB line 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 equivalence of geometric shapes and congruent to ≅ size ∆ABC ≅ ∆XYZ ~ similarity same shapes, not same size ∆ABC ~ ∆XYZ triangle triangle shape ABC ≅ BCD | x-y | distance distance between points x and y | x-y | = 5 π = 3.141592654... π pi constant is the ratio between the circumference c = π·d = 2· π·r and diameter of a circle rad radians radians angle unit 360º = 2 π rad grad grads grads angle unit 360º = 400 grad
2.3 ALGEBRA SYMBOLS
Symbol Symbol Name Meaning / definition Example x x variable unknown value to find when 2 x = 4, then x = 2 ≡ equivalence identical to ≜ equal by definition equal by definition := equal by definition equal by definition ~ approximately equal weak approximation 11 ~ 10 ≈ approximately equal approximation sin (0.01) ≈ 0.01
∝ proportional to proportional to f(x) ∝ g(x) ∞ lemniscate infinity symbol ≪ much less than much less than 1 ≪ 1000000 ≫ much greater than much greater than 1000000 ≫ 1 ( ) parentheses calculate expression inside first 2 * (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 { } braces set ⌊x⌋ floor brackets rounds number to lower integer ⌊4.3 ⌋= 4 ⌈x⌉ ceiling brackets rounds number to upper integer ⌈4.3 ⌉= 5 x! exclamation mark factorial 4! = 1*2*3*4 = 24 | x | single vertical bar absolute value | -5 | = 5 f (x) function of x maps values of x to f(x) f (x) = 3 x+5
(f ◦g) function composition (f ◦g) (x) = f (g(x)) f (x)=3 x, g(x)= x-1 ⇒(f ◦g)( x)=3( x-1)
(a,b) open interval (a,b) ≜ {x | a < x < b} x ∈ (2,6) [a,b] closed interval [a,b] ≜ {x | a ≤ x ≤ b} x ∈ [2,6]
∆ delta change / difference ∆t = t1 - t 0 ∆ discriminant = b2 - 4 ac summation - sum of all values in range of ∑ sigma ∑ x = x +x +...+x series i 1 2 n
Page 34 of 330 ∑∑ sigma double summation
product - product of all values in range of ∏ capital pi ∏ x =x x ... x series i 1 2 n e constant / Euler's e e = 2.718281828... e = lim (1+1/ x)x , x→∞ number Euler-Mascheroni γ γ = 0.527721566... constant φ golden ratio golden ratio constant
2.4 LINEAR ALGEBRA SYMBOLS
Symbol Symbol Name Meaning / definition Example dot scalar product a b × cross vector product a × b A⊗B tensor product tensor product of A and B A ⊗ B inner product [ ] brackets matrix of numbers ( ) parentheses matrix of numbers | A | determinant determinant of matrix A det( A) determinant determinant of matrix A || x || double vertical bars norm
T T A transpose matrix transpose (A )ij = ( A)ji
† A Hermitian matrix matrix conjugate transpose (A†)ij = ( A)ji
* A Hermitian matrix matrix conjugate transpose (A*)ij = ( A)ji A -1 inverse matrix A A -1 = I rank( A) matrix rank rank of matrix A rank( A) = 3
dim( U) dimension dimension of matrix A rank( U) = 3
2.5 PROBABILITY AND STATISTICS SYMBOLS
Symbol Symbol Name Meaning / definition Example P(A) probability function probability of event A P(A) = 0.5 probability of events P(A ∩ B) probability that of events A and B P(A∩B) = 0.5 intersection probability of events probability that of events A or B P(A ∪ B) union P(A∪B) = 0.5 conditional probability probability of event A given event B P(A | B) P(A | B ) = 0.3 function occured probability density f (x) P(a ≤ x ≤ b ) = ∫ f (x) dx function (pdf) cumulative distribution F(x) F(x) = P(X ≤ x ) function (cdf) population mean mean of population values = 10 E(X) expectation value expected value of random variable X E(X) = 10 expected value of random variable X E(X | Y ) conditional expectation E(X | Y=2 ) = 5 given Y Page 35 of 330 var (X) variance variance of random variable X var (X) = 4 σ2 variance variance of population values σ2 = 4 standard deviation of random variable std (X) standard deviation std (X) = 2 X standard deviation value of random σ standard deviation σ = 2 X variable X X median middle value of random variable x covariance of random variables X and cov (X,Y) covariance cov (X,Y ) = 4 Y correlation of random variables X and corr (X,Y) correlation corr (X,Y ) = 3 Y correlation of random variables X and ρ correlation ρ = 3 X,Y Y X,Y summation - sum of all values in range ∑ summation of series
∑∑ double summation double summation
value that occurs most frequently in Mo mode population
MR mid-range MR = ( xmax +xmin )/2 Md sample median half the population is below this value
Q1 lower / first quartile 25% of population are below this value 50% of population are below this value Q median / second quartile 2 = median of samples
Q3 upper / third quartile 75% of population are below this value x sample mean average / arithmetic mean x = (2+5+9) / 3 = 5.333 2 2 s sample variance population samples variance estimator s = 4 sample standard population samples standard deviation s s = 2 deviation estimator
zx standard score zx = ( x-x) / s x X ~ distribution of X distribution of random variable X X ~ N (0,3) N( ,σ2) normal distribution gaussian distribution X ~ N (0,3) U(a,b) uniform distribution equal probability in range a,b X ~ U (0,3) exp (λ) exponential distribution f (x) = λe-λx , x≥0 gamma (c, λ) gamma distribution f (x) = λ c x c-1e-λx / Γ(c), x≥0
χ 2 (k) chi-square distribution f (x) = x k/2-1e-x/2 / ( 2 k/2 Γ(k/2) )
F (k1, k 2) F distribution k n-k Bin (n,p) binomial distribution f (k) = nCk p (1 -p)
Poisson (λ) Poisson distribution f (k) = λke-λ / k!
Geom (p) geometric distribution f (k) = p (1-p) k hyper-geometric HG (N,K,n) distribution Bern (p) Bernoulli distribution
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2.6 COMBINATORICS SYMBOLS
Symbol Symbol Name Meaning / definition Example n! factorial n! = 1·2·3·...· n 5! = 1·2·3·4·5 = 120
nPk permutation 5P3 = 5! / (5-3)! = 60
nCk
combination 5C3 = 5!/[3!(5-3)!]=10
2.7 SET THEORY SYMBOLS
Symbol Symbol Name Meaning / definition Example { } set a collection of elements A={3,7,9,14}, B={9,14,28} objects that belong to set A and set intersection A ∩ B B A ∩ B = {9,14} objects that belong to set A or set union A ∪ B B A ∪ B = {3,7,9,14,28} subset has less elements or equal to subset A ⊆ B the set {9,14,28} ⊆ {9,14,28} proper subset / strict subset has less elements than the A ⊂ B subset set {9,14} ⊂ {9,14,28} A ⊄ B not subset left set not a subset of right set {9,66} ⊄ {9,14,28} set A has more elements or equal superset A ⊇ B to the set B {9,14,28} ⊇ {9,14,28} proper superset / strict set A has more elements than set B A ⊃ B superset {9,14,28} ⊃ {9,14} A ⊅ B not superset set A is not a superset of set B {9,14,28} ⊅ {9,66} 2A power set all subsets of A Ƅ (A) power set all subsets of A A = B equality both sets have the same members A={3,9,14}, B={3,9,14}, A=B all the objects that do not belong to Ac complement set A objects that belong to A and not to A \ B relative complement A={3,9,14}, B={1,2,3}, A-B={9,14} B objects that belong to A and not to A - B relative complement A={3,9,14}, B={1,2,3}, A-B={9,14} B objects that belong to A or B but A ∆ B symmetric difference A={3,9,14}, B={1,2,3}, A ∆ B={1,2,9,14} not to their intersection objects that belong to A or B but symmetric difference A ⊖ B not to their intersection A={3,9,14}, B={1,2,3}, A ⊖ B={1,2 ,9,14} a∈A element of set membership A={3,9,14}, 3 ∈ A x∉A not element of no set membership A={3,9,14}, 1 ∉ A (a,b) ordered pair collection of 2 elements set of all ordered pairs from A and A×B cartesian product B |A| cardinality the number of elements of set A A={3,9,14}, |A|=3 #A cardinality the number of elements of set A A={3,9,14}, #A=3 Page 37 of 330 aleph infinite cardinality א Ø empty set Ø = { } C = {Ø} U universal set set of all possible values natural numbers set (with ℕ ∈ ℕ ℕ0 zero) 0 = {0,1,2,3,4,...} 0 0 natural numbers set ℕ ∈ ℕ ℕ1 (without zero) 1 = {1,2,3,4,5,...} 6 1 ℤ integer numbers set ℤ = {...-3,-2,-1,0,1,2,3,...} -6 ∈ ℤ ℚ rational numbers set ℚ = { x | x =a/b, a,b∈ℕ} 2/6 ∈ ℚ ℝ real numbers set ℝ = { x | -∞ < x <∞} 6.343434 ∈ ℝ ℂ = { z | z=a +bi , -∞ 2.8 LOGIC SYMBOLS Symbol Symbol Name Meaning / definition Example · and and x · y ^ caret / circumflex and x ^ y & ampersand and x & y + plus or x + y ∨ reversed caret or x ∨ y | vertical line or x | y x' single quote not - negation x' x bar not - negation x ¬ not not - negation ¬ x ! exclamation mark not - negation ! x ⊕ circled plus / oplus exclusive or - xor x ⊕ y ~ tilde negation ~ x ⇒ implies ⇔ equivalent if and only if ∀ for all ∃ there exists ∄ there does not exists ∴ therefore ∵ because / since Page 38 of 330 2.9 CALCULUS & ANALYSIS SYMBOLS Symbol Symbol Name Meaning / definition Example limit limit value of a function represents a very small number, near ε epsilon zero ε → 0 e e constant / Euler's number e = 2.718281828... e = lim (1+1/ x)x , x→∞ y ' derivative derivative - Leibniz's notation (3 x3)' = 9x2 y '' second derivative derivative of derivative (3 x3)'' = 18 x y(n) nth derivative n times derivation (3 x3)(3) = 18 derivative derivative - Lagrange's notation d(3 x3)/ dx = 9 x2 second derivative derivative of derivative d2(3 x3)/ dx 2 = 18 x nth derivative n times derivation time derivative derivative by time - Newton notation time second derivative derivative of derivative partial derivative ∂(x2+y2)/ ∂x = 2 x ∫ integral opposite to derivation ∬ double integral integration of function of 2 variables ∭ triple integral integration of function of 3 variables ∮ closed contour / line integral ∯ closed surface integral ∰ closed volume integral [a,b] closed interval [a,b] = { x | a ≤ x ≤ b } (a,b) open interval (a,b) = { x | a < x < b } i imaginary unit i ≡ √-1 z = 3 + 2 i z* complex conjugate z = a +bi → z*= a-bi z* = 3 + 2 i z complex conjugate z = a +bi → z = a -bi z = 3 + 2 i ∇ nabla / del gradient / divergence operator ∇f (x,y,z) vector unit vector x * y convolution y(t) = x(t) * h(t) ℒ Laplace transform F(s) = ℒ{f (t)} ℱ Fourier transform X(ω) = ℱ{f (t)} δ delta function Page 39 of 330 PART 3: AREA, VOLUME AND SURFACE AREA 3.1 AREA: 1 1 a 2 sin Bsin C Triangle: A = bh = ab sin C = = (ss − a )(s − b )(s − c ) 2 2 2sin A Rectangle: A = lw Square: A = a2 Parallelogram: A = bh = ab sin A Rhombus: A = a 2 sin A a + b Trapezium: A = h s ∠AB + ∠CD Quadrilateral: A = (s − a )(s − b )(s − c )(s − d )− abcd × cos 2 2 dd sin I A = 21 2 Rectangle with rounded corners: A = lw − r 2 (4 − π ) 3 3 × a 2 Regular Hexagon: A = 2 Regular Octagon: A = (12 + 2)× a 2 na 2 Regular Polygon: A = 180 tan4 n 3.2 VOLUME: Cube: V = a3 Cuboid: V = abc Prisim: V = (bA )× h 1 Pyramid: V = × ()bA × h 3 2 Tetrahedron: V = × a 3 12 2 Octahedron: V = × a 3 3 15 + 7 5 Dodecahedron: V = × a 3 4 (35 + 5) Icosahedron: V = × a 3 12 3.3 SURFACE AREA: Cube: SA = 6a 2 Cuboids: SA = 2(ab + bc + ca ) Page 40 of 330 Tetrahedron: SA = 3 × a 2 Octahedron: SA = 2× 3 × a 2 Dodecahedron: SA = 3× 25 +10 5 × a 2 Icosahedron: SA = 5× 3 × a 2 Cylinder: SA = 2π (hr + r) 3.4 MISCELLANEOUS: Diagonal of a Rectangle d = l 2 + w2 Diagonal of a Cuboid d = a 2 + b 2 + c 2 a Longest Diagonal (Even Sides) = 180 sin n a Longest Diagonal (Odd Sides) = 90 sin2 n Total Length of Edges (Cube): = 12 a Total Length of Edges (Cuboid): = 4(a + b + c) Circumference C = 2πr = πd Perimeter of rectangle P = 2(a + b) P Semi perimeter s = 2 Euler’s Formula Faces + Verticies = Edges + 2 3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4) A=area a=side ‘a’ b=base b=side ‘b’ C=circumference C=central angle c=side ‘c’ d=diameter d=diagonal d1=diagonal 1 d2=diagonal 2 E=external angle h=height I=internal angle l=length n=number of sides P=perimeter r=radius Page 41 of 330 r1=radius 1 s=semi-perimeter SA=Surface Area V=Volume w=width Page 42 of 330 PART 4: ALGEBRA & ARITHMETIC 4.1 POLYNOMIAL FORMULA: Qudaratic: Where ax 2 + bx + c = 0 , − b ± b 2 − 4ac x = 2a Cubic: Where ax 3 + bx 2 + cx + d = 0 , b Let, x = y − 3a b 3 b 2 b ∴a y − + b y − + c y − + d = 0 3a 3a 3a 2 3 3 + − b + + 2b − bc = ay c y d 2 0 3a 27 a 3a 2 3 − b + 2b − bc c d 2 3a 27 a 3a y 3 + y + = 0 a a 2 3 − b + 2b − bc c d 2 3a 27 a 3a y 3 + y −= a a b 2 c − 3a Let, A = = 3st ... ()1 a 3 + 2b − bc d 2 27 a 3a Let, B −= = s 3 − t 3 ... ()2 a ∴ y 3 + Ay = B y 3 + 3sty = s 3 − t 3 Solution to the equation = s − t Let, y = s − t ∴(s − t)3 + 3st (s − t) = s 3 − t 3 ()()s 3 − 3 2ts + 3st 2 − t 3 + 3 2ts − 3st 2 = s 3 − t 3 Solving (1) for s and substituting into (2) yields: Let, u = t 3 3 2 A ∴ u + Bu − = 0 27 Page 43 of 330 ie :αu 2 + βu + γ = 0 α = 1 β = B 3 γ −= A 27 − β ± β 2 − 4αγ u = 2α 4A3 − B ± B 2 + 27 u = 2 4A3 − B ± B 2 + 3 27 ∴t = 3 u = 2 Substituting into (2) yields: 3 4A3 − B ± B 2 + 3 3 3 27 s = B + t = B + 2 3 4A3 − B ± B 2 + 3 ∴ = + 27 s 3 B 2 Now, y = s − t 3 4A3 4A3 − B ± B 2 + − B ± B 2 + 3 3 ∴ = + 27 − 27 y 3 B 2 2 b Now, x = y − 3a 3 4A3 4A3 − B ± B 2 + − B ± B 2 + 3 27 3 27 b x = B + − − 3 2 2 3a 2 3 − b + 2b − bc c d 2 3a 27 a 3a Where, A = & B −= a a Page 44 of 330 4.2 FUNDAMENTALS OF ARITHMETIC: (r −1)+ (r +1) Rational Numbers: Every rational number can be written as r = 2 1 isx rational Irrational Numbers: lim lim ()cos 2n ()m!×π × x = m ∞→ n ∞→ 0 isx irrational 4.3 ALGEBRAIC EXPANSION: Babylonian Identity: (c1800BC) Common Products And Factors : Binomial Theorem: For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by: Binomial Expansion: For any power of n, the binomial (a + x) can be expanded Page 45 of 330 This is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression. There will always be n+1 terms and the general form is: Difference of two squares: Brahmagupta–Fibonacci Identity: Also, Degen's eight-square identity: Note that: Page 46 of 330 and, 4.4 ROOT EXPANSIONS: 1 2 2 ( kx ± ky ) = ( x ± y ) k 1 2 x ± y = ()kx ± ky k 2 ± = x ± x y y 1 y 1 2 x ± y = ()x ± xy x 2 ± = x ± y x y k k k 4.5 LIMIT MANIPULATIONS: ( ± ) = ( ( ))± ( ( )) lim an bn lim an lim bn n→∞ n→∞ n→∞ ()()= () lim ka n k lim an n→∞ n→∞ ()()= ()()() lim ba nn lim an lim bn n→∞ n→∞ n→∞ ()()() = () lim af n f lim an n→∞ n→∞ L’Hopital’s Rule: f x)(' If lim ( xf )( ) = lim (g x)( ) = 0 or ± ∞, and lim exists (ie g x)(' ≠ ,0 x = a ), then it follows that →ax →ax →ax g x)(' xf )( f x)(' lim = lim →ax g x)( →ax g x)(' Proof: xf )( xf )( ÷ (x − a) ( xf )( − af )( )÷ (x − a) lim = lim = lim →ax g x)( →ax g x)( ÷ ()x − a →ax ()()g x)( − g a)( ÷ x − a ()xf )( − af )( lim →ax ()x − a f x)(' f a)(' = = lim = ()g x)( − g a)( →ax g x)(' g a)(' lim →ax ()x − a Page 47 of 330 4.6 SUMMATION MANIPULATIONS: , where C is a constant 4.7 COMMON FUNCTIONS: Constant Function: y=a or f (x)=a 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 m. Where the gradient is between any two points (x1, y1 (&) x2 , y2 ) Page 48 of 330 rise y − y m = = 2 1 − run x2 x1 = + − Also, y y1 m(x x1 ) The equation of the line with gradient m . and passing through the point (x1, y1 ) . Parabola/Quadratic Function: y = (xa − h)2 + k The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h,k). y = ax 2 + bx + c The graph is a parabola that opens up if a > 0 or down if a < 0 and has a − b − b vertex at , f . 2a 2a x = ay 2 + by + c The graph is a parabola that opens right if a > 0 or left if a < 0 and has a − b − b vertex at g , . This is not a function. 2a 2a Circle: (x − h)2 + (y − k)2 = r 2 Graph is a circle with radius r and center (h,k). Ellipse: (x − h)2 (y − k)2 + = 1 a 2 b2 Graph is an ellipse with center (h,k) with vertices a units right/left from the center and vertices b units up/down from the center. Hyperbola: (x − h)2 (y − k)2 − =1 a 2 b2 Graph is a hyperbola that opens left and right, has a center at (h,k) , vertices a units left/right of center and asymptotes that pass through b center with slope ± . a (y − k)2 (x − h)2 − = 1 b2 a2 Page 49 of 330 Graph is a hyperbola that opens up and down, has a center at (h,k) , vertices b units up/down from the center and asymptotes that pass b through center with slope ± . a 4.8 LINEAR ALGEBRA: Vector Space Axioms: A real vector space is a set X with a special element 0, and three operations: • Addition: Given two elements x, y in X, one can form the sum x+y, which is also an element of X. • Inverse: Given an element x in X, one can form the inverse -x, which is also an element of X. • Scalar multiplication: Given an element x in X and a real number c, one can form the product cx, which is also an element of X. These operations must satisfy the following axioms: Additive axioms. For every x,y,z in X, we have 1. x+y = y+x. 2. (x+y)+z = x+(y+z). 3. 0+x = x+0 = x. 4. (-x) + x = x + (-x) = 0. Multiplicative axioms. For every x in X and real numbers c,d, we have 5. 0x = 0 6. 1x = x 7. (cd)x = c(dx) Distributive axioms. For every x,y in X and real numbers c,d, we have 8. c(x+y) = cx + cy. 9. (c+d)x = cx +dx. A normed real vector space is a real vector space X with an additional operation: • Norm: Given an element x in X, one can form the norm ||x||, which is a non-negative number. This norm must satisfy the following axioms, for any x,y in X and any real number c: 10. ||x|| = 0 if and only if x = 0. 11. || cx || = |c| ||x||. 12. || x+y || <= ||x|| + ||y|| A complex vector space consists of the same set of axioms as the real case, but elements within the vector space are complex. The axioms are adjusted to suit. Subspace: When the subspace is a subset of another vector space, only axioms (a) and (b) need to be proved to show that the subspace is also a vector space. Common Spaces: Page 50 of 330 Real Numbers ℜ,ℜ2 ,ℜ3 ,..., ℜn (n denotes dimension) Complex Numbers: C, C 2 ,C 3 ,..., C n (n denotes dimension) Polynomials P1, P2 , P3 ,..., Pn (n denotes the highest order of x) All continuous functions C[ ,ba ](a & b denote the interval) (This is never a vector space as it has infinite dimensions) Rowspace of a spanning set in R n Stack vectors in a matrix in rows Use elementary row operations to put matrix into row echelon form The non zero rows form a basis of the vector space Columnspace of a spanning set in R n Stack vectors in a matrix in columns Use elementary row operations to put matrix into row echelon form Columns with leading entries correspond to the subset of vectors in the set that form a basis Nullspace: Solutions to xA = 0 A Using elementary row operations to put matrix into row echelon form, columns with no leading entries are assigned a constant and the remaining variables are solved with respect to these constants. Nullity: The dimension of the nullspace Columns A)( = Nullity A)( + Rank A)( Linear Dependence: + + + = rc 11 rc 22 ... rc nn 0 = = = Then, c1 c2 cn 0 If the trivial solution is the only solution, r1 , r2 ,... rn are independent. r A)( ≠ r A b)|( : No Solution r A)( = r A b)|( = n : Unique Solution r A)( = r A b)|( < n : Infinite Solutions Basis: S is a basis of V if: S spans V S is linearly dependant = { } S u1,u2 ,u3 ,..., un Page 51 of 330 x y The general vector within the vector space is: w = z ... = + + + + w uc 11 uc 22 uc 33 ... uc nn u11 u21 u31 ... un1 c1 u u u ... u c Therefore, 12 22 32 n2 2 []w = u u u ... u c 13 23 33 n3 3 ...... ... u1n u2n u3n ... unn cn If the determinant of the square matrix is not zero, the matrix is invertible. Therefore, the solution is unique. Hence, all vectors in w are linear combinations of S. Because of this, S spans w. Standard Basis: 1 0 0 0 0 1 0 0 Real Numbers S(ℜn ) = 0,0,1,..., 0 ... ... ... ... 0 0 0 1 = { 2 3 n } Polynomials S(Pn ) ,,1 xx , x ,..., x Any set the forms the basis of a vector space must contain the same number of linearly independent vectors as the standard basis. Orthogonal Complement: ⊥ { } W is the nullspace of A, where A is the matrix that contains v1, v2 , v3 ,..., vn in rows. dim( W ⊥ ) = nullity A)( Orthonormal Basis: A basis of mutually orthogonal vectors of length 1. Basis can be found with the Gram- Schmidt process outline below. 0 i ≠ j < >= v ,v ji 1 i = j u =< , vu > v <+ , vu > v <+ , vu > v + ... <+ , vu > v ) In an orthonormal basis: 1 1 2 2 3 3 n n = + + + + u vc 11 vc 22 vc 33 ... vc nn ) Gram-Schmidt Process: This finds an orthonormal basis recursively. Page 52 of 330 = { } In a basis B u1,u2 ,u3 ,..., un = q1 u1 ^ = = q1 v1 q1 q1 Next vector needs to be orthogonal to v1 , = <− > q2 u2 u2 , v1 v1 Similarly = − < > − < > q3 u3 u3 ,v1 v1 u3 ,v2 v2 = <− > <− > − <− > qn un u3 ,v1 v1 u3 ,v2 v2 ... u3 ,vn vn ^ = = qn vn q n qn = { } Therefore, orthogonal basis is: B' v1,v2 ,v3 ,..., vn Coordinate Vector: = + + + If v ec 11 ec 22 ... ec nn c1 c v = 2 B ... cn For a fixed basis (usually the standard basis) there is 1 to 1 correspondence between vectors and coordinate vectors. Hence, a basis can be found in R n and then translated back into the general vector space. Dimension: Real Numbers dim(ℜn ) = n = + Polynomials dim( Pn ) n 1 = × Matricis dim( M ,qp ) p q If you know the dimensions and you are checking if a set forms a basis of the vector space, only Linear Independence or Span needs to be checked. 4.9 COMPLEX VECTOR SPACES: + a1 ib 1 a + ib Form: C n = 2 2 ... + an ib n Dot Product: _ _ _ • = + + + u v u1 v1 u2 v2 ... u vnn Where: Page 53 of 330 u • v = v • u ≠ v • u (u + v) • w = u • w + v • w su • v = (us • v), s ∈ C u • u ≥ 0 u • u = 0 iff u = 0 Inner Product: u = u • u = u 2 + u 2 + ... + u 2 1 2 n d vu ),( = u − v Orthogonal if u • v = 0 Parallel if u = sv , s ∈ C 4.10 LINEAR TRANSITIONS & TRANSFORMATIONS: Transition Matrix: From 1 vector space to another vector space = + + + + T u)( T ( uc 11 uc 22 uc 33 ... uc nn ) = + + + + T u)( 1Tc (u1 ) 2Tc (u2 ) 3Tc (u3 ) ... nTc (un ) Nullity(T)+Rank(T)=Dim(V)=Columns(T) Change of Basis Transition Matrix: − v = M 1M v B' B' BB = vB' CBB 'vB = { } For a general vector space with the standard basis: S s1,s2 ,..., sn = [ ] M B (v1 S (|...|) v ) Sn = [] M B' (u1 S (|...|) u )Sm Transformation Matrix: From 1 basis to another basis = { } V span ( v1,v2 ,v3 ,..., vn ) = {} B1 v1,v2 ,v3 ,..., vn = {} U span ( u1,u2 ,u3 ,..., um ) = {} B2 u1,u2 ,u3 ,..., um = []()()() A T (v1 ) B2 | T (v2 ) B2 |...| T (vn ) B2 = −1 A' C 'BB AC 'BB 4.11 INNER PRODUCTS: Definition: An extension of the dot product into a general vector space. Axioms: 1. < u,v >=< v,u > 2. < u,v + w >=< u,v > + < u,w > 3. < ku ,v >= k < ,vu > Page 54 of 330 < ,uu >≥ 0 4. < ,uu >= 0 iff u = 0 ^ u Unit Vector: u = u Cavchy-Schuarz Inequality: < ,vu >2 ≤< ,uu <×> ,vv > Inner Product Space: 1 u =< ,uu > 2 = < ,uu > u 2 =< ,uu > 2 < > < > < >2 ≤ 2 × 2 ,vu ≤ − ≤ ,vu ≤ ,vu u v ⇒ 1⇒ 1 1 u v u v u ≥ ,0 u = 0 iff u = 0 ku = k u u + v = u + v Angle between two vectors: As defined by the inner product, < ,vu > cos ()θ = u v Orthogonal if: < ,vu >= 0 Distance between two vectors: As defined by the inner product, d vu ),(= u − v Generalised Pythagoras for orthogonal vectors: u + v 2 = u 2 + v 2 4.12 PRIME NUMBERS: + 3 1 N 1 if N is odd and prime Determinate: ∆(N) = = N +1 0 if N is odd and composite 2 2k +1 N 1+ × ∑ + k=1 N 2k 1 List of Prime Numbers: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 Page 55 of 330 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 Fundamental Theory of Arithmetic: That every integer greater than 1 is either prime itself or is the product of a finite number of prime numbers. Laprange’s Theorem: That every natural number can be written as the sum of four square integers. Eg: 59 = 72 + 32 +12 + 02 = 2 + 2 + 2 + 2 ∈ Ie: x a b c d dcba ,,,,; x N 0 Additive primes: Primes: such that the sum of digits is a prime. 2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131… Annihilating primes: Primes: such that d(p) = 0, where d(p) is the shadow of a sequence of natural numbers 3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199 Bell number primes: Primes: that are the number of partitions of a set with n members. 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. Carol primes: Of the form (2 n−1) 2 − 2. 7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 Centered decagonal primes: Of the form 5( n2 − n) + 1. 11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, 6301, 6661, 7411, 9461, 9901, 12251, 13781, 14851, 15401, 18301, 18911, 19531, 20161, 22111, 24151, 24851, 25561, 27011, 27751 Centered heptagonal primes: Of the form (7 n2 − 7n + 2) / 2. Page 56 of 330 43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, 4663, 5741, 8233, 9283, 10781, 11173, 12391, 14561, 18397, 20483, 29303, 29947, 34651, 37493, 41203, 46691, 50821, 54251, 56897, 57793, 65213, 68111, 72073, 76147, 84631, 89041, 93563 Centered square primes: Of the form n2 + (n+1) 2. 5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681 Centered triangular primes: Of the form (3 n2 + 3n + 2) / 2. 19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589 Chen primes: Where p is prime and p+2 is either a prime or semiprime. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 Circular primes: A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10). 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 Cousin primes: Where (p, p+4) are both prime. (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) Cuban primes: 3 3 x − y 3 Of the form , x = y +1⇒ ()y +1 − y3 x − y 7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 x3 − y3 (y + 2)3 − y3 Of the form , x = y + 2 ⇒ x − y 2 13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 Cullen primes: Of the form n×2 n + 1. 3, 393050634124102232869567034555427371542904833 Dihedral primes: Primes: that remain prime when read upside down or mirrored in a seven-segment display. 2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 Double factorial primes: Of the form n!! + 1. Values of n: Page 57 of 330 0, 1, 2, 518, 33416, 37310, 52608 Of the form n!! − 1. Values of n: 3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318 Double Mersenne primes: p−1− A subset of Mersenne primes: of the form 22 1 −1 for prime p. 7, 127, 2147483647, 170141183460469231731687303715884105727 Eisenstein primes without imaginary part: Eisenstein integers that are irreducible and real numbers (primes: of the form 3n − 1). 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 Emirps: Primes which become a different prime when their decimal digits are reversed. 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 Euclid primes: Of the form pn# + 1 (a subset of primorial prime:). 3, 7, 31, 211, 2311, 200560490131 Even prime: Of the form 2n. 2 Factorial primes: Of the form n! − 1 or n! + 1. 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 Fermat primes: n+1 Of the form 22 +1 3, 5, 17, 257, 65537 Fibonacci primes: Primes in the Fibonacci sequence 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 Fortunate primes: Fortunate numbers that are prime 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 Gaussian primes: Prime elements of the Gaussian integers (primes: of the form 4n + 3). 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503 Generalized Fermat primes base 10: Of the form 10 n + 1, where n > 0. 11, 101 Page 58 of 330 Genocchi number primes: The only positive prime Genocchi number is 17 Gilda's primes: Gilda's numbers that are prime. 29, 683, 997, 2207, 30571351 Good primes: 2 Primes: pn for which pn > pn−i pn+i for all 1 ≤ i ≤ n−1, where pn is the nth prime. 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 Happy primes: Happy numbers that are prime. 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093 Harmonic primes: Primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where ωpis the Wolstenholme quotient. 5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349 Higgs primes for squares: Primes p for which p−1 divides the square of the product of all earlier terms. 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 Highly cototient number primes: Primes that are a cototient more often than any integer below it except 1. 2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 Irregular primes: Odd primes p which divide the class number of the p-th cyclotomic field. 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613, 617, 619 (p, p−5) irregular primes: Primes p such that (p, p−5) is an irregular pair. 37 (p, p−9) irregular primes: Primes p such that (p, p−9) is an irregular pair 67, 877 Isolated primes: Primes p such that neither p−2 nor p+2 is prime. 2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 Kynea primes: Of the form (2 n + 1) 2 − 2. 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207 Page 59 of 330 Left-truncatable primes: Primes that remain prime when the leading decimal digit is successively removed. 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 Leyland primes: Of the form xy + yx, with 1 < x ≤ y. 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 Long primes: b p−1 −1 Primes p for which, in a given base b, gives a cyclic number. They are also called full reptend primes:. p Primes p for base 10: 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 Lucas primes: Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2 . 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 Lucky primes: Lucky numbers that are prime. 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 Markov primes: Primes p for which there exist integers x and y such that x2 + y2 + p2 = 3xyp . 2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 1686049, 2922509, 3276509, 94418953, 321534781, 433494437, 780291637, 1405695061, 2971215073, 19577194573, 25209506681 Mersenne primes: Of the form 2 n − 1. 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 Mersenne prime exponents: Primes p such that 2 p − 1 is prime. 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583 Mills primes: n Of the form θ 3 , where θ is Mills' constant. This form is prime for all positive integers n. 2, 11, 1361, 2521008887, 16022236204009818131831320183 Minimal primes: Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:: 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 Motzkin primes: Primes that are the number of different ways of drawing non-intersecting chords on a circle between n points. 2, 127, 15511, 953467954114363 Page 60 of 330 Newman–Shanks–Williams primes: Newman–Shanks–Williams numbers that are prime. 7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 Non-generous primes: Primes p for which the least positive primitive root is not a primitive root of p 2. 2, 40487, 6692367337 Odd primes: Of the form 2n − 1. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199... Padovan primes: Primes in the Padovan sequence P(0) = P(1) = P(2) = 1, P(n) = P(n−2) + P(n−3). 2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 Palindromic primes: Primes that remain the same when their decimal digits are read backwards. 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 Palindromic wing primes: a(10 m −1) m Primes of the form ± b ×10 2 9 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 Partition primes: Partition numbers that are prime. 2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 Pell primes: Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2 . 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 Permutable primes: Any permutation of the decimal digits is a prime. 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 Perrin primes: Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, P(n) = P(n−2) + P(n−3). 2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 Pierpont primes: Of the form 2 u3v + 1 for some integers u,v ≥ 0. These are also class 1- primes:. 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 Page 61 of 330 Pillai primes: Primes p for which there exist n > 0 such that p divides n!+ 1 and n does not divide p−1. 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 Primeval primes: Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number. 2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 Primorial primes: Of the form pn# −1 or pn# + 1. 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 Proth primes: Of the form k×2 n + 1, with odd k and k < 2n. 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 Pythagorean primes: Of the form 4n + 1. 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 Prime quadruplets: Where ( p, p+2, p+6, p+8) are all prime. (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) Primes of binary quadratic form: Of the form x2 + xy + 2y2, with non-negative integers x and y. 2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821 Quartan primes: Of the form x4 + y4, where x,y > 0. 2, 17, 97, 257, 337, 641, 881 Ramanujan primes: Integers Rn that are the smallest to give at least n primes: from x/2 to x for all x ≥ Rn (all such integers are primes:). 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 Regular primes: Primes p which do not divide the class number of the p-th cyclotomic field. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 Repunit primes: Primes containing only the decimal digit 1. 11, 1111111111111111111, 11111111111111111111111 The next have 317 and 1,031 digits. Primes in residue classes: Of the form an + d for fixed a and d. Also called primes: congruent to d modulo a. Page 62 of 330 Three cases have their own entry: 2n+1 are the odd primes:, 4n+1 are Pythagorean primes:, 4n+3 are the integer Gaussian primes:. 2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 10n+d (d = 1, 3, 7, 9) are primes: ending in the decimal digit d. Right-truncatable primes: Primes that remain prime when the last decimal digit is successively removed. 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 Safe primes: Where p and (p−1) / 2 are both prime. 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907 Self primes in base 10: Primes that cannot be generated by any integer added to the sum of its decimal digits. 3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 Sexy primes: Where (p, p+6) are both prime. (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) Smarandache–Wellin primes: Primes which are the concatenation of the first n primes: written in decimal. 2, 23, 2357 The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes: which end with 719. Solinas primes: Of the form 2 a ± 2b ± 1, where 0 < b < a. 3, 5, 7, 11, 13 Sophie Germain primes: Where p and 2p+1 are both prime. 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953 Star primes: Of the form 6n(n − 1) + 1. 13, 37, 73, 181, 337, 433, 541, 661, 937, 1093, 2053, 2281, 2521, 3037, 3313, 5581, 5953, 6337, 6733, 7561, 7993, 8893, 10333, 10837, 11353, 12421, 12973, 13537, 15913, 18481 Page 63 of 330 Stern primes: Primes that are not the sum of a smaller prime and twice the square of a nonzero integer. 2, 3, 17, 137, 227, 977, 1187, 1493 Super-primes: Primes with a prime index in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime). 3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991 Supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 Swinging primes: Primes which are within 1 of a swinging factorial: n ≀ ±1. 2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011 Thabit number primes: Of the form 3×2 n − 1. 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 Prime triplets: Where (p, p+2, p+6) or (p, p+4, p+6) are all prime. (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353) Twin primes: Where (p, p+2) are both prime. (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463) Two-sided primes: Primes which are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:: 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 Ulam number primes: Ulam numbers that are prime. 2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897 Unique primes: The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period). 3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 Wagstaff primes: Of the form (2 n+1) / 3. 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 Page 64 of 330 Wall-Sun-Sun primes: 2 p A prime p > 5 if p divides the Fibonacci number F p , where the Legendre symbol is defined as p− 5 5 p 1 if p ±≡ 1 (mod 5) = As of 2011, no Wall-Sun-Sun primes: are known. 5 −1 if p ±≡ 2 ()mod 5 Wedderburn-Etherington number primes: Wedderburn-Etherington numbers that are prime. 2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 Weakly prime numbers Primes that having any one of their (base 10) digits changed to any other value will always result in a composite number. 294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 Wieferich primes: Primes p for which p2 divides 2 p−1 − 1. 1093, 3511 Wieferich primes: base 3 (Mirimanoff primes:) Primes p for which p2 divides 3 p−1 − 1. 11, 1006003 Wieferich primes: base 5 Primes p for which p2 divides 5 p−1 − 1 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 Wieferich primes: base 6 Primes p for which p2 divides 6 p−1 − 1. 66161, 534851, 3152573 Wieferich primes: base 7 Primes p for which p2 divides 7 p−1 − 1. 5, 491531 Wieferich primes: base 10 Primes p for which p2 divides 10 p−1 − 1. 3, 487, 56598313 Wieferich primes: base 11 Primes p for which p2 divides 11 p−1 − 1. 71 Wieferich primes: base 12 Primes p for which p2 divides 12 p−1 − 1. 2693, 123653 Wieferich primes: base 13 Primes p for which p2 divides 13 p−1 − 1 863, 1747591 Wieferich primes: base 17 Primes p for which p2 divides 17 p−1 − 1. 3, 46021, 48947 Page 65 of 330 Wieferich primes: base 19 Primes p for which p2 divides 19 p−1 − 1 3, 7, 13, 43, 137, 63061489 Wilson primes: Primes p for which p2 divides ( p−1)! + 1. 5, 13, 563 Wolstenholme primes: − 2 p 1 4 Primes p for which the binomial coefficient ≡ 1 (mod p ) p −1 16843, 2124679 Woodall primes: Of the form n×2 n − 1. 7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 4.13 GENERALISATIONS FROM PRIME NUMBERS: Perfect Numbers: A perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. Even perfect numbers are of the form 2 p−1 (2 p−1), where (2 p−1) is prime and by extension p is also prime. It is unknown whether there are any odd perfect numbers. List of Perfect Numbers: Rank p Perfect number Digits Year Discoverer 1 2 6 1 Known to the Greeks 2 3 28 2 Known to the Greeks 3 5 496 3 Known to the Greeks 4 7 8128 4 Known to the Greeks First seen in the medieval manuscript, 5 13 33550336 8 1456 Codex Lat. Monac. 6 17 8589869056 10 1588 Cataldi 7 19 12 1588 Cataldi 8 31 19 1772 Euler 9 61 37 1883 Pervushin 10 89 54 1911 Powers 11 107 65 1914 Powers 12 127 77 1876 Lucas 13 521 314 1952 Robinson 14 607 366 1952 Robinson 15 1279 770 1952 Robinson 16 2203 1327 1952 Robinson 17 2281 1373 1952 Robinson 18 3217 1937 1957 Riesel 19 4253 2561 1961 Hurwitz 20 4423 2663 1961 Hurwitz 21 9689 5834 1963 Gillies 22 9941 5985 1963 Gillies Page 66 of 330 23 11213 6751 1963 Gillies 24 19937 12003 1971 Tuckerman 25 21701 13066 1978 Noll & Nickel 26 23209 13973 1979 Noll 27 44497 26790 1979 Nelson & Slowinski 28 86243 51924 1982 Slowinski 29 110503 66530 1988 Colquitt & Welsh 30 132049 79502 1983 Slowinski 31 216091 130100 1985 Slowinski 32 756839 455663 1992 Slowinski & Gage 33 859433 517430 1994 Slowinski & Gage 34 1257787 757263 1996 Slowinski & Gage 35 1398269 841842 1996 Armengaud, Woltman, et al. 36 2976221 1791864 1997 Spence, Woltman, et al. 37 3021377 1819050 1998 Clarkson, Woltman, Kurowski, et al. 38 6972593 4197919 1999 Hajratwala, Woltman, Kurowski, et al. 39 13466917 8107892 2001 Cameron, Woltman, Kurowski, et al. 40 20996011 12640858 2003 Shafer, Woltman, Kurowski, et al. 41 24036583 14471465 2004 Findley, Woltman, Kurowski, et al. 42 25964951 15632458 2005 Nowak, Woltman, Kurowski, et al. Cooper, Boone, Woltman, Kurowski, et 43 30402457 18304103 2005 al. Cooper, Boone, Woltman, Kurowski, et 44 32582657 19616714 2006 al. 45 37156667 22370543 2008 Elvenich, Woltman, Kurowski, et al. 46 42643801 25674127 2009 Strindmo, Woltman, Kurowski, et al. 47 43112609 25956377 2008 Smith, Woltman, Kurowski, et al. Amicable Numbers: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. List of Amicable Numbers: Amicable Pairs Amicable Pairs Amicable Pairs 220 284 1,328,470 1,483,850 8,619,765 9,627,915 1,184 1,210 1,358,595 1,486,845 8,666,860 10,638,356 2,620 2,924 1,392,368 1,464,592 8,754,130 10,893,230 5,020 5,564 1,466,150 1,747,930 8,826,070 10,043,690 6,232 6,368 1,468,324 1,749,212 9,071,685 9,498,555 10,744 10,856 1,511,930 1,598,470 9,199,496 9,592,504 12,285 14,595 1,669,910 2,062,570 9,206,925 10,791,795 17,296 18,416 1,798,875 1,870,245 9,339,704 9,892,936 63,020 76,084 2,082,464 2,090,656 9,363,584 9,437,056 66,928 66,992 2,236,570 2,429,030 9,478,910 11,049,730 67,095 71,145 2,652,728 2,941,672 9,491,625 10,950,615 69,615 87,633 2,723,792 2,874,064 9,660,950 10,025,290 79,750 88,730 2,728,726 3,077,354 9,773,505 11,791,935 100,485 124,155 2,739,704 2,928,136 10,254,970 10,273,670 122,265 139,815 2,802,416 2,947,216 10,533,296 10,949,704 122,368 123,152 2,803,580 3,716,164 10,572,550 10,854,650 141,664 153,176 3,276,856 3,721,544 10,596,368 11,199,112 142,310 168,730 3,606,850 3,892,670 10,634,085 14,084,763 171,856 176,336 3,786,904 4,300,136 10,992,735 12,070,305 Page 67 of 330 176,272 180,848 3,805,264 4,006,736 11,173,460 13,212,076 185,368 203,432 4,238,984 4,314,616 11,252,648 12,101,272 196,724 202,444 4,246,130 4,488,910 11,498,355 12,024,045 280,540 365,084 4,259,750 4,445,050 11,545,616 12,247,504 308,620 389,924 4,482,765 5,120,595 11,693,290 12,361,622 319,550 430,402 4,532,710 6,135,962 11,905,504 13,337,336 356,408 399,592 4,604,776 5,162,744 12,397,552 13,136,528 437,456 455,344 5,123,090 5,504,110 12,707,704 14,236,136 469,028 486,178 5,147,032 5,843,048 13,671,735 15,877,065 503,056 514,736 5,232,010 5,799,542 13,813,150 14,310,050 522,405 525,915 5,357,625 5,684,679 13,921,528 13,985,672 600,392 669,688 5,385,310 5,812,130 14,311,688 14,718,712 609,928 686,072 5,459,176 5,495,264 14,426,230 18,087,818 624,184 691,256 5,726,072 6,369,928 14,443,730 15,882,670 635,624 712,216 5,730,615 6,088,905 14,654,150 16,817,050 643,336 652,664 5,864,660 7,489,324 15,002,464 15,334,304 667,964 783,556 6,329,416 6,371,384 15,363,832 16,517,768 726,104 796,696 6,377,175 6,680,025 15,938,055 17,308,665 802,725 863,835 6,955,216 7,418,864 16,137,628 16,150,628 879,712 901,424 6,993,610 7,158,710 16,871,582 19,325,698 898,216 980,984 7,275,532 7,471,508 17,041,010 19,150,222 947,835 1,125,765 7,288,930 8,221,598 17,257,695 17,578,785 998,104 1,043,096 7,489,112 7,674,088 17,754,165 19,985,355 1,077,890 1,099,390 7,577,350 8,493,050 17,844,255 19,895,265 1,154,450 1,189,150 7,677,248 7,684,672 17,908,064 18,017,056 1,156,870 1,292,570 7,800,544 7,916,696 18,056,312 18,166,888 1,175,265 1,438,983 7,850,512 8,052,488 18,194,715 22,240,485 1,185,376 1,286,744 8,262,136 8,369,864 18,655,744 19,154,336 1,280,565 1,340,235 Sociable Numbers: Sociable numbers are generalisations of amicable numbers where a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself. The sequence must be cyclic, eventually returning to its starting point . List of Sociable Numbers: C4s 1264460 1547860 1727636 1305184 2115324 3317740 3649556 2797612 2784580 3265940 3707572 Page 68 of 330 3370604 4938136 5753864 5504056 5423384 7169104 7538660 8292568 7520432 C5 Poulet 1918 5D 12496 2^4*11*71 14288 2^4*19*47 15472 2^4*967 14536 2^3*23*79 14264 2^3*1783 C6 Moews&Moews 1992 11D 21548919483 3^5*7^2*13*19*17*431 23625285957 3^5*7^2*13*19*29*277 24825443643 3^2*7^2*13*19*11*20719 26762383557 3^4*7^2*13*19*27299 25958284443 3^2*7^2*13*19*167*1427 23816997477 3^2*7^2*13*19*218651 C6 Moews&Moews 1995 11D/12D 90632826380 2^2*5*109*431*96461 101889891700 2^2*5^2*31*193*170299 127527369100 2^2*5^2*31*181*227281 159713440756 2^2*31*991*1299709 129092518924 2^2*31*109*9551089 106246338676 2^2*17*25411*61487 C6 Needham 2006 13D 1771417411016 2^3*11*20129743307 1851936384424 2^3*7*1637*20201767 2118923133656 2^3*7*863*43844627 2426887897384 2^3*59*5141711647 2200652585816 2^3*43*1433*4464233 2024477041144 2^3*253059630143 C6 Needham 2006 13D 3524434872392 2^3*7*17*719*5149009 4483305479608 2^3*89*6296777359 4017343956392 2^3*13*17*3019*752651 4574630214808 2^3*607*6779*138967 4018261509992 2^3*31*59*274621481 3890837171608 2^3*61*22039*361769 Page 69 of 330 C6 Needham 2006 13D 4773123705616 2^4*7*347*122816069 5826394399664 2^4*101*3605442079 5574013457296 2^4*53*677*1483*6547 5454772780208 2^4*53*239*2971*9059 5363145542992 2^4*307*353*3093047 5091331952624 2^4*318208247039 C8 Flammenkamp 1990 Brodie ? 10D 1095447416 2^3*7*313*62497 1259477224 2^3*43*3661271 1156962296 2^3*7*311*66431 1330251784 2^3*43*3867011 1221976136 2^3*41*1399*2663 1127671864 2^3*11*61*83*2531 1245926216 2^3*19*8196883 1213138984 2^3*67*2263319 C8 Flammenkamp 1990 Brodie ? 10D 1276254780 2^2*3*5*1973*10781 2299401444 2^2*3*991*193357 3071310364 2^2*767827591 2303482780 2^2*5*67*211*8147 2629903076 2^2*23*131*218213 2209210588 2^2*13^2*17*192239 2223459332 2^2*131*4243243 1697298124 2^2*907*467833 C9 Flammenkamp 1990 9D/10D 805984760 2^3*5*7*1579*1823 1268997640 2^3*5*17*61*30593 1803863720 2^3*5*103*367*1193 2308845400 2^3*5^2*11544227 3059220620 2^2*5*2347*65173 3367978564 2^2*841994641 2525983930 2*5*17*367*40487 2301481286 2*13*19*4658869 1611969514 2*805984757 C28 Poulet 1918 5D/6D 14316 2^2*3*1193 19116 2^2*3^4*59 31704 2^3*3*1321 47616 2^9*3*31 83328 2^7*3*7*31 177792 2^7*3*463 295488 2^6*3^5*19 629072 2^4*39317 589786 2*294893 294896 2^4*7*2633 Page 70 of 330 358336 2^6*11*509 418904 2^3*52363 366556 2^2*91639 274924 2^2*13*17*311 275444 2^2*13*5297 243760 2^4*5*11*277 376736 2^5*61*193 381028 2^2*95257 285778 2*43*3323 152990 2*5*15299 122410 2*5*12241 97946 2*48973 48976 2^4*3061 45946 2*22973 22976 2^6*359 22744 2^3*2843 19916 2^2*13*383 17716 2^2*43*103 This list is exhaustive for known social numbers where C>4 4.14 GOLDEN RATIO & FIBONACCI SEQUENCE: Relationship: ϕ =1.61803398 8749894848 2045868343 6563811772 0309179805 8 Infinite Series: Continued Fractions: Page 71 of 330 Trigonometric Expressions: Fibonacci Sequence: ϕ n − 1( −ϕ)n ϕ n − (−ϕ)−n F n)( = = 5 5 n n + − = 1 1 5 − 1 5 F n)( 5 2 2 4.15 FERMAT’S LAST THEOREM: a n + bn ≠ c n for integers ,ba & c and n > 2 Proposed by Fermat in 1637 as an extension of Diophantus’s explanation of the case when n=2. Case when n=3 was proved by Euler (1770) Case when n=4 was proved by Fermat Case when n=5 was proved by Legendre and Dirichlet (1885) Case when n=7 was proved by Gabriel Lamé (1840) General case when n>2 was proved by Andrew Wiles (1994). The proof is too long to be written here. See: http://www.cs.berkeley.edu/~anindya/fermat.pdf 4.16 BOOLEAN ALGEBRA: Axioms: Axiom Dual Name B = 0 if B ≠ 1 B =1if B ≠ 0 Binary Field NOT 0 =1 1= 0 0• 0 = 0 1+1 =1 AND/OR 1•1 = 1 0 + 0 = 0 AND/OR 0•1=1•0 = 0 0 +1 =1+ 0 =1 AND/OR Page 72 of 330 Theorems of one variable: Theorem Dual Name B •1 = B B + 0 = B Identity B • 0 = 0 B +1 =1 Null Element B • B = B B + B = B Idempotency Involution B = B Complements B • B = 0 B + B =1 Theorems of several variables: Theorem Dual Name B •C = C • B B + C = C + B Commutativity (B •C)• D = B • (C • D) (B + C)+ D = B + (C + D) Associativity (B •C)+ (B • D) = B • (C + D) (B + C)•(B + D) = B + (C • D) Distributivity B • (B + C) = B B + (B •C) = B Covering Combining (B •C)+ (B •C)= B (B + C)• (B + C) = B Consesnus (B • C) + (B • D) + (C • D) = B • C + B • D (B • C)+ (B • D)+ (C • D) = B • C + • • = ( + + ) + + = ( • • ) De Morgan’s B1 B2 ... B1 B2 ... B1 B2 ... B1 B2 ... Page 73 of 330 PART 5: COUNTING TECHNIQUES & PROBABILITY 5.1 2D (nn +1) Triangle Number T = n 2 2 = + n Tn Tn−1 = 2 Square Number Tn n n(3n −1) Pentagonal Number T = n 2 5.2 3D n3 + 3n 2 + 2n Tetrahedral Number T = n 6 2n3 + 3n 2 + n Square Pyramid Number T = n 6 5.3 PERMUTATIONS Permutations: = n! n! Permutations (with repeats): = ()()groupA !× groupB ×...! 5.4 COMBINATIONS n! Ordered Combinations: = ()n − p ! n n! Unordered Combinations: = = p p!()n − p ! Ordered Repeated Combinations: = n p (p + n − )!1 Unordered Repeated Combinations: = p!×()n − !1 − − − n n n1 n n1 n2 n! Grouping: = ... = n1 n2 n3 n1!n2!n3!... nr! 5.5 MISCELLANEOUS: Total Number of Rectangles and Squares from a a x b rectangle: = × ∑ Ta Tb = Number of Interpreters: TL−1 (cc +1) Max number of pizza pieces: = +1 2 (cc + 3) Max pieces of a crescent: = +1 2 c 3 + 5c Max pieces of cheese: = +1 6 Page 74 of 330 (3ll +1) Cards in a card house: = 2 Different arrangement of dominos: = 2 −dn × n! − b a MOD a 1 a Unit Fractions: = + b b + b + INT 1 b INT 1 a a θ = − Angle between two hands of a clock: 5.5 m 30 h = ( + ) Winning Lines in Noughts and Crosses: 2 a 1 P Bad Restaurant Spread: = 1− s + n − n = 1 1 5 − 1 5 Fibonacci Sequence: 5 2 2 ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) a=side ‘a’ b=side ‘b’ c=cuts d=double dominos h=hours L=Languages l=layers m=minutes n= n th term n=n number P=Premium/Starting Quantity p=number you pick r=number of roles/turns s=spread factor T=Term θ=the angle 5.6 FACTORIAL: Definition: n!= n× (n − )1 × (n − )2 ×... × 2×1 Table of Factorials: 0! 1 (by definition) 1! 1 11! 39916800 2! 2 12! 479001600 3! 6 13! 6227020800 4! 24 14! 87178291200 5! 120 15! 1307674368000 6! 720 16! 20922789888000 7! 5040 17! 355687428096000 Page 75 of 330 8! 40320 18! 6402373705728000 9! 362880 19! 121645100408832000 10! 3628800 20! 2432902008176640000 1 n+ Approximation: n!≈ 2π × n 2 ×e−n (within 1% for n>10) 5.7 THE DAY OF THE WEEK: This only works after 1753 = + + 31 m + y − y + y MOD 7d y 12 4 100 400 d=day m=month y=year SQUARE BRAKETS MEAN INTEGER DIVISION INT=Keep the integer MOD=Keep the remainder 5.8 BASIC PROBABILITY: Axiom’s of Probability: 1. P(Ω) =1for the eventspace 2. P(A)∈[ 1,0 ]for any event A. ( ∪ ) = ( )+ ( ) 3. If A1 and A2 are disjoint, then P A1 A2 P A1 P A2 ∞ ∞ = () Generally, if Ai are mutually disjoint, then PU Ai ∑ P Ai i=1 i=1 A ∪ B = B ∪ A Commutative Laws: A ∩ B = B ∩ A (A ∪ B)∪C = A ∪ (B ∪C) Associative Laws: ()()A ∩ B ∩C = A ∩ B ∩C (A ∪ B)∩ C = (A∩C)∪ (B ∩ C) Distributive Laws: ()()()A ∩ B ∪ C = A∪C ∩ B ∪ C 1 if point is ∈ D Ι = Indicator Function: D 0 if point is ∉ D 5.9 VENN DIAGRAMS: Complementary Events: 1− P(A) = P(A) Null Set: P(Φ) = 0 m = P A)( ∑ P A |( Bi P() Bi ) Totality: i=1 = + + P A)( P A |( B1 P() B1 ) P A |( B2 P() B2 ) ... ∩ = Φ ≠ where Bi B j for i j P A)( = P(A∩ B) + P(A∩ B )' Page 76 of 330 P(A ∩ B) Conditional Probability: P()A | B = P()B P(A∩ B) = P(B)⋅ P(A | B) = P(A)⋅ P(B | A) (Multiplication Law) Union: P(A ∪ B) = P(A)+ P(B)− P(A ∩ B) (Addition Law) Independent Events: P(A ∩ B) = P(A)⋅ P(B) P(A ∪ B) = P(A)+ P(B)− P(A)⋅ P(B) P(B | A) = P(B) ∩ ∩ ∩ = ( ) ( ) ( ) P(A1 A2 ... Ak ) P A1 P A2 ... P Ak Mutually Exclusive: P(A ∩ B) = 0 P(A∩ B') = P A)( P(A∪ B) = P(A)+ P(B) P(A∪ B') = P(B') Subsets: if A ⊂ B then P(A) ≤ P(B) Baye’s Theorem: P A |( B P B)() P A|( B P B)() P B A)|( = = P A)( P A B P B)()|( + P A B P B )'()'|( m = ∩ Event’s Space: P A)( ∑ P(A Bi ) i=1 5.11 BASIC STATISTICAL OPERATIONS: Variance: v = σ 2 µ − a a b µ a + b Arithmetic Mean: = = = ⇒ µ = b − µ a b µ 2 ∑ xi µ = ns b − µ µ b Geometric Mean: = = ⇒ µ = ab µ − a a µ b − µ b 2 Harmonic Mean: = ⇒ µ = µ − a a /1 a + /1 b x − µ Standardized Score: z = i σ Confidence Interval: th Quantile: The p of quantile of the distribution F is defined to be the value x p such that = ≤ = F(x p ) p or P(X x p ) p ∴ = −1 ( ) x p F p x0.25 is the lower quartile x0.5 is the median x0.75 is the upper quartile 5.12 DISCRETE RANDOM VARIABLES: ( − )2 ∑ xi x Standard Deviation: σ = ns Expected Value: Page 77 of 330 i = × E[X ] ∑ P(xi ) xi 1 E[aX + b] = aE [X ]+ b Variance: ( − )2 ∑ xi x v = ns v = ()E[]x − E x][ 2 v = E[x2 ]− ()E x][ 2 var[ aX + b] = a2 var[ X ] Probability Mass Function: P x)( = xf )( = P(X = x) Cumulative Distribution Function: F x)( = P(X ≤ x) F(x + )0 = F x)( 5.13 COMMON DRVs: Bernoulli Trial: Definition: 1 trial, 1 probability that is either fail or success Example: Probability of getting a 6 from one roll of a die = { } Outcomes: S X 1,0 p x =1 = Probability: PX x)( 1− p x = 0 Expected Value: E[X ] = p Variance: Var [X ] = p − p2 = p(1− p) Binomial Trial: Definition: Repeated Bernoulli Trials Example: Probability of getting x 6s from n rolls of a die = { } Outcomes: S X ,...3,2,1,0 n n − ()()()= ⋅ x ⋅ − xn Probability: PX x p 1 p x Expected Value: E[X ] = np Variance: Var [X ] = np (1− p) n=number to choose from p=probability of x occurring x=number of favorable results Poisson Distribution: Definition: The limit of the binomial distribution as n → ∞, p → 0 Example: Probability of getting x 6s from n rolls of a die = { } Outcomes: S X ,...3,2,1,0 n x λ −λ Probability: P ()x = e ,λ = np X x! Expected Value: E[X ] = np = λ Variance: Var [X ] = np = λ Page 78 of 330 Geometric Binomial Trial: Definition: Number of Bernoulli Trials to get 1 st Success. Example: Probability of getting the first 6s from n rolls of a die = { } Outcomes: S X ,...3,2,1,0 ( ) = ( − )x−1 Probability: PX x p 1 p 1 Expected Value: E[X ] = p 1− p Variance: Var [X ] = p 2 Negative Binomial Trial: Definition: Number to 1 st get to n successes. Example: Probability of getting the first n 6s from x rolls of a die x −1 − ()()= x − xn Probability: PX x p 1 p n −1 k Expected Value: E[X ] = p kn (1− p) Variance: Var [X ] = p2 Hypergeometric Trial: Definition: Number of choosing k out of m picks from n possiblities without replacement Example: An urn with black balls and other coloured balls. Finding the probability of getting m black balls out of n without replacement. r n − r k m − k Probability: P ()x = X n m mr Expected Value: E[X ] = n n = number of balls r = number of black balls m = number of balls drawn from urn without replacement k = number of black balls drawn from urn 5.14 CONTINUOUS RANDOM VARIABLES: Probability Density Function: = xf )( ∞ If ∫ )( dxxf = &1 xf )( ≥ 0 for − ∞ ≤ x ≤ ∞ ∞− x Cumulative Distribution Function: = F (x) = P( X ≤ x) = ∫ f (x)dx ∞− b Interval Probability: P(a ≤ X ≤ b) = F b)( − F a)( = ∫ )( dxxf a Page 79 of 330 ∞ Expected Value: E x)( = ∫ x × )( dxxf ∞− ∞ E g x))(( = ∫ g x)( × )( dxxf ∞− Variance: Var (X ) = E(X 2 ) − E(( X )) 2 5.15 COMMON CRVs: Uniform Distribution: Declaration: X ~ Uniform ba ),( 1 a ≤ x ≤ b PDF: xf )( = b − a 0 otherwise 0 x < a x x − a F (x) = f (x)dx = a ≤ x ≤ b CDF: ∫ − ∞− b a 1 x > b a + b Expected Value: = 2 (b − a)2 Variance: = 12 Exponential Distribution: Declaration: X ~ Exponentia l λ)( Page 80 of 330 0 x < 0 = PDF: xf )( −λ λe x x ≥ 0 x 0 x < 0 CDF: F x)( = )( dxxf = ∫ − −λx ≥ ∞− 1 e x 0 1 Expected Value: = λ 1 Variance: = λ2 Normal Distribution: Declaration: X ~ Normal µ,( σ 2 ) x − µ Standardized Z Score: Z = σ −1 x−µ 2 − 1 2 1 σ 1 z PDF: xf )( = e 2 = e 2 σ 2π σ 2π CDF: Φ(Z ) (The integration is provided within statistic tables) Expected Value: = µ Variance: =σ 2 5.16 BIVARIABLE DISCRETE: Probability: P(X = ,Yx = y) = yxf ),( P(X ≤ ,Yx ≤ y) = ∑ yxf ),( over all values of x & y Page 81 of 330 Marginal Distribution: ( ) = f X x ∑ (xf i y), y = fY y)( ∑ ,( yxf i ) x Expected Value: = × E[X ] ∑ x f X x)( x = × E Y ][ ∑ y fY y)( y = × × E[X Y ], ∑∑ x y f ,YX yx ),( x y = × Independence: yxf ),( f X x)( fY y)( Covariance: Cov = E[X Y], − E[X ]× E Y ][ 5.17 BIVARIABLE CONTINUOUS: Conditions: ∞ ∞ ()()≥ = f ,YX , yx 0 & ∫ ∫ f ,YX , yx dxdy 1 ∞− ∞− Probability: y x ≤ ≤ = () = P(X ,Yx y) F ,YX , yx ∫ ∫ f ,YX yx ),( dxdy ∞−∞− x < = < −∞ < < ∞ = P(X x) P(X x, Y ) ∫ f X )( dxx ∞− y < = −∞ < < ∞ < = P(Y y) P( X ,Y y) ∫ fY )( dyy ∞− Where the domain is: D = {( , yx ): x∈( ba ),, y ∈(α(x),β (x))} β () b x f ()(), yx dxdy = f , yx dydx ∫D ,YX ∫ ∫ ,YX a α ()x Where the domain is: D = {( , yx ): x∈(γ (x),δ (x)), y ∈( ,dc )} δ () d x f ()(), yx dxdy = f , yx dxdy ∫D ,YX ∫ ∫ ,YX c γ ()x Where the domain is the event space: b β (x) d δ (x) () = () = ∫ ∫ f ,YX , yx dydx ∫ ∫ f ,YX , yx dxdy 1 a α ()x c γ ()x Marginal Distribution: b () = f X x ∫ f ,YX ),( dyyx where a & b are bounds of y a Page 82 of 330 b = fY y)( ∫ f ,YX ),( dxyx where a & b are bounds of x a Measure: ()()= ( ) = mes D f ,YX , yx dxdy where f , yx 1 ∫D ,YX mes ()D = dxdy (In this case, mes(D) is the area of D) ∫D Expected Value: ∞ = × E[X ] ∫ x f X )( dxx ∞− ∞ = × E Y ][ ∫ y fY )( dyy ∞− ∞ ∞ = × × E[X Y ], ∫ ∫ x y f ,YX yx ),( dxdy ∞− ∞− = × Independence: yxf ),( f X x)( fY y)( f yx ),( = ,YX Conditional: f |YX x y)|( fY y)( = ( = ) f |YX x y)|( f X x | Y y ()()∈ = = = P X A | Y y f X x | Y dxy ∫A Covariance: Cov = E[X Y], − E[X ]× E Y ][ Cov (X Y), Correlation Coefficient: ρ = ,YX σ σ X Y Bivariate Unifrom Distribution: k if (x, y) ∈ D ()()= Ι× = f ,YX , yx k D , yx 0 if (x, y) ∉ D y x () = F ,YX , yx ∫ ∫ f ,YX yx ),( dxdy ∞−∞− 1 1 k = = dxdy mes (D) ∫D Multivariate Uniform Distribution: k if ( x , x ),... ∈ D f ()()x , x ,... = k Ι× x , x ,... = 1 2 , XX 21 ,... 1 2 D 1 2 ∉ 0 if ( x1, x2 ),... D ()()= F , XX ,... x1, x2 ,... f , XX ,... x1, x2 ,... dx 1dx 2 ... 21 ∫D 21 1 k = mes (D) Bivariate Normal Distribution: − ()−µ 2 ()−µ 2 ρ (−µ )(−µ ) 1 x X + y Y − 2 x X y Y 1 ()12 −ρ 2 σ 2 σ 2 σσ () = × X Y YX f ,YX , yx e πσ σ − ρ 2 2 X Y 1 µ µ ∈ℜ σ σ > ρ ∈(− ) where X , Y & X , Y 0 & 1,1 Page 83 of 330 ℜ2 ( ) = ρ = Note that the curves in where f ,YX , yx k are circles if 0 and ρ ≠ σ = σ (µ µ ) elipses if 0 provided X Y . The centre is at the point X , Y 5.18 FUNCTIONS OF RANDOM VARIABLES: Sums (Discrete): Z = X + Y Z = z iff X = ,Yx = z − x ( ) = ( = ) = ( = = − ) = ( − ) fZ z P Z z ∑ P X ,Yx z x ∑ f ,YX , zx x x x ( ) ( ) If X & Y are independent: (ie: the convolution of f X x & fY x ) ( ) = ( − ) = ( ) ( − ) fZ z ∑ f ,YX , zx x ∑ f X fx Y z x x x Sums (Continuous): Z = X + Y Z ≤ z iff (X ,Y ) is in the region {x + y ≤ z} ∞ − xz ()()()= ≤ = FZ z P Z z ∫ ∫ f ,YX , yx dydx ∞− ∞− ∞ − ∞ − ∞ dF d xz d xz () = Z () = () = ()() = − fZ z z ∫ ∫ f ,YX , yx dydx ∫ ∫ f ,YX , dyyx dx ∫ f ,YX , zx dxx z dz ∞− ∞− ∞− dz ∞− ∞− ( ) ( ) If X & Y are independent: (ie: the convolution of f X x & fY x ) ∞ ∞ ()()()()= − = − fZ z ∫ f ,YX , zx dxx ∫ f X fx Y z dxx ∞− ∞− Quotients (Discrete): Z = Y / X Z = z iff X = ,Yx = zx ( ) = ( = ) = ( = = ) = ( ) fZ z P Z z ∑ P X ,Yx zx ∑ f ,YX , zxx x x If X & Y are independent: ( ) = ( − ) = ( ) ( ) fZ z ∑ f ,YX , zx x ∑ f X fx Y zx x x Quotients (Continuous): Z = Y / X Z ≤ z iff (X ,Y ) is in the region {y / x ≤ z} If x > 0 , then y < zx . If x < 0 , then y > zx . Thus, 0 ∞ ∞ zx ()()()()= ≤ = + FZ z P Z z ∫∫ f ,YX , yx dydx ∫ ∫ f ,YX , yx dydx ∞− zx 0 ∞− Page 84 of 330 0 ∞ ∞ zx () = dF Z () = d () + d () fZ z z ∫∫ f ,YX , yx dydx ∫ ∫ f ,YX , yx dydx z dz ∞− zx dz 0 ∞− 0 ∞ ∞ zx 0 ∞ = d () + d ()()() = − + ∫ ∫ f ,YX , dyyx dx ∫ ∫ f ,YX , dyyx dx ∫ xf ,YX , zxx dx ∫ xf ,YX , zxx dx ∞− dz zx 0 dz ∞− ∞− 0 ∞ () = () fZ z ∫ fx ,YX , zxx dx ∞− If X & Y are independent: ∞ () = ()() fZ z ∫ fx X fx Y zx dx ∞− Maximum: Assuming that X1, X 2 ,..., X n are independant random variables with cdf F and density f. = ( ) Z max X1, X 2 ,..., X n ≤ ≤ ∀ Z z iff X i z, i ( ) = ( ≤ ) = ( ≤ ≤ ≤ ) = ( ≤ )× ( ≤ )× × ( ≤ ) = ( ( ))n FZ z P Z z P X1 z, X 2 z,..., X n z P X1 z P X 2 z ... P X n z FX z d − f ()z = ()()()F ()z n = nF z n 1 f z Z dz X X X Minimum: Assuming that X1, X 2 ,..., X n are independant random variables with cdf F and density f. = ( ) Z min X1, X 2 ,..., X n > > ∀ Z z iff X i z, i ( ) = ( ≤ ) = − ( > ) = − ( > > > ) FZ z P Z z 1 P Z z 1 P X1 z, X 2 z,..., X n z = − ()()()()> × > × × > = − ()− n 1 P X1 z P X 2 z ... P X n z 1 1 FX z d − f ()z = (1− ()1− F ()z n )= n()()1− F ()z n 1 f z Z dz X X X Order Statistics: Assuming that X1, X 2 ,..., X n are independant random variables with cdf F and density f. Sorting X i in non decreasing order: ≤ ≤ ≤ X ()()()1 X 2 ... X n The k th order statistic of a statistical sample is equal to its k th smallest value. = ( ) = ( ) Particularly: X ()1 min X1, X 2 ,..., X n and X ()n max X1, X 2 ,..., X n < < < ≤ + Let 1 k n . The event x X ()k x dx occurs if: 1. k −1 observations are less than x 2. one observation is in the interval ( , xx + dx ] 3. n − k observations are greater than x + dx The probability of any particular arrangment of this type is: k−1 −kn = f (x)F(x) (1− F(x)) dx By the combination law: n! k −1 −kn f ()k = ()()()xf F x ()1− F x K ()()k − !1 n − k ! Page 85 of 330 5.19 TRANSFORMATION OF THE JOINT DENSITY: Bivariate Functions: ( ) Let X and Y have joint density f ,YX , yx ℜ2 → ℜ ℜ2 → ℜ Let g1 : & g2 : = ( ) = ( ) U g1 X ,Y &V g2 X ,Y ℜ2 → ℜ ℜ2 → ℜ Assuming that g1 & g2 can be inverted. There exist h1 : & h2 : so that = ( ) = ( ) X h1 U,V &Y h2 U,V Multivariate Functions: ( ) Let X1,..., X n have joint density f X x1,.., xn ℜn → ℜ Let gi : = ( ) Yi gi X1,..., X n ℜn → ℜ Assuming that gi can be inverted. There exist hi : so that = ( ) X i hi Y1,..., Yn Jacobian: ∂g ∂g 1 (), yx 1 (), yx ∂x ∂y ∂g ∂g ∂g ∂g (), yxJ = = 1 (), yx × 2 (), yx − 1 (), yx × 2 (), yx ∂g ∂g ∂ ∂ ∂ ∂ 2 (), yx 2 (), yx x y y x ∂x ∂y ∂g ∂g 1 ()x ... 1 ()x x x 1 n (), yxJ = ...... ∂g ∂g n ()x ... n ()x x1 xn There is an assumption that the derivateves exist and that ( , yxJ ) ≠ 0∀ , yx Joint Density: f (h ( vu ),, h ( ,vu )) f (),vu = ,YX 1 2 ,VU ()()() hJ 1 vu ,, h2 ,vu f (h (x ,..., x ),..., h (x ,..., x )) f ()y ,..., y = X 11 n n 1 n Y 1 n ()()() hJ x11 ,..., xn ,..., hn x1,..., xn ABBREVIATIONS σ = Standard Deviation = mean ns = number of scores p = probability of favourable result v = variance xi = Individual x score x = mean of the x scores Page 86 of 330 z = Standardized Score Page 87 of 330 PART 6: STATISTICAL ANALYSIS 6.1 GENERAL PRINCIPLES: y = = .. y.. ∑∑ yij , y.. i=1 j = 1 N y = = i. yi. ∑ yij , yi. j=1 n1 y = = . j y. j ∑ yij , y. j i=1 a SS Mean Square Value of x: MS = x x df )x( MS F-Statistic of x: F = x MSE SS ÷ (a −1) F-Statistic of the Null Hypothesis: F = TREATMENT ~ F 0 ÷ ()− a− ,1 −aN SS ERROR N a = > P-Value: p P(F F0 ) Relative Efficiency: Multiplication of CRD observations that need to be carried out to ignore the effect of a block (or similar). 6.2 CONTINUOUS REPLICATE DESIGN (CRD): Treatments: = a Factors: = 1 Replications per treatment: = n i = = = + + + Total Treatments: N nT n1 n2 ... nt Mathematical Model: i = 2,1 ,..., a y = µ + ε , ij i ij = j 2,1 ,..., ni i = 2,1 ,..., a y = µ +τ + ε , ij i ij = j 2,1 ,..., ni µ +τ σ 2 yij ~ N( i , ) a: number of levels of the factor, th ni :is the number of observations on the i level of the factor, th yij :is the ij observation and Page 88 of 330 εij :random experimental error, normally independently distributed with mean 0 and variance σ2 a µ ∑ i = : overall mean = i 1 a i: treatment mean a th = − τ = τi: i treatment effect yi. y.. where ∑ i 0 i=1 Test for Treatment Effect: ANOVA: Source of Variation Degrees of Freedom Sum of Squares a a 2 2 2 y y Treatment t - 1 = ()− = i. − .. SS TREATMENT ∑n yii . y.. ∑ i=1 i=1 ni N a ni = ()− 2 = − Error nT - 1 SS ERROR ∑∑ yij yi. SS T SS TREATMENT i=1j = 1 2 a ni a ni 2 2 y = ()− = − .. Total nT - t SS T ∑∑ yij y.. ∑∑ yij i=1j = 1 i=1j = 1 N 6.3 RANDOMISED BLOCK DESIGN (RBD): Treatments: = a Factors: = 1 Replications per treatment: = b Total Treatments: = N = ab Mathematical Model: i = 2,1 ,..., a = µ + ε yij ij ij , j = 2,1 ,..., b i = 2,1 ,..., a y = µ +τ + β + ε , ij i j ij = j 2,1 ,..., ni a: number of levels of the factor, b :is the number of blocks th yij :is the ij observation and εij :random experimental error, normally independently distributed with mean 0 and variance σ2 : overall mean Page 89 of 330 th τi: i treatment effect h βj: j block effect Test for Treatment Effect: Test for Block Effect: MSE (b −1)MS + (ab −1)MSE Relative Efficiency: RE = CRD = BLOCK RBD / CRD ()− MSE RBD ba 1 MSE ANOVA: Source of Degrees of Sum of Squares Variation Freedom a a 2 2 y = ()− = 1 2 − .. Treatment a-1 SS TREATMENT b∑ yi. y.. ∑ yi. i=1 b i=1 N b b 2 2 y = ()− = 1 2 − .. Block b-1 SS BLOCK a∑ y. j y.. ∑ y. j i=1 a i=1 N a b = ( − − + )2 = − − Error (a-1)(b-1) SS ERROR ∑∑ yij yi. y. j y.. SS T SS TREATMENT SS BLOCKS i=1j = 1 a b a b 2 2 2 y Total ab-1 = ()− = − .. SS T ∑∑ yij y.. ∑∑ yij i=1j = 1 i=1j = 1 N 6.4 LATIN SQUARE DESIGN (LSD): a=4 design 1 2 3 4 1 y111 y221 y331 y441 2 y212 y322 y432 y142 3 y313 y423 y133 y243 4 y414 y124 y234 y344 Treatments: = a Factors: = 1 Replications per treatment: = a Total Treatments: = N = a 2 Mathematical Model: i = 2,1 ,..., a = µ + ε j = 2,1 ,..., a yijk ijk ijk , k = 2,1 ,..., a i = 2,1 ,..., a = µ +τ + β + γ + ε j = 2,1 ,..., a yij i j k ij , k = 2,1 ,..., a a: number of levels of the factor, th yijk :is the ijk observation and Page 90 of 330 εijk :random experimental error, normally independently distributed with mean 0 and variance σ2 : overall mean th τi: i treatment effect th βj: j row effect th γ k: k column effect Test for Treatment Effect: MSE MS + MS + (a −1)MSE Relative Efficiency: RE = CRD = ROWS COLUMNS LSD / CRD ()+ MSE LSD a 1 MSE ANOVA: Source of Degrees of Sum of Squares Variation Freedom a = ( − )2 Treatment a-1 SS TREATMENT a∑ yi.. y... i=1 a = ( − )2 Rows a-1 SS ROWS a∑ y j.. y... i=1 a = ( − )2 Columns a-1 SS COLUMNS a∑ y.. k y... i=1 a a a 2 2 = ( − − − + ) Error a − 3a + 2 SS ERROR ∑∑∑ yijk yi.. y j.. y.. k 2y... i=1j = 1k = 1 a a a 2 2 = ( − ) Total a −1 SS T ∑∑∑ yijk y... i=1j = 1k = 1 6.5 ANALYSIS OF COVARIANCE: Mathematical Model: i = 2,1 ,..., a y = µ +τ + β ()x − x + ε , ij i ij ij = j 2,1 ,..., ni a: number of levels of the factor, th ni :is the number of observations on the i level of the factor, th yij :is the ij observation and εij :random experimental error, normally independently distributed with mean 0 and variance σ2 : overall mean th τi: i treatment effect β: linear regression coefficient indicating dependency of yij on xij th xij :is the ij covariate Assumptions: • Treatment do not influence the covariate o May be obvious from the nature of the covariates. o Test through ANOVA on Covariates. • The regression coefficient β is the same for all treatments o Perform analysis of variance on covariates • The relationship between the response y and covariate x is linear. Page 91 of 330 o For each treatment fit a linear regression model and assess its quality 6.6 RESPONSE SURFACE METHODOLOGY: Definition: Creating a design such that it will optimise the response 1st order: 2nd order: Common Designs : • Designs for first order model o 2 level factorial designs + some centre points • Designs for second order model o 2 level factorial designs + some centre points + axial runs (central composite designs) o 3 levels designs; 2 level factorial designs and incomplete block designs (Box-Behnken designs) • Latin Hypercube Designs o Space filling Multilevel designs Criterion for determining the optimatility of a design: X Matrix is determined as the coefficients of β in whatever order studied. 6.7 FACTORIAL OF THE FORM 2 n: General Definition: A 2 n consists of n factors each studied at 2 levels. Contrasts for a 2 2 design: ab − b + a − I ab − a + b − I ab − a − b + I M = ; M = ; M = A 2n B 2n AB 2n Sum of Squares for a 2 2 design: (ab − b + a − I )2 (ab − a + b − I )2 (ab − a − b + I )2 SS = ; SS = ; SS = A 22 n B 22 n AB 22 n = + + SS TREATMENT SS A SS B SS AB Hypothesis for a CRD 2 2 design: Test for interaction effect of factor AB: Page 92 of 330 And only if the interaction is not significant: Test for Main effect of factor A: Test for Main effect of factor B: Hypothesis for a RBD 2 2 design: Test for block effect: Test for interaction effect of factor AB: Test for Main effect of factor A: Test for Main effect of factor B: 6.8 GENERAL FACTORIAL: General Definition: Factor A at x A levels, Factor B at x B levels, Factor C at x C levels, etc. × × × Order: x A xB xC ... Degrees of freedom for Main Effects: df(A) = x A - 1 df(B) = x B – 1 df(C) = x C – 1 etc. Degrees of freedom for Higher Order Effects: df(AB)= df(A) df(B) etc. df(ABC)= df(A) df(B) df(C) etc. 6.9 ANOVA ASSUMPTIONS: Assumptions: • Normality o ANOVA applied to the absolute deviation of a response from the mean o Plot of observed values against expected value => Linear plot will imply normality • Constant Variance o Plot of residual agains predicted values => Random plot with no vertical funnelling structure will imply constant variance o Levene’s Test • Independence o Plot of residuals in time sequence => Random plot will imply independence Page 93 of 330 Levene’s Test: i = 2,1 ,..., a d = y −ψ ij ij i = j 2,1 ,..., ni ψ Where i = o Mean of observations at the i th level o Or Median of observations at the i th level o Or 10% trimmed mean of observations at the i th level. 6.10 CONTRASTS: a = α µ + α µ + + α µ α = Linear Contrast: l 1 1 2 2 ... n n ;∑ i 0 i=1 ^^ a = α µ Estimated Mean of Contrast: lE )( ∑ i i i=1 ^^ a α 2 Estimated Variance of Contrast: lv )( = s 2 ∑ i where s 2 is MSE of corresponding i=1 ni a ^^ α µ ∑ i i lE )( = F of Contrast: f = = i 1 ^^ a α 2 lv )( s2 ∑ i i=1 ni α β = Orthogonal Contrasts: if ∑ ii 0 i 6.11 POST ANOVA MULTIPLE COMPARISONS: When the null hypothesis is rejected and the optimum treatment combination needs to be found, Bonderroni Method: • Let m be the number of different treatments performed α • Perform each individual test at level of significance m • Difference of the contrast is significant if F(l) > Critical Value of F using new level of significance Fisher’s Least Significant Difference: 1 1 1 1 • LSD = tα s + = tα MSE + ij ,2/ df TREATMENT ,df ERROR ,2/ df TREATMENT ,df ERROR ni n j ni n j • − > Difference is significant if yi. y j. LSD Tukey’s W Procedure: MSE • Difference is significant if y − y ≥ qα vt ),( i. j. n • q is a conservative statistic read from the tables Scheffe’s Method: ^^ • S = lv )( (a − )1 Fα ,, dfdf 21 • a is the number of treatments, df 1 = a-1, df 2 = degrees of freedom for MSE Page 94 of 330 ^^ • Difference is significant if lE )( > S Page 95 of 330 PART 7: PI 7.1 AREA: πd 2 Cd Circle: A = πr 2 = = 4 4 Cyclic Quadrilateral: (s − a)(s − b)(s − c)(s − d ) Q Area of a sector (degrees) A = ×πr 2 360 1 Area of a sector (radians) A = r 2θ 2 r 2 Q Area of a segment (degrees) A = ×π − sin Q 2 180 w 2 Area of an annulus: A = π ()r 2 − r 2 = π 2 1 2 π Ellipse: A = lw = π rr 4 21 7.2 VOLUME: Cylinder: V = πr 2 h 4 Sphere: V = πr 3 3 1 Cap of a Sphere: V = πh(3r 2 + h 2 ) 6 1 1 Cone: V = πr 2 h 3 1 Ice-cream & Cone: V = πr 2 ()h + 2r 3 π 2 Doughnut: V = 2π 2 rr 2 = (b + a )(b − a )2 12 4 πw2 w Sausage: V = l − 4 3 4 Ellipsoid: V = π rrr 3 321 7.3 SURFACE AREA: Sphere: SA = 4πr 2 Hemisphere: SA = 3πr 2 = π 2 = π 2 ( 2 − 2 ) Doughnut: SA 4 rr 12 b a Sausage: SA = πwl Cone: SA = π (rr + r 2 + h 2 ) Page 96 of 330 7.4 MISELANIOUS: Q Q Length of arc (degrees) l = ×C = ×πr 360 180 Q Length of chord (degrees) l = 2r × sin = 2 r 2 − h 2 2 2 3()r − r 1+ 1 2 ()r + r 2 ≈ π ()+ 1 2 Perimeter of an ellipse P r1 r2 3()r − r 2 10 + 4 − 1 2 ()+ 2 r1 r2 7.6 PI: π ≈ 3.14159265 3589793238 4626433832 7950288... π = C d θ θ Archimedes’ Bounds: 2k nsin < π < 2k nsin 2k 2k π 2 2 4 4 6 6 8 8 ∞ 4n2 John Wallis: = × × × × × × × ×... = ∏ 2 − 2 1 3 3 5 5 7 7 9 n=1 4n 1 π 1 1 1 1× 3 1 1× 3× 6 1 Isaac Newton: = + + + + ... 6 2 2 3× 23 2× 4 5× 25 2× 4× 6 7 × 27 π 1 1 1 1 1 1 1 James Gregory: = 1− + − + − + − ... 4 3 5 7 9 11 13 15 π 1 1 1 Schulz von Strassnitzky: = arctan + arctan + arctan 4 2 5 8 π 1 1 John Machin: = 4arctan − arctan 4 5 239 π 2 1 1 1 1 Leonard Euler: = + + + + ... 6 12 22 32 42 π 3 5 7 11 13 17 19 23 29 31 = × × × × × × × × × ×... 4 4 4 8 12 12 16 20 24 28 32 where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. 1 1 1 1 1 1 1 1 1 1 1 1 `π =1+ + + − + + + + − + + − +... 2 3 4 5 6 7 8 9 10 11 12 13 If the denominator is a prime of the form 4m - 1, the sign is positive; if the denominator is 2 or a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors. 1 1 n n 4 n ()1 + i − ()1 − i Jozef Hoene-Wronski: π = lim n → ∞ i 2 2 2 + 2 2 + 2 + 2 Franciscus Vieta: = × × ×... π 2 2 2 Page 97 of 330 Integrals: Infinite Series: 1 ∞ (−1)n 25 1 28 26 22 22 1 − − + − − − + = π 6 ∑ 10 n + + + + + + + 2 n=0 2 4n 1 4n 3 10 n 1 10 n 3 10 n 5 10 n 7 10 n 9 See also: Zeta Function within Part 17 Page 98 of 330 Continued Fractions: 7.7 CIRCLE GEOMETRY: a 2 + b 2 Radius of Circumscribed Circle for Rectangles: r = 2 a Radius of Circumscribed Circle for Squares: r = 2 a Radius of Circumscribed Circle for Triangles: r = 2sin A Radius of Circumscribed Circle for Quadrilaterals: 1 (ab + cd )(ac + bd )(ad + bc ) r = × 4 (s − a )(s − b )(s − c )(s − d ) a Radius of Inscribed Circle for Squares: r = 2 A Radius of Inscribed Circle for Triangles: r = s a Radius of Circumscribed Circle: r = 180 sin2 n Page 99 of 330 a Radius of Inscribed Circle: r = 180 tan2 n 7.8 ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7): A=Angle ‘A’ A=Area a=side ‘a’ B=Angle ‘B’ b=side ‘b’ B=Angle ‘B’ c=side ‘c’ C=circumference d=diameter d=side ‘d’ h=shortest length from the center to the chord r=radius < r1=radius 1 ( r1 r2 ) < r2=radius 2 ( r2 r3 ) r3=radius 3 l=length n=number of sides P=perimeter Q=central angle s=semi-perimeter w=width w=length of chord from r1 7.9 CRESCENT GEOMETRY: 1 Area of a lunar crescent: A = πcd 4 Area of an eclipse crescent: 2 + 2 − 2 2 + 2 − 2 π −1 w l b −1 w l b 2 cos sin 2cos 2wl 2wl A = w2 π − + 360 2 2 + 2 − 2 2 + 2 − 2 π −1 w l b −1 w l b 2 cos sin 2cos 2wl 2wl − b 2 π − + 360 2 Page 100 of 330 7.10 ABBREVIATIONS (7.9): A=Area b=radius of black circle c=width of the crescent d=diameter l=distance between the centres of the circles w=radius of white circle Page 101 of 330 PART 8: APPLIED FIELDS: 8.1 MOVEMENT: v 2 Stopping distance: s = − 2a v 2 Centripetal acceleration: a = r mv 2 Centripetal force: F = ma = C r 2h Dropping time : t = g = ma Force: F 3 2 2 − v 1 2 c 1 Kinetic Energy: E = mv 2 k 2 (u sin θ )2 Maximum height of a cannon: h = g l Pendulum swing time: t = 2π g = Potential Energy: E p mgh 2u sin θ Range of a cannon: s = ()ut cos θ = × ()u cos θ g 2u sin θ Time in flight of a cannon: t = g m m Universal Gravitation: F = G 21 r 2 ABBREVIATIONS (8.1): a=acceleration (negative if retarding) c=speed of light ( 3×10 8 ms -1) Ek=Kinetic Energy Ep=potential energy F=force g=gravitational acceleration ( ≈9.81 on Earth) G=gravitational constant = 67.6 ×10 −11 h=height l=length of a pendulum m=mass m1=mass 1 m2=mass 2 Page 102 of 330 r=radius r=distance between two points s=distance t=time u=initial speed v=final speed θ=the angle 8.2 CLASSICAL MECHANICS: Newton’s Laws: First law: If an object experiences no net force, then its velocity is constant; the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero). Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma. Third law: When two bodies interact by exerting force on each other, these forces (termed the action and the reaction) are equal in magnitude, but opposite in direction. Inertia: Page 103 of 330 Moments of Inertia: Description Diagram Formulae Two point masses, M and m, with reduced mass and separated by a distance, x. Rod of length L and mass m (Axis of rotation at the end of the rod) Rod of length L and mass m Thin circular hoop of radius r and mass m Thin circular hoop of radius r and mass m Thin, solid disk of radius r and mass m Page 104 of 330 Thin cylindrical shell with open ends, of radius r and mass m Solid cylinder of radius r, height h and mass m Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius or when defining the normalized thickness tn = t/r and letting r2, length h and mass m r = r2, then Sphere (hollow) of radius r and mass m Ball (solid) of radius r and mass m Page 105 of 330 Right circular cone with radius r, height h and mass m About a diameter: Torus of tube radius a, cross-sectional radius b and mass m. About the vertical axis: Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass m Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate) Thin rectangular plate of height h and of width w and mass m Solid cuboid of height h, width w, and depth d, and mass m Page 106 of 330 Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis. Plane polygon with vertices , , , ..., and mass uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. Infinite disk with mass normally distributed on two axes around the axis of rotation (i.e. Where : is the mass-density as a function of x and y). Velocity and Speed: ∆P v = AVE ∆t Acceleration: ∆V a = AVE ∆t Trajectory (Displacement): Page 107 of 330 Kinetic Energy: Centripetal Force: Circular Motion: , or , Angular Momentum: Page 108 of 330 Torque: Work: Laws of Conservation: Momentum: = Energy: ∑ EIN ∑ EOUT = = = = Force: ∑ FNET 0 ⇒ ∑ FUP ∑ FDN ,∑ FL ∑ FR ,∑cm ∑acm ABBREVIATIONS (8.2) a=acceleration EK=Kinetic Energy Er=rotational kinetic energy F=force I=mass moment of inertia J=impulse L=angular momentum m=mass P=path p=momentum t=time v=velocity W=work τ=torque 8.3 RELATIVISTIC EQUATIONS: Kinetic Energy: Page 109 of 330 Momentum: Time Dilation: Length Contraction: Relativistic Mass: . 8.4 ACCOUNTING: Profit: p = s − c p Profit margin: m = ×100 c Simple Interest: = P(1+ tr ) Compound Interest: = P(1+ r)t Continuous Interest: = Pe rt ABBREVIATIONS (8.4): c=cost I=interest m=profit margin (%) p=profit P=premium r=rate s=sale price t=time 8.5 MACROECONOMICS: GDP: y = AE = AD = C + I + G + NX y = Summation of all product quantities multiplied by cost RGDP: RGDP = Summation of all product quantities multiplied by base year cost NGDP: NGDP = Summation of all product quantities multiplied by current year cost RGDP − RGDP Growth: Growth = CURRENT BASE ×100 RGDP BASE Net Exports: NX = X - M Working Age Population: WAP = Labor Force + Not in Labor Force Page 110 of 330 Labor Force: LF = Employed + Unemployed Unemployment: UE = Frictional + Structural + Cyclical Natural Unemployment: NUE = Frictional + Structural UE Unemployment Rate: UE% = ×100 LF E Employment Rate: E% = ×100 LF LF UE + E Participation Rate: P% = ×100 = ×100 WAP WAP CPI: CPI = Indexed Average Price of all Goods and Services CPI − CPI Inflation Rate: Inflation Rate = CURRENT BASE ×100 CPI BASE ABBREVIATIONS (8.5) AD=Aggregate Demand AE=Aggregate Expenditure C=Consumption CPI=Consumer Price Index E=Employed G=Government I=Investment LF=Labor Force M=Imports NGDP=Nominal GDP NUE=Natural Unemployment NX=Net Export P=Participation RGDP=Real GDP (Price is adjusted to base year) UE=Unemployed WAP=Working Age Population X=Exports Y=GDP Page 111 of 330 PART 9: TRIGONOMETRY 9.1 CONVERSIONS: Degrees 30° 60° 120° 150° 210° 240° 300° 330° Radians 33 ⅓ 66 ⅔ 133 ⅓ 166 ⅔ 233 ⅓ 266 ⅔ 333 ⅓ 366 ⅔ Grads grad grad grad grad grad grad grad grad Degrees 45° 90° 135° 180° 225° 270° 315° 360° Radians Grads 50 grad 100 grad 150 grad 200 grad 250 grad 300 grad 350 grad 400 grad 9.2 BASIC RULES: sin θ tan θ = cos θ a b c sin A sin B sin C Sin Rule: = = or = = sin A sin B sin C a b c b 2 + c 2 − a 2 Cos Rule: cos A = or a 2 = b 2 + c 2 − 2bc cos A 2bc Tan Rule: Auxiliary Angle: Pythagoras Theorem: a 2 + b2 = c 2 Periodicy: sin (nπ ) = ,0 n∈ Ζ cos ()()nπ = −1 n ,n∈ Ζ Page 112 of 330 9.3 RECIPROCAL FUNCTIONS 1 sec θ = cos θ 1 csc θ = sin θ 1 cos θ cot θ = = tan θ sin θ 9.4 BASIC IDENTITES: Pythagorean Identity: 9.5 IDENTITIES BETWEEN RELATIONSHIPS: tan ()θ 1 sec 2 (θ )−1 1 sin ()()θ ±= 1− cos 2 θ ±= = ±= ±= 1+ tan 2 ()θ csc ()θ sec ()θ 1+ cot 2 ()θ 1 csc 2 (θ ) −1 1 cot ()θ cos ()()θ ±= 1− sin 2 θ ±= ±= = ±= 1+ tan 2 ()θ csc ()θ sec ()θ 1+ cot 2 ()θ Page 113 of 330 sin ()θ 1− cos 2 (θ ) 1 1 tan ()θ ±= ±= ±= ±= sec 2 ()θ −1 = 1− sin 2 ()θ cos ()θ csc 2 ()θ −1 cot ()θ 1 1 1+ tan 2 (θ ) sec ()θ csc ()θ = ±= ±= ±= ±= 1+ cot 2 ()θ sin ()θ 1− cos 2 ()θ tan ()θ sec 2 ()θ −1 1− sin 2 (θ ) cos ()θ 1 1 cot ()θ ±= ±= = ±= csc 2 ()θ −1 ±= sin ()θ 1− cos 2 ()θ tan ()θ sec 2 ()θ −1 1 1 k arctan = arctan + arctan x x + k x 2 + kx +1 9.6 ADDITION FORMULAE: Sine: Cosine: Tangent: Arcsine: Arccosine: Arctangent: 9.7 DOUBLE ANGLE FORMULAE: Sine: Generally, n n k −kn 1 sin ()()()nx = ∑ cos x sin x sin ()n − k π k=0 k 2 Cosine: Generally, n n k −kn 1 cos ()()()nx = ∑ cos x sin x cos ()n − k π k=0 k 2 Page 114 of 330 Tangent: Generally. n n k −kn 1 ∑ cos ()()x sin x sin ()n − k π sin ()nx k=0 k 2 tan ()nx = = () n cos nx n k −kn 1 ∑ cos ()()x sin x cos ()n − k π k=0 k 2 Cot: 9.8 TRIPLE ANGLE FORMULAE: Sine: Cosine: Tangent: Cot: 9.9 HALF ANGLE FORMULAE: Sine: Cosine: Tangent: Cot: Page 115 of 330 9.10 POWER REDUCTION: Sine: If n is even: If n is odd: Cosine: If n is even: If n is odd: Page 116 of 330 Sine & Cosine: 9.11 PRODUCT TO SUM: 9.12 SUM TO PRODUCT: 9.13 HYPERBOLIC EXPRESSIONS: Hyperbolic sine: Hyperbolic cosine: Hyperbolic tangent: Hyperbolic cotangent: Page 117 of 330 Hyperbolic secant: Hyperbolic cosecant: 9.14 HYPERBOLIC RELATIONS: 9.15 MACHIN-LIKE FORMULAE: Form: Formulae: Page 118 of 330 Identities: for , for , for , for . 9.16 SPHERICAL TRIANGLE IDENTITIES: 1 1 sin ()A − B tan ()a − b 2 = 2 1 1 sin ()A+ B tan c 2 2 1 1 sin ()a −b tan ()A − B 2 = 2 1 1 sin ()a + b cot c 2 2 1 1 cos ()A − B tan ()a + b 2 = 2 1 1 cos ()A + B tan c 2 2 1 1 cos ()a − b tan ()A + B 2 2 = 1 1 cos ()a + b cot c 2 2 9.17 ABBREVIATIONS (9.1-9.16) A=Angle ‘A’ Page 119 of 330 a=side ‘a’ B=Angle ‘B’ b=side ‘b’ B=Angle ‘B’ c=side ‘c’ Page 120 of 330 PART 10: EXPONENTIALS & LOGARITHIMS 10.1 FUNDAMENTAL THEORY: e = 2.71828182 8459045235 3602874713 5266249775 7247093700 0 10.2 EXPONENTIAL IDENTITIES: xx ba = x +ba a x − = x ba xb x y aa = ()xy a b a ()()x a = x ab = xb a x b = b x a − 1 x a = x a 10.3 LOG IDENTITIES: Page 121 of 330 10.4 LAWS FOR LOG TABLES: 10.5 COMPLEX NUMBERS: 10.6 LIMITS INVOLVING LOGARITHMIC TERMS Page 122 of 330 PART 11: COMPLEX NUMBERS 11.1 GENERAL: Fundamental: i2 −= 1 Standard Form: z = a + bi Polar Form: z = rcis θ = r(cos θ + isin θ ) b Argument: arg (z) = θ , where tan θ = a Modulus: mod (z) = r = z = a + bi = a 2 + b2 Conjugate: z = a − bi Exponential: z = ⋅er iθ De Moivre’s Formula: z = rcis θ 1 1 θ + 2kπ , k=0,1,…,(n-1) z n = r n cis n π Euler’s Identity: ei +1= 0 (Special Case when n=2) n−1 2 πki ∑e n = 0 (Generally) k=0 11.2 OPERATIONS: Addition: (a + bi) + (c + di) = (a + c) + (b + d)i Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i. Multiplication: (a + bi)(c + di) = ac + bci + adi + bdi 2 = (ac - bd) + (bc + ad)i. bi) + (a + bi) (a + bi)(c - di) ac + bci - adi + bd ac + bd bc - ad Division: = = = + i. di) + (c + di) (c + di)(c - di) (c + di)(c - di) c2 + d2 c2 + d2 Sum of Squares: 11.3 IDENTITIES: Exponential: Logarithmic: Trigonometric: Page 123 of 330 Hyperbolic: Page 124 of 330 PART 12: DIFFERENTIATION For Differential Equations, see Functions 12.1 GENERAL RULES: Plus Or Minus: y = f ()()()± g ± h ... x x x = ± ± y' f '()()()x g' x h' x ... Product Rule: y = uv y'= 'vu + uv ' Quotient Rule: u y = v 'vu − uv ' y'= v 2 Power Rule: n y = ( f () ) x = ()n−1 × y' fn ()x f '()x Chain Rule: dy = dy × du × dv dx du dv dx Blob Rule: y = e f()x = × f()x y' f '()x e Base A Log: = y log a f ()x f '() y'= x × () f ()x ln a Natural Log: = ( ) y a ln f ()x f '() y'= a × x f ()x Exponential (X): y = k x y = ln' k × k x First Principles: f ()()+ − f f ' = hx x ()x lim h →0 h Page 125 of 330 Angle of intersection between two curves: = = Two curves y f1 x &)( y f 2 x)( intersect at x0 − f (' x ) − f (' x ) α = tan 1 2 0 1 0 + × 1 f 2 (' x0 ) f 1 (' x0 ) 12.2 EXPONETIAL FUNCTIONS: 12.3 LOGARITHMIC FUNCTIONS: 12.4 TRIGONOMETRIC FUNCTIONS: Page 126 of 330 −1 ()arc cot( x ') = x 2 +1 1 ()arc sec( x ') = 1 x 2 1− x 2 −1 ()arc csc( x ') = 1 x 2 1− x 2 12.5 HYPERBOLIC FUNCTIONS: −1 ()arc coth( x ') = x 2 − 0 −1 ()arc sec xh ')( = x 2 − x 4 −1 ()arc csc xh ')( = x 2 + x 4 12.5 PARTIAL DIFFERENTIATION: First Principles: Page 127 of 330 ie: Gradient: Total Differential: Chain Rule: Page 128 of 330 Implicit Differentiation: Therefore, Higher Order Derivatives: Page 129 of 330 PART 13: INTEGRATION 13.1 GENERAL RULES: n+1 n [f() ] f ' []f dx = x + C ∫ ()()x x + Power Rule: n 1 n+1 n [f ] [] = ()x + a∫ f '()()x f x dx a C n +1 udv = uv − vdu By Parts: ∫ ∫ xf )( Constants: ∫ kdt = kf x)( 0 13.2 RATIONAL FUNCTIONS: Page 130 of 330 For 13.3 TRIGONOMETRIC FUNCTIONS (SINE): Page 131 of 330 13.4 TRIGONOMETRIC FUNCTIONS (COSINE): Page 132 of 330 13.5 TRIGONOMETRIC FUNCTIONS (TANGENT): 13.6 TRIGONOMETRIC FUNCTIONS (SECANT): Page 133 of 330 13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT): 13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE): Page 134 of 330 also: also: also: Page 135 of 330 also: also: 13.9 TRIGONOMETRIC FUNCTIONS (SINE & TANGENT): 13.10 TRIGONOMETRIC FUNCTIONS (COSINE & TANGENT): 13.11 TRIGONOMETRIC FUNCTIONS (SINE & COTANGENT): 13.12 TRIGONOMETRIC FUNCTIONS (COSINE & COTANGENT): 13.13 TRIGONOMETRIC FUNCTIONS (ARCSINE): Page 136 of 330 13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE): 13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT): 13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT): Page 137 of 330 13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT): 13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT): 13.19 EXPONETIAL FUNCTIONS for Page 138 of 330 (erf is the Error function) where where when b≠0, λ≠ 0 and for , which is the logarithmic mean Page 139 of 330 (!! is the double factorial) (I0 is the modified Bessel function of the first kind) 13.20 LOGARITHMIC FUNCTIONS Page 140 of 330 13.21 HYPERBOLIC FUNCTIONS Page 141 of 330 Page 142 of 330 13.22 INVERSE HYPERBOLIC FUNCTIONS Page 143 of 330 13.23 ABSOLUTE VALUE FUNCTIONS 13.24 SUMMARY TABLE Page 144 of 330 13.25 SQUARE ROOT PROOFS Page 145 of 330 ∫ a 2 + x 2 dx x Let x = a tan θ ∴dx = asec 2 θdθ → tan θ = a = ∫ a 2 + (a tan θ )2 × asec 2 θdθ = ∫ a 2 + a 2 tan 2 θ ×asec 2 θdθ = ∫ a 2 + a 2 (sec 2 θ − )1 × asec 2 θdθ = ∫ a 2 + a 2 sec 2 θ − a 2 ×asec 2 θdθ = ∫ a 2 sec 2 θ × asec 2 θdθ = ∫ asec θ × asec 2 θdθ = ∫ a 2 sec 3 θdθ = a 2 ∫sec θ ×sec 2 θdθ u = sec θ,dv = sec 2 θdθ du = sec θ tan θdθ,v = tan θ ∴a 2 ∫sec 3 θdθ = sec θ × tan θ − ∫ tan θ ×sec θ tan θdθ a 2 ∫sec 3 θdθ = sec θ tan θ − ∫ tan 2 θ sec θdθ a 2 ∫sec 3 θdθ = sec θ tan θ − ∫(sec 2 θ − sec)1 θdθ a 2 ∫sec 3 θdθ = sec θ tan θ − ∫sec 3 θ −sec θdθ a 2 ∫sec 3 θdθ = sec θ tan θ − ∫sec 3 θdθ + ∫sec θdθ 2a 2 ∫sec 3 θdθ = sec θ tan θ + ∫sec θdθ 1 sec 3 θdθ = ()sec θ tan θ + sec θdθ ∫ 2a 2 ∫ 1 sec 3 θdθ = ()sec θ tan θ + ln sec θ + tan θ + C ∫ 2a 2 1 a 2 + x 2 s a 2 + x 2 s ∴ a 2 + x 2 dx = × + ln + + C ∫ 2 2a a a a a Page 146 of 330 ∫ a 2 − x 2 dx x Let x = asin θ ∴dx = acos θdθ → sin θ = a = ∫ a 2 − ()asin θ 2 × acos θdθ = ∫ a 2 − a 2 sin 2 θ × acos θdθ = ∫ a 2 − a 2 ()1− cos 2 θ × acos θdθ = ∫ a 2 − a 2 + a 2 cos 2 θ ×acos θdθ = ∫ a 2 cos 2 θ × acos θdθ = ∫ acos θ × acos θdθ = ∫ a 2 cos 2 θdθ = a 2 ∫cos 2 θdθ 1+ cos ()2θ = a 2 dθ ∫ 2 a 2 = 1+ cos ()2θ dθ 2 ∫ a 2 sin ()2θ = θ + + C 2 2 a 2 sin2 θ cos θ = θ + + C 2 2 a 2 = []θ + sin θ cos θ + C 2 a 2 x x a 2 − x 2 = arcsin + × + C 2 a a a Page 147 of 330 ∫ x 2 − a 2 dx x Let x = asec θ ∴dx = asec θ tan θdθ → sec θ = a = ∫ ()asec θ 2 − a 2 × asec θ tan θdθ = ∫ a 2 sec 2 θ − a 2 × asec θ tan θdθ = ∫ a 2 ()1+ tan 2 θ − a 2 × asec θ tan θdθ = ∫ a 2 + a 2 tan 2 θ − a 2 × asec θ tan θdθ = ∫ a 2 tan 2 θ × asec θ tan θdθ = ∫ a tan θ × asec θ tan θdθ = ∫ a 2 tan 2 θ sec θdθ = a 2 ∫ tan 2 θ sec θdθ = a 2 ∫()sec 2 θ −1 sec θdθ = a 2 ∫sec 3 θ − sec θdθ = a 2 ()∫sec 3 θdθ − ∫sec θdθ 2 1 = a ()sec θ tan θ + ln sec θ + tan θ − ()ln sec θ + tan θ + C 2a 2 1 = ()sec θ tan θ + ln sec θ + tan θ − a 2 ()ln sec θ + tan θ + C 2 1 x x 2 − a 2 x x 2 − a 2 x x 2 − a 2 = × + ln + − a 2 ln + + C 2 a a a a a a 1 x× x 2 − a 2 1 x x 2 − a 2 x x 2 − a 2 = + ln + − ln + + C 2 a 2 2 a a a a × 2 − 2 2 − 2 = 1 x x a + 1 − 2 x + x a + 2 a ln C 2 a 2 a a 13.26 CARTESIAN APPLICATIONS b = Area under the curve: A ∫ f x)( dx a b Volume: V = ∫ A a b b = π []2 = π []2 Volume about x axis: Vx ∫ y dx ∫ f x)( dx a a Page 148 of 330 d = π []2 Volume about y axis: Vy ∫ x dy c b = π + ()2 Surface Area about x axis: SA 2 ∫ f x)( 1 f ' x)( dx a b dy 2 Length wrt x-ordinates: L = ∫ 1+ dx a dx 2 d dx Length wrt y-ordinates: L = ∫ 1+ dy (Where the function is continually increasing) c dy t 2 dx 2 dy 2 Length parametrically: L = ∫ + dt t1 dt dt Line Integral of a Scalar Field: n = = ∆ L f x zy ),,( ds lim ∑ f (xi , y y , z y ) si ∫C n ∞→ i=1 where x = tx ),( y = ty ),( z = tz ),( a ≤ t ≤ b dx 2 dy 2 dz 2 ds = + + dt dt dt dt 2 2 2 b dx dy dz L = ∫ f x zy ),,( ds = ∫ f x zy ),,( + + dt C a dt dt dt Line Integral of a Vector Field: W = F zyx ),,( •T ),,( dszyx ∫C where x = tx ),( y = ty ),( z = tz ),( a ≤ t ≤ b C = tr )( = )( itx + )( jty + )( ktz tr )(' tT )( = tr )(' dx 2 dy 2 dz 2 ds = + + dt = )(' dttr dt dt dt = • = b • tr )(' = b • W ∫ F zyx ),,( T ),,( dszyx ∫ F tr )(( )(' dttr ∫ F tr )(( )(' dttr C a tr )(' a Area of a Surface: m n = = * ∆ A ),,( dSzyxf lim ∑∑ f (Pij ) Sij ∫∫S ,nm ∞→ i=1j = 1 2 ∂z 2 ∂z A = ∫∫ ),,( dSzyxf = ∫∫ gyxf yx )),(,,( + +1dA S D ∂x ∂y Where D is the projection of the surface S on the (x,y) plane 13.27 HIGHER ORDER INTEGRATION Page 149 of 330 Properties of Double Integrals: Volume using Double Integrals: V = ),( dAyxf ∫∫R : If f(x, y) is continuous on the rectangle (Fubini’s Theorem) R = { yx |),( a ≤ x ≤ ,cb ≤ y ≤ d}, then b d V = ),( dAyxf = yxf ),( dxdy ∫∫R ∫∫ a c If f(x, y) is continuous on the region = { ≤ ≤ ≤ ≤ } R yx |),( a x b, g1 x)( y g2 x)( b 2 xg )( V = ),( dAyxf = yxf ),( dydx ∫∫R ∫ ∫ a 1 xg )( If f(x, y) is continuous on the region = { ≤ ≤ ≤ ≤ } R yx |),( g1 y)( x g2 y),( c y d b 2 yh )( V = yxf ),( A = yxf ),( dxdx ∫∫R ∫ ∫ a 1 yh )( Page 150 of 330 If a function is continuous on a polar region R = { r θ |),( a ≤ r ≤ b,α ≤ θ ≤ β} β b V = yxf ),( A = (rf cos θ,r sin θ )rdrd θ ∫∫R ∫∫ α a Volume using Triple Integrals: V = ),,( dVzyxf ∫∫∫B If f(x,y,z) is continuous on the rectangular box (Fubini’s Theorem) B = { zyx |),,( a ≤ x ≤ ,cb ≤ y ≤ ,rd ≤ z ≤ s}, then s d b V = ),,( dVzyxf = zyxf ),,( dxdydz ∫∫∫B ∫∫∫ r c a If f(x,y,z) is represented as a projection = { ∈ ≤ ≤ } B zyx yx ),(|),,( D,u1 yx ),( z u2 yx ),( , then 2 yxu ),( V = ∫∫∫ ),,( dVzyxf = ∫∫ ∫ ),,( dzzyxf dA B D 1 yxu ),( Page 151 of 330 If f(x,y,z) is continuous on the region = { ≤ ≤ ≤ ≤ ≤ ≤ } B zyx |),,( a x , gb 1 x)( y g2 x),( u1 yx ),( z u2 yx ),( b 2 xg )( 2 yxu ),( V = ∫∫∫ ),,( dVzyxf = ∫ ∫ ∫ zyxf ),,( dzdydx B a 1 xg )( 1 yxu ),( If a function is continuous within the cylindrical coordinate system = { θ α ≤θ ≤ β θ ≤ ≤ θ ≤ ≤ } B r z |),,( ,h1 )( r h2 ),( u1 yx ),( z u2 yx ),( β h θ )( ru θ r θ )sin,cos( V = ),,( dVzyxf = 2 2 (rf cos θ,r sin θ z), rdzdrd θ ∫∫∫ ∫α ∫θ ∫ θ θ B h1 )( 1 ru r )sin,cos( If a function is continuous within the spherical coordinate system Page 152 of 330 = { ρ φ θ θ ≤ θ ≤ θ θ ≤ φ ≤ θ φ θ ≤ ρ ≤ φ θ } B |),,( 1 2 ,h1 )( h2 ),( g1 ),( g2 ),( = ρ φ θ ρ φ θ ρ φ = ρ φ θ zyxf ),,( f ( sin cos , sin sin , cos ) f0 ),,( θ h θ )( g θφ )),( V = ),,( dVzyxf = 2 2 2 f ρ φ θ ),,( ρ 2 sin φdρdφdθ ∫∫∫ ∫θ ∫θ ∫ θφ 0 B 1 h1 )( g1 ),( Centre of Mass: Of a laminate: m = ρ ),( dAyx ∫∫R 1 x = ∫∫ x× ρ ),( dAyx m R 1 y = ∫∫ y × ρ ),( dAyx m R Of a general solid: m = ρ ),,( dVzyx ∫∫∫B 1 x = ∫∫∫ x× ρ ),,( dVzyx m B 1 y = ∫∫∫ y × ρ ),,( dVzyx m B 1 z = ∫∫∫ z × ρ ),,( dVzyx m B 13.28 WORKING IN DIFFERENT COORDINATE SYSTEMS: Cartesian: dA = dxdy Polar: dA = rdrd θ Cylindrical: dV = rdzdrd θ Page 153 of 330 Spherical: dV = ρ 2 sin φdρdφdθ Cartesian to Polar: r 2 = x 2 + y 2 y (2D) tan θ = x Polar to Cartesian: x = r cos θ (2D) y = rsin θ Cartesian to Cylindrical: r 2 = x2 + y 2 y tan θ = (3D) x z = z Cylindrical to Cartesian: x = r cos θ y = rsin θ (3D) z = z Spherical to Cartesian: x = ρ sin φ cos θ y = ρ sin φ sin θ (3D) z = ρ cos φ Page 154 of 330 PART 14: FUNCTIONS 14.1 ODD & EVEN FUNCTIONS: Definitions: Even: f (− x) = (xf ),∀x Odd: f (− x) −= (xf ),∀x Composite Functions: Odd ± Odd = Odd Odd ± Even = Neither Even ± Even = Even Odd x Odd = Even Odd / Odd = Even Even x Even = Even Even / Even = Even Even of Odd = Even Even of Even = Even Even of Neither = Neither Odd of Odd = Odd Odd of Even = Even Odd of Neither = Neither Basic Integration: a If f(x) is odd: ∫ )( dxxf = 0 −a a a If f(x) is even: ∫ )( dxxf = 2∫ )( dxxf −a 0 14.2 MULTIVARIABLE FUNCTIONS: ( )= ( )= ( ) Limit: lim f yx ),( lim f mxx ),( lim f mxx ),( yx → )0,0(),( mxx → )0,0(),( x → )0()( 2 Discriminant: D() = z z − (z ) , yx 00 yyxx xy = Critical Points: z f yx ),( = zx 0 Solve for: = z y 0 If the critical point (x 0,y 0) is a local maximum, then D(x 0,y 0) >= 0 fxx (x 0,y 0) <= 0 and fyy (x 0,y 0) <= 0 If D(x 0,y 0) > 0 , and either fxx (x 0,y 0) < 0 or fyy (x 0,y 0) < 0 then the critical point (x 0,y 0) is a local maximum. Page 155 of 330 If the critical point (x 0,y 0) is a local minimum, then D(x 0,y 0) >= 0 fxx (x 0,y 0) >= 0 and fyy (x 0,y 0) >= 0 If D(x 0,y 0) > 0 , and either fxx (x 0,y 0) > 0 or fyy (x 0,y 0) > 0 then the critical point (x 0,y 0) is a local minimum. If the critical point (x 0,y 0) is a saddle point, then D(x 0,y 0) <= 0 If D(x 0,y 0) < 0 , then the critical point (x 0,y 0) is a saddle point. 14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS: Separable : dy = xf )( dx g y)( g )( dyy = )( dxxf ∫ g )( dyy = ∫ )( dxxf Linear : dy + P x)( × y = Q x)( dx ∫ )( dxxP xI )( = e 1 y = ()xI )( ×Q )( dxx xI )( ∫ Homogeneous : f (λx,λy) = yxf ),( dy y = yxf ),( = F dx x y dy dv Let xv )( = ,∴ = v + x x dx dx Page 156 of 330 dv ∴v + x = F v)( dx dv x = F v)( − v dx dv = dx F v)( − v x dv = dx ∫ F v)( − v ∫ x Exact : dy = yxf ),( → M ),( dxyx + N ),( dyyx = 0 dx = If: M y N x = = When: FX M & FY N Therefore, F = ∫ M x,( y)dx =Φ x,( y) + g ( y) ∂ F = ()Φ + g ( y) = Φ + g (' y) = N Y ∂y Y ∴ g ( y) = ... So: F( , yx ) = Φ yx ),( + g y)( = C Bernoulli Form: dy + P()()yx = Q yx n dx Let: v = y1−n dv − dy ∴ = ()1− yn n dx dx n ∴ dy = y dv dx 1− n dx y n dv ∴ + P()()yx = Q yx n 1− n dx 1 dv − ⇒ + P()()yx 1 n = Q x 1− n dx dv ⇒ + ()()()()1− n P vx = 1− n Q x dx This is a linear 1 st ODE Page 157 of 330 dv + P x)( ×v = Q x)( dx 2 2 ∫ 2 )( dxxP xI )( = e 1 v = ()xI )( ×Q )( dxx xI )( ∫ 2 ()()− ()1−n () 1 1 ∫ 1 dxxPn y = v = 1−n ()xI )( ×Q )( dxx = ()1−n e ×()()1− n Q dxx ∫ 2 ∫ ()()1− dxxPn ∫ xI )( e 14.4 SECOND ORDE, FIRST DEGREE, DIFFERENTIAL EQUATIONS: d 2 y dy a + b + cy = xf )( Where dx 2 dx ay '' +by '+cy = xf )( Greg’s Lemma: (D −α )( ( )exp αx )= p'( )ex αx ()D −α ()xe αx = eαx ()D −α 2 ()2ex αx = 2eαx ()D −α n ()nex αx = !en αx Homogeneous : ay '' +by '+cy = 0 ⇒ am 2 + bm + c = 0 − b ± b2 − 4ac m = 2a There are three possible outcomes: ≠ = 1xm + 2xm = { 1xm 2 xm } 1) m1,m2 where m1 m2 ⇒ yh Ae Be S e ,e = = ( + ) 1xm = { 1xm 1xm } 2) m1,m2 where m1 m2 ⇒ yh A Bx e S e , xe = α ± β = αx ( (β ) + (β )) = { αx (β ) αx (β )} 3) m 2,1 j ⇒ yh e Acos x B sin x S e cos x , e sin x Undetermined Coefficients ay '' +by '+cy = xf )( , where xf )( is in the form of 1) A polynomial 2) α sin (kx ) 3) αekt NB: Multiplication is OK: eg: xf )( = 3 3ex x Step 1: Solve for the homogeneous case ay '' +by '+cy = 0 ∴ = + yh yc 11 yc 22 = {} S y1, y2 Step 2: Determine the spanning set Page 158 of 330 = n + n−1 + + = { n n−1 } 1) A polynomial ⇒ y p An x An−1x 1xA A0 T x , x ,... α ( ) = + = { ( ) ( )} 2) sin kx ⇒ y p Asin( kx ) B cos( kx ) T cos kx sin, kx α kt = kt = { kt } 3) e ⇒ y p Ae T e NB: Multiplication is OK: eg: xf )( = 3 3ex x = (x )(3 + 2 + + ) y p Ae Bx Cx Dx E = (x )(3 + 2 + + ) y p e Bx Cx Dx E Step 3: Determine the form of y p with arbitary coefficients Case 1: When S ∩ T = Φ Then y p is a linear combination of the elements of T Case 2: When S ∩ T ≠ Φ (ie: y p is part of y h) Multiply each element in T by x to get a new set T’ and then recheck if S ∩ T '= Φ . This is now a case 1, otherwise repeat the process until S ∩ T '= Φ Step 4: Determine the coefficients of y p by substituting y p y p y '',', p back into the orginal differential equation and comparing the coefficients with xf )( Then, = + + + + y yh y p1 y p2 y p3 ... If the intital values are given, go on to solve for the constants within the homogeneous part. Variation of Parameters d 2 y dy + b + cy = xf )( Where dx 2 dx and the method of undetermined coefficients is not suitable. y '' +by '+cy = xf )( Step 1: Solve for the homogeneous case ay '' +by '+cy = 0 ∴ = + yh yc 11 yc 22 Step 2: Assume that there is a solution of the differential equation of the form: = + = + y v1 )( ux 1 x)( v2 )( ux 2 x)( uv 11 uv 22 + = + = Such that v1 )(' ux 1 x)( v2 )(' ux 2 x)( v 'u11 v2 'u2 0 Therefore, = + y uv 11 uv 22 = + + + = + y' v 'u11 uv 11 ' v2 'u2 uv 22 ' uv 11 ' uv 22 ' = + + + y '' uv 11 '' v u11 '' v2 u2 '' uv 22 '' Substituting into the differential equation: = ( + + + )+ ( + )+ ( + ) xf )( uv 11 '' v u11 '' v2 u2 '' uv 22 '' uvb 11 ' uv 22 ' uvc 11 uv 22 = + + + + + + + xf )( u1 '' v1 bu 'v11 cu v11 u2 '' v2 bu 2 'v2 cu v22 u v11 '' u2 v2 '' = ()()+ + + + + + + xf )( u1 '' bu 1 ' cu v11 u2 '' bu 2 ' cu 2 v2 u v11 '' u2 v2 '' = + xf )( u v11 '' u2 v2 '' Page 159 of 330 + + = + + = As u1 '' bu 1 ' cu 1 0 & u2 '' bu 2 ' cu 2 0 + = + = We know that vu 11 ' vu 22 ' 0 and u v11 '' u2 v2 '' xf )( Step 3: Solving the equations above by Cramer’s Rule. u1 u2 v1 ' 0 = () u1 ' u2 'v2 ' xf 0 u2 () xf u2 ' − ()uxf v '= = 2 1 () u1 u2 W u1,u2 u1' u2 ' u1 0 () u1 ' xf ()uxf v '= = 1 2 () u1 u2 W u1,u2 u1' u2 ' Step 4: 0 u2 () xf u2 ' − ()uxf v ()x = v 'dx = dx = 2 dx 1 ∫ 1 ∫ ∫ () u1 u2 W u1,u2 u1 ' u2 ' u1 0 () u1' xf ()uxf v ()x = v 'dx = dx = 1 dx 2 ∫ 2 ∫ ∫ () u1 u2 W u1,u2 u1 ' u2 ' = + yh uc 11 x)( uc 22 x)( = + y p v1 )( ux 1 x)( v2 )( ux 2 x)( = + y yh y p Euler Type Of the form: ax 2 y '' +bxy '+cy = ,0 x > 0 (b −1) c Characteristic Equation: (ra )(r −1)+ (rb )+ c = 0 or r 2 + r + y = 0 a a There are three possible outcomes: ≠ = r1 + r2 > 1) ,rr 21 where r1 r2 ⇒ yh C1x C2 x , x 0 = = r1 + r1 ( ) > 2) ,rr 21 where r1 r2 ⇒ yh C1x C2 x ln x , x 0 = α ± β = α ( (β ( ))+ (β ( ))) > 3) r 2,1 j ⇒ yh x Acos ln x Bsin ln x , x 0 Reduction of Order + + = = Of the form: f1 yx '')( f 2 yx ')( f3 )( yx 0 where a solution is known y1 g x)( Page 160 of 330 y = gxv x)()( ∴ y'= v gx x)()(' + gxv x)(')( Therefore, let ∴ y '' = v gx x)()('' + v gx x)(')(' + v gx x)(')(' + gxv x)('')( ∴ y '' = v gx x)()('' + v gx x)(')('2 + gxv x)('')( Substituting Yields: ( + + )+ ( + )+ ( ) = f1 x v gx x)()('')( v gx x)(')('2 gxv x)('')( f 2 x v gx x)()(')( gxv x)(')( f3 x)( gxv x)()( 0 This should cancel down to only include v'& v '' ( + )+ ( ) = f1 x v gx x)()('')( v gx x)(')('2 f 2 x v gx x)()(')( 0 Therefore, let u = v'& u'= v '' ( + )+ ( ) = f1 x u x g x)()(')( u x g x)(')(2 f2 x)( u x g x)()( 0 + + = f1 ux x g x)()(')( 2f1 ux x g x)(')()( f2 ux x g x)()()( 0 ()()+ + = u x)(' f1 x g x)()( u x 2)( f1 x g x)(')( f2 x g x)()( 0 −()+ u x)(' 2f1 x g x)(')( f2 x g x)()( = () u x)( f1 x g x)()( −()+ u x)(' 2f1 x g x)(')( f2 x g x)()( ∫ du = ∫ () dx u x)( f1 x g x)()( −()+ () 2f1 x g x)(')( f2 x g x)()( ln u x)( = ∫ () dx f1 x g x)()( −()+ 1 2 xgxfxgxf )()()(')(2 ∫ () dx u x)( =e 1 xgxf )()( −()+ 1 2 xgxfxgxf )()()(')(2 ∫ () dx v x)(' =e 1 xgxf )()( −()+ 1 2 xgxfxgxf )()()(')(2 ∫ () dx xv )( = ∫e 1 xgxf )()( dx −()+ 1 )(')(2 2 xgxfxgxf )()( ∫ () dx y = ∫e 1 xgxf )()( dx ×g x)( = + General solution is of the form: y yc 11 yc 22 Note: Constants can be combined into a single constant to simplify working if required. Power Series Solutions: 2 () d y + () dy + () = P x 2 Q x R yx 0 Where dx dx about a point x0 P()()()yx '' +Q yx '+R yx = 0 Case 1: About an Ordinary Point ( ) ≠ Where P x0 0 and analytic at x0 ∞ ∞ ∞ = ()()()()− n = − n−1 = − − n−2 Assume y ∑ an x x0 y', ∑ na n x x0 y '', ∑ nn 1 an x x0 n=0 n=1 n=1 • Substitute into the differential equation. • Adjust the indicies in terms of xn • Obtain a recurance relation of an in terms of a0 & a1 Page 161 of 330 • Express an in terms of a0 & a1 ∞ = ()− n = + y ∑ an x x0 ya 10 ya 21 n=0 Case 2: About a Regular Singular Point Q(x) 2 R(x) Where P(x ) = 0 , lim ()x − x exists and lim ()x − x exists. 0 → 0 → 0 xx 0 P()x xx 0 P()x = For convienance, taking about x0 0 Q(x) ∞ x = xp ()x = xp n () ∑ n P x n=0 R()x ∞ x2 = 2 ()xqx = xq n () ∑ n P x n=0 This is convergent on x < R x2 y '' + (xpx (x))y'+( 2 (xqx ))y = 0 ∞ ∞ 2 + n + n = x y '' x∑ n xp y' ∑ n xq y 0 n=0 n=0 2 + ()()+ + 2 + + + + 2 + = x y '' px 0 1xp 2 xp ... y' q0 1xq 2 xq ... y 0 = → As near x0 0 , x 0 , the equation behaves like 2 + + = x y '' xp 0 y' 0 yq 0 This is an Euler form. ( − )+ + = = ≥ Indicial Equation: rr 1 0rp q0 0 , Thus r ,rr 21 | r1 r2 ∞ = r ()+ + 2 + = +nr ≠ > Therefore, y x a0 1xa 2 xa ... ∑ n xa for a0 ,0 x 0 n=0 From the indicial equation, there are three different forms of the general solution. { − }∉{Ζ} Form 1: r1 r2 ∞ = − r1 ()− n y1 x x0 ∑an x x0 n=0 ∞ = − r2 ()− n y2 x x0 ∑bn x x0 n=0 = + y yc 11 yc 22 = Form 2: r1 r2 ∞ = r1 + n y1 x 1 ∑ n 1xra n=1 ∞ = ()+ r1 + n y2 y1 ln x x 1 ∑ n 1xrb n=1 = + y yc 11 yc 22 Where the coefficients of the series are found by substituting into the differential equation. { − }∈{Ζ} Form 3: r1 r2 Page 162 of 330 ∞ = r1 + n y1 x 1 ∑ n 1xra n=1 ∞ = ()+ r2 + n y2 ay 1 ln x x 1 ∑ n 2 xrb n=1 = + y yc 11 yc 22 Where the coefficients of the series are found by substituting into the differential equation. Case 3: About an Irregular Singular Point Q(x) 2 R(x) Where P(x ) = 0 , lim ()x − x does not exist or lim ()x − x does not exist. 0 → 0 → 0 xx 0 P()x xx 0 P()x 14.5 ORDINARY DIFFERENTIAL EQUATIONS USING MATRICES: Derivation of Methods: • = + + + x 1 t)( a 111 tx )( a12 x2 t)( ... a1 xnn t)( • x 2 t)( = a tx )( + a x t)( + ... + a x t)( • • 121 22 2 2 nn ⇒ x t)( = txA )( ⇒ x = xA ... • = + + + x n t)( an 11 tx )( an x22 t)( ... ann xn t)( • x 1 t)( • x 2 t)( where A = []a , x = ∈ℜn ij ... • x n t)( ∴ = + + + x 1 xc 1 2 xc 2 ... n xc n c1 c2 x = []x | x |...| x 1 2 n ... cn tx )( = Φ )( ct Fundamental Matrix: λ λ λ 1t 2t nt v11 e v12 e ... vn1e λ λ λ 1t 2t nt v21 e v22 e ... vn2e General Solution: tx )( = Φ )( ct = c ...... λ λ λ 1t 2t nt vn1e vn2e ... vnn e = Φ−1 Particular Solution: c t 0)( x0 Homogeneous Solution: • x = xA General Solution: x = e At c = Pe Jt P −1 c Page 163 of 330 x1 )0( x2 )0( Particular Solution: c = ... xn )0( Inhomogeneous Solution: • x (t) = (txA )+ (tu ) − General Solution: ()()tx = Φ ct + Φ()()t ∫Φ 1 ()dttut ()()tx = Φ ct + Φ()()t ∫Φ − ()dttut x1 )0( x2 )0( Particular Solution: c = ... xn )0( nth Order linear, constant coefficient ODE: Page 164 of 330 Page 165 of 330 14.6 APPLICATIONS OF FUNCTIONS Terminology: Gradient Vector of a Scalar Field: ∂f ∂f ∂f ∂f ∂f ∂f grad zyxf )],,([ = ∇ zyxf ),,( = , , = i + j + k ∂x ∂y ∂z ∂x ∂x ∂x Directional Derivatives: = + + = For a function zyxf ),,( and unit vector u 1iu u2 j 3 ku , u 1, the directional derivative of f = ( ) at the point P0 x0 , y0 , z0 in the domain of the function and in the direction u is: ( + + + )− = xf 0 hu 1, y0 hu 2 , z0 hu 3 (xf 0 , y0 , z0 ) Du (xf 0 , y0 , z0 ) lim h→0 h = ∇ • Du (xf 0 , y0 , z0 ) zyxf ),,( u Optimising the Directional Derivative: • The function zyxf ),,( increase most rapidly at any point P in its domain in the direction of the gradient vector ∇f P)( . The directional derivative in this direction is ∇f P)( . Therefore, the gradient ∇f always points in the direction of the most rapid increase of the function, which is referred to as the steepest ascent direction. Thus, if we want to maximize a function, it is important to move in the gradient direction. • The function zyxf ),,( decreases most rapidly at any point P in its domain in the negative direction of the gradient vector ∇f P)( . The directional derivative in this direction is − ∇f P)( . Therefore, the antigradient − ∇f always points in the direction of the most rapid decrease of the function, which is referred to as the steepest descent direction. Thus, if we want to minimize a function, it is important to move in the negative gradient direction. • The rate of change is zero in the direction perpendicular to ∇f P)( 14.7 ANALYTIC FUNCTIONS If functions are analytic at a point x 0: Analytic ± Analytic = Analytic Analytic x Analytic = Analytic Analytic / Analytic = Analytic Page 166 of 330 PART 15: MATRICIES 15.1 BASIC PRINICPLES: Size = i × j , i=row, j=column = [ ] A aij 15.2 BASIC OPERTAIONS: + = [ + ] Addition: A B aij bij − = [ − ] Subtraction: A B aij bij = [ ] Scalar Multiple: kA ka ij [ T ] = Transpose: A ij Aji eg: (A + B + C + ... )T = AT + B T + C T + ... ()ABCD ... T = ... D T C T B T AT b1 b Scalar Product: a • b = []a a a ... 2 1 2 3 b 3 ... Symmetry: AT = A Cramer’s Rule: Ax = B det( A ) where Ai = column i replaced by B x = i i det( A) Least Squares Solution −1 In the form xA = b , x = (AT A) AT b + = For a linear approximation: r0 1xr b + + 2 = For a quadratic approximation: r0 1xr 2 xr b Etc. + + When columns are not Linearly Independent: x = A b = VS U T b 15.3 SQUARE MATRIX: Page 167 of 330 Diagonal: Lower Triangle Matrix: Upper Triangle Matrix: 15.4 DETERMINATE: 2x2: det ( A) = ad − bc 3x3: det ( A) = aei + bfg + cdh − afh − bdi − ceg n n + + = = × ()− ()1+ j nxn: det ( A) = a11 C11 a12 C12 a1n C1n ∑ a1 Cijj ∑ a1 j Mi 1 j 1 j=1 j=1 Rules : Page 168 of 330 Page 169 of 330 15.5 INVERSE −1 a b 1 d − b 2x2: = c d ad − bc − c a − a b c 1 ei − fh ch − bi bf − ce 1 3x3: d e f = fg − di ai − cg cd − af aei − afh − bdi + bfg + cdh − ceg g h i dh − eg bg − ah ae − bd Minor: Mij = Determinate of Sub matrix which has deleted row i and column j a b c = A d e f g h i b c M 21 = h f Page 170 of 330 ( + ) Cofactor: Cij = Mij × (−1) ji Adjoint Method for Inverse: adj A)( = CT − 1 A 1 = adj A)( det( A) Left Inverse: CA = I −1 C = ()AT A AT (when rows(A)>columns(A)) Right Inverse: AC = I −1 C = AT ()AA T (when rows(A) Pseudo inverse: For any matrix A, dim(A) = n x m. A+ is the pseudo inverse. dim(A +) = m x n + + A = USV T , A = VS U T dim( A) = m× n;dim( A+ ) = n× m A× A+ × A = A A+ × A× A+ = A+ T ()A× A+ = A× A+ T ()A+ × A = A+ × A 15.6 LINEAR TRANSFORMATION Page 171 of 330 Axioms for a linear transformation: If F(u + v) = F u)( + F v)( [Preserves Addition] And F(λu) = λF u)( [Preserves Scalar Multiplication] Transition Matrix: The matrix that represents the linear transformation = + + + T v)( 1Tc (v1 ) 2Tc (v2 ) ... nTc (vn ) T x)( = Ax = [ ] A T(e1 T(|) e2 T(|...|) e3 ) (With m columns and n rows) (T :V →W ,dim( V ) = m,dim( W ) = n ) Zero Transformation: T v)(= ,0 ∀vεV Identity Transformation: T v)( = v,∀vεV 15.7 COMMON TRANSITION MATRICIES Rotation (Clockwise): Rotation (Anticlockwise): Scaling: Shearing (parallel to x-axis): Shearing (parallel to y-axis): 15.8 EIGENVALUES AND EIGENVECTORS Definitions: All solutions of Ax = λx Eigenvalues: All solutions of λ of det(A-λI)=0 Eigenvectors: General solution of [A-λI][X]=0 (ie: the nullspace) Characteristic Polynomial: The function p λ)( = det( A − λI) Page 172 of 330 Algebraic Multiplicity: The number of times a root is repeated for a given eigenvalue. ∑ of all algebraic multiplicity = degree of the characteristic polynomial. Geometric Multiplicity: The number of linearly independent eigenvectors you get from a given eigenvalue. T :V →V Transformation: T x)( = λx The same process for an ordinary matrix is used. Linearly Independence: The set of eigenvectors for distinct eigenvalues is linearly independent. Digitalization: For a nxn matrix with n distinct eigenvalues; if and only if there are n Linearly Independent Eigenvectors: D = P −1 AP = [ ] Where P P1 | P2 |...| Pn , Pn is an eigenvector. λ 1 0 0 ... 0 λ 0 2 0 ... 0 D = 0 0 λ ... 0 3 ...... 0 λ 0 0 0 0 n Cayley-Hamilton Theorem: Every matrix satisfies its own polynomial: n n−1 P λ)( = a λ + a − λ ... + a λ +a = 0 n n 1 1 0 λ = n + n−1 + + = P )( an A an−1 A ... a1 A a0 0 Orthonormal Set: The orthonomal basis of a matrix A can be found = [ ] with P P1 | P2 |...| Pn , the orthonormal set will be P P P B = 1 , 2 ,... n P1 P2 Pn QR Factorisation: A = [u | u |...| u ] = QR 1 2 n dim( A) = n × , kk ≤ n All columns are Linearly Independent = [ ] Q v1 | v2 |...| vn by the Gram-Schmidt Process Page 173 of 330 T T T T q1 u2 v1 u3 v1 u4 v1 ... uk v1 T T T 0 q2 u3 v2 u4 v2 ... uk v2 0 0 q u T v ... u T v = 3 4 3 k 3 R T 0 0 0 q4 ... uk v4 ...... 0 0 0 0 ... qk • • • • q1 u2 v1 u3 v1 u4 v1 ... uk v1 • • • 0 q2 u3 v2 u4 v2 ... uk v2 0 0 q u • v ... u • v R = 3 4 3 k 3 • 0 0 0 q4 ... uk v4 ...... 0 0 0 0 ... qk 15.9 JORDAN FORMS Generalised Diagonlisation: P −1 AP = J A = PJP −1 Jordan Block: λ 1 0 ... 0 0 λ 0 1 ... 0 0 0 0 λ ... 0 0 = J B ...... 0 0 0 ... λ 1 0 0 0 ... 0 λ J1 0 ... 0 0 J ... 0 Jordan Form: J = 2 ...... 0 0 ... J n Algebraic Multiplicity: The number of times λ appears on main diagonal Geometric Multiplicity: The number of times λ appears on main diagonal without a 1 directly above it = { } Generalised Chain: um ,um−1,..., u2 ,u1 , where u1 is an eigenvector u = (A − λ )uI + k k 1 = []− λ uk+1 A I | uk = [ ] P P1 | P2 |...| Pm ...| , for every eigenvector of A Page 174 of 330 Powers: Ak = PJ k P −1 k k J1 0 ... 0 J 0 ... 0 1 0 J ... 0 0 J k ... 0 J k = 2 = 2 ...... ...... k 0 0 ... J n 0 0 ... J n k k k−1 k k−2 λ λ λ ... 1 2 k k = λk λk−1 J B 0 ... 1 0 0 λk ... ...... 15.12 SINGULAR VALUE DECOMPOSITION Fundamentally: A = USV T , U and V are orthogonal Size: dim( A) = m× n dim( U ) = m× m dim( S) = m× n dim( V ) = n× n + + Pseudo inverse: A = VS U T Procedure: STEP 1: Find the symmetric matrix AT A . This has orthogonal diagonalisation. STEP 2: Find a set of eigenvalues of AT A . These are all non-negative. λ = ,,, dcba ... STEP 3: Arrange eigenvalues in decreasing order λ > λ > λ > 1 2 3 ... STEP 4: Find σi σ = λ i i ∴σ > σ > σ > 1 2 3 ... STEP 5: Find S dim(S) = m x n S 0 = 1 S where S 1 is a matrix with the diagonal equal to σ ,σ ,σ ,... and all other 0 0 1 2 3 elements 0. STEP 6: Find a set of eigenvectors of AT A Page 175 of 330 = p1, p2 , p3 ,... STEP 7: Normalise each eigenvector ^ = = pi vi pi pi STEP 8: Form V = [ ] V v1 | v2 | v3 ...| STEP 9: Generate first columns of u corresponding to non-zero eigenvalues 1 x = Av i σ i i STEP 10: Generate remaining columns of U such that it is an orthogonal square matrix. x1 x2 x + = 0 ... a 1 xa (ie: xa+1 belongs to the nullspace of the matrix that is made from the found vectors of x arranged in rows.) STEP 11: Normalise each vector of U = xi ui , this only needs to be done for xa+1 onwards xi STEP 12: Form U = [ ] U u1 | u2 | u3 ...| 15.11 COMPLEX MATRICIS: Conjugate Transpose: A∗ = AT ∗ A∗ = A ∗ ∗ ∗ ()A + B = A + B ∗ ()zA = Az ∗ ∗ ()AB = B A∗∗ Hermitian Matrix: (Similar to Symmetric Matricis in the real case) A square matrix such that A *=A Eigenvalues of A are purely real Eigenvectors from distinct eigenvalues are orthogonal. This leads to a unitary digitalisation of the Hermitian matrix. These are normal Skew-Hermitian: A square matrix such that A *=-A Eigenvalues of A are purely imaginary Eigenvectors from distinct eigenvalues are orthogonal. Page 176 of 330 ∗ ∗ If A is Skew-Hermitian, iA is normal as: (iA ) = Ai = (− i)(− A) = iA These are normal Unitary Matrix: (Similar to Orthogonal Matricis in the real case) A square matrix such that A*A=I Columns of A form an orthonormal set of vectors Rows of A from an orthonormal set of vectors Normal Matrix: ∗ ∗ Where AA = A A These will have unitary diagonalisation ∗ ∗ All Hermition and Skew-Hermitian matricis are normal ( A A = AA = AA ) Diagonalisation: For a nxn matrix with n distinct eigenvalues; if and only if there are n Linearly Independent Eigenvectors: − D = P 1 AP = [ ] Where P P1 | P2 |...| Pn , Pn is an eigenvector. λ 1 0 0 ... 0 λ 0 2 0 ... 0 D = 0 0 λ ... 0 3 ...... 0 λ 0 0 0 0 n − ∗ If A is Hermitian, D = P 1 AP = P AP as P are an orthonormal set of vectors. Spectral Theorem: For a nxn Normal matrix and eigenvectors form an orthonormal set = [ ] P P1 | P2 |...| Pn = λ * + λ * + + λ * A PP 111 P P222 ... P Pnnn Therefore, A can be represented as a sum of n matricis, all of rank 1. Therefore, A can be approximated as a sum of the dominant eigenvalues 15.12 NUMERICAL COMPUTATIONS: Rayleigh Quotient: if ( λ;v ) is an eigenvalue/eigenvector pair of A, then Page 177 of 330 Page 178 of 330 Power method: If A is a nxn matrix with Linearly Independent Eigenvectors, and distinct eigenvectors λ ≥ λ ≥ ≥ λ { } arranged such that: 1 2 ... n and the set of eigenvectors are: v1,v2 ,..., vn Any vector “w” can be written as: = + + + w0 vc 11 vc 22 ... vc nn = = + + + = λ + λ + + λ w1 Aw 0 1 Avc 1 c2 Av 2 ... cn Av n c v111 c v222 ... c vnnn λ s λ s w = Aw = c λ sv + c λ sv + ... + c λ sv = λ vc + c 2 v +... + c n v s s−1 111 222 nn n 1 11 2 λ 2 n λ n 1 1 λ λ s i i As <1, lim = 0 λ s→∞ λ 1 1 ∴ → λ s ws c v111 Appling this with the Rayleigh Quotient: 1 w − = s 1 λ = can be any vector usually ws A , R(ws ), w0 0 ws−1 ... 15.13 POWER SERIES: 2 3 k t At A A A Jt −1 e = I + A + + + ... + + ... = Pe P !2 !3 k! λ e 1t 0 ... 0 t λ 2 3 k 2t Dt D D D 0 e ... 0 e = I + D + + + ... + + ... = !2 !3 k! ...... λ 0 0 ... e nt e 1tJ 0 ... 0 t 2 3 k 2tJ Jt J J J 0 e ... 0 e = I + J + + + ... + + ... = !2 !3 k! ...... 0 0 ... e ntJ 2 λt λt t λt e te e ... 2 3 k t !2 J J J λt λt ntJ = + + n + n + + n + = 0 e te ... e I J n ...... !2 !3 k! λ 0 0 e t ... ...... Page 179 of 330 PART 16: VECTORS 16.1 BASIC OPERATIONS: + a1 b1 + = + Addition: a b a2 b2 + a3 b3 − a1 b1 − = − Subtraction: a b a2 b2 − a3 b3 = ⇔ = = = Equality: a b a1 b1 ,a2 b2 , a3 b3 ak + bl = λa + µb ⇒ k = λ,l = µ ka 1 Scalar Multiplication: = ak ka 2 ka 3 Parallel: a = bk ⇔ ba = ()2 + ()2 + ()2 Magnitude: a a1 a 2 a 3 ^ a Unit Vector: = a a Zero Vector: A vector with no magnitude and no specific direction Dot Product: a • b = a ⋅ b ⋅ cos θ • = + + a b a b11 a b 22 a b33 a • b Angle Between two Vectors: cos θ = a ⋅ b a b + a b + a b cos θ = 11 22 33 a 2 + a 2 + a 2 ⋅ b 2 + b 2 + b 2 1 2 3 1 2 3 a 1 a cos( α ) ^ a2 Angle of a vector in 3D: a = = cos( β ) a cos( γ ) a3 a Perpendicular Test: a • b = 0 Page 180 of 330 ^ Scalar Projection: a onto b: P = a • b ^^ 1 Vector Projection: a onto b: P = a •bb = ()a • bb b 2 × = − − − Cross Product: a b a b32 ba 23 , ba 13 ba 31 , ba 21 a b12 a×b = a ⋅ b ⋅sin θ ⋅ n a × b = a ⋅ b ⋅ sin θ a ×b −= b × a a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a × b) i j k a a a a a a × = = 2 3 − 1 3 + 1 2 a b det a1 a2 a3 i det j det k det b b b b b b 2 3 1 3 1 2 b1 b2 b3 16.2 Lines r = a + λb , where a is a point on the line, and b is a vector parallel to the line = + λ x a1 b1 = + λ y a2 b2 = + λ z a3 b3 x − a y − a z − a λ = 1 = 2 = 3 b1 b2 b3 16.3 Planes Generally: n • AR = 0 n • r = n • a n • r = k Where: n = ,, cba & r = ,, zyx : ax + by + cz = k Tangent Plane: We define the tangent plane at a point P ( x 0 , y 0 , z 0 ) on a surface S given by the equation zyxf ),,( = c as a plane which contains all tangent lines at P to curves in S through P. Therefore, → 0 = ∇ (xf , y , z )• PQ 0 0 0 = • − − − f x (x0 , y0 , z0 ), f y (x0 , y0 , z0 ), f z (x0 , y0 , z0 ) x x0 , y y0 , z z0 Page 181 of 330 Normal Line: The normal line to S at P is the line passing through P and perpendicular to the tangent plan. The equation of the normal line is: 16.4 Closest Approach Two Points: d = PQ ^ Point and Line: d = PQ × a ^ Point and Plane: d = PQ • n ^ Two Skew Lines: d = PQ • n = PQ • ()a × b [r (t)− r (t)]• [v − v ] = 0 Solving for t: b a b a []() • [] = r ba t v ba 0 16.5 Geometry AB × AC Area of a Triangle: A = 2 Area of a Parallelogram: A = AB × AC Area of a Parallelepiped: A = AD • (AB × AC ) 16.6 Space Curves Where: tr )( = )( itx + )( jty + )( ktz Page 182 of 330 Velocity: tv )( = tr )(' = )(' itx + )(' jty + )(' ktz Acceleration: ta )( = tv )(' = r t)('' = x )('' it + y )('' jt + z )('' kt Definition of “s”: The length of the curve from r to r+ r dr tr )(' T = = Unit Tangent: ds tr )(' T =1 dr dr ds Chain Rule: = × dt ds dt dr dr ds As =1, = =speed ds dt dt Normal: T • T =1 d ()T • T = 0 ds dT dT • T + T • = 0 ds ds dT 2T • = 0 ds dT T • = 0 ds As T is tangent to the curve, dT is normal ds Page 183 of 330 dT ds N = dT ds dT dT Curvature: = N = κN ds ds tr )(' × r t)('' tv )( × ta )( ∴κ = dT = = ds tr )(' 3 tv )( 3 Unit Binomial: B = T × N dB Tortion: τ = ds ABBREVIATIONS λ = a scalar value = a scalar value θ = the angle between the vectors a = a vector b = a vector k = a scalar value l = a scalar value n = the normal vector r = the resultant vector Page 184 of 330 PART 17: SERIES 17.1 MISCELLANEOUS n = + + + + + = General Form: Sn a1 a2 a2 a4 ... an ∑ an n=1 ∞ = + + + + = Infinite Form: S∞ a1 a2 a2 a4 ... ∑ an n=1 i = + + + + + = Partial Sum of a Series: Si a1 a2 a2 a4 ... ai ∑ an n=1 1 2 3 9 1 1 1 10 0.99…=1: 999.0 = 9 + 9 + 9 + ... = =1 1 10 10 10 1− 10 17.2 TEST FOR CONVERGENCE AND DIVERGENCE Test For Convergence: (S ) = L , if L exists, it is convergent lim n n ∞→ Test For Divergence: (a ) ≠ 0 lim n n ∞→ Geometric Series ∞ ≥ − Divergent , r 1 ar n 1 ∑ < n=1 Convergent , r 1 P Series ∞ 1 Divergent , p ≤1 ∑ p > n=1 x Convergent , p 1 The Sandwich Theorem ≤ ≤ If there is a positive series so that an bn cn If (a ) = (c ) = L then, (b ) = L lim n lim n , lim n n→∞ n→∞ n→∞ Hence, if an & cn are convergent, bn must also be convergent The Integral Test = If an f x)( if f x)( is continuous, positive and decreasing ∞ If S∞ or ∫ f x)( is true, then the other is true 1 1 1 a = = f = = f n n n)( x x)( Eg: ∞ ∞ ∴ = 1 = []∞ = ∫ f x)( dx ∫ dx ln x 1 D N E... 1 1 x Page 185 of 330 ∴ an is divergent The Direct Comparison Test If we want to test an , and know the behaviour of bn , where an is a series with only non-negative terms ≤ If bn is convergent and an bn , then an is also convergent The Limit Comparison Test ∞ a ∞ If there is a convergent series c , then if n < ∞ , then a converges ∑ n lim ∑ n n=1 n ∞→ cn n=1 ∞ a ∞ If there is a divergent series d , then if n > 0 , then a diverges ∑ n lim ∑ n n=1 n→∞ dn n=1 D’almbert’s Ratio Comparison Test FOR POSITIVE TERMS: a n+1 < Converges: lim 1 n ∞→ an an+1 > Diverges: lim 1 n ∞→ an an+1 = Not enough information: lim 1 n ∞→ an The nth Root Test ∞ For a where a ≥ 0 , then if n a , ∑ n n lim n n=1 n→∞ Converges: n a <1 lim n n→∞ Diverges: n a > 1 lim n n→∞ Not enough information: n a = 1 lim n n→∞ Abel’s Test: ∞ ∞ If ∑ an is positive and decreasing, and ∑ cn is a convergent series. n=1 n=1 ∞ × Then ∑an cn converges n=1 Negative Terms ∞ ∞ If ∑ an converges, then ∑an is said to be absolutely convergent n=1 n=1 Alternating Series Test ∞ ∞ = − n × This is the only test for an alternating series in the form ∑ an ∑( )1 bn n=1 n=1 Page 186 of 330 Let b be the sequence of positive numbers. If b < b and b = 0 , then the series is n n+1 n lim n n→∞ convergent. Alternating Series Error = − ≤ th Rn S sn bn+1 , where Rn is the error of the partial sum to the n term. 17.3 ARITHMETIC PROGRESSION: Definition: ,aa + d,a + d,2 a + 3d,... Nth Term: = a + d(n −1) n n Sum Of The First N Terms: ∑ a = ()2a + d()n −1 a=1 2 17.4 GEOMETRIC PROGRESSION: Definition: ,ara ,ar 2 ,ar 3 ,... Nth Term: = ar n−1 n ( − n ) = = a 1 r Sum Of The First N Terms: S n ∑ a a=1 1− r a(1− r n ) a Sum To Infinity: S = = (given r < 1) ∞ lim − − n ∞→ 1 r 1 r P, A,Q,... A Q Geometric Mean: = r, = r P A A Q ∴ = ⇒ A2 = PQ ⇒ A = PQ P A 17.5 SUMMATION SERIES n (nn +1) Linear: 1+2+3+4+… ∑ a = a=1 2 n (nn +1)(2n +1) Quadratic: 12+2 2+3 2+4 2+… ∑ a 2 = a=1 6 n (nn +1)2 Cubic: 13+2 3+3 3+4 3+… ∑ a 3 = a=1 2 17.6 APPROXIMATION SERIES Taylor Series ∞ ∞ n)( f x )( = − n = 0 − n = + − + − 2 + − 3 + f x)( ∑ an (x x0 ) ∑ (x x0 ) a0 a1 (x x0 ) a2 (x x0 ) a3 (x x0 ) ... n=0 n=0 n! n)( f x )( where, a = 0 n n! Page 187 of 330 Maclaurun Series = Special case of the Taylor Series where x0 0 Linear Approximation: 1 1 n)( f x )( ≈ = − n = 0 − n = + − f x)( L x)( ∑an (x x0 ) ∑ (x x0 ) a0 a1(x x0 ) n=0 n! n=0 Quadratic Approximation: 2 2 n)( f x )( ≈ = − n = 0 − n = + − + − 2 f x)( Q x)( ∑an (x x0 ) ∑ (x x0 ) a0 a1(x x0 ) a2 (x x0 ) n=0 n! n=0 Cubic Approximation: 3 3 n)( f x )( ≈ = − n = 0 − n = + − + − 2 + − 3 f x)( C x)( ∑ an (x x0 ) ∑ (x x0 ) a0 a1(x x0 ) a2 (x x0 ) a3 (x x0 ) n=0 n! n=0 17.7 MONOTONE SERIES > an+1 > Strictly Increasing: an+1 an 1 an ≥ Non-Decreasing: an+1 an < an+1 < Strictly Decreasing: an+1 an 1 an ≤ Non-Increasing: an+1 an Convergence: A monotone sequence is convergent if it is bounded, and hence the limit → ∞ exists when an 17.8 RIEMANN ZETA FUNCTION ∞ ζ () = 1 Form: n ∑ n k=1 k Euler’s Table: ∞ π 2 ζ () = 1 = + 1 + 1 + = n=2 2 ∑ 2 1 ... k=1 k 4 9 6 ∞ π 4 ζ () = 1 = + 1 + 1 + 1 + = n=4 4 ∑ 4 1 ... k=1 k 16 81 256 90 ∞ π 6 ζ () = 1 = + 1 + 1 + 1 + = n=6 6 ∑ 6 1 ... k=1 k 64 729 4096 945 π 8 n=8 ζ ()8 = 9450 Page 188 of 330 π 10 n=10 ζ ()10 = 93555 691 π 12 n=12 ζ ()12 = 638512875 2π 14 n=14 ζ ()14 = 18243225 3617 π 16 n=16 ζ ()16 = 3256415662 50 43867 π 18 n=18 ζ ()18 = 3897929548 0125 174611 π 20 n=20 ζ ()20 = 1531329465 290625 155366 π 22 n=22 ζ ()22 = 1344785694 0643125 236364091 π 24 n=24 ζ ()24 = 2019195719 6375652187 5 1315862 π 26 n=26 ζ ()26 = 1109448197 6030578125 Alternating Series: Proof for n=2: x3 x5 x7 Taylor Series Expansion: sin( x) = x − + − +... 3! 5! 7! Page 189 of 330 sin( x) = (xx − π )(x + π )(x − 2π )(x + 2π )... Polynomial Expansion: sin( x) = (xx 2 − π 2 )(x 2 − 4π 2 )(x 2 − 9π 2 )... 2 2 2 = − x − x − x sin( x) Ax 1 2 1 2 2 1 2 2 ... π 2 π 3 π sin( x) lim =1 = A x→0 x x3 x5 x7 x2 x 2 x2 − + − + = − − − x ... x1 2 1 2 2 1 2 2 .. !3 !5 !7 π 2 π 3 π 1 1 1 1 1 − −= − − − ... !3 π 2 22 π 2 32 π 2 42 π 2 Comparing the Coefficient of x 3: π 2 1 1 1 =1+ + + ... 6 22 32 42 17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS where denotes a Bernoulli number 17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS Where x ≠ 1 Page 190 of 330 (m < n) (geometric series starting at 1) (special case when x = 2) (special case when x = 1/2) where is the Touchard polynomials. 17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS Page 191 of 330 Page 192 of 330 [ 17.12 INFINITE SUMMATIONS TO PI 17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS ABBREVIATIONS a = the first term d = A.P. difference r = G.P. ratio 17.14 POWER SERIES EXPANSION Exponential: Page 193 of 330 Trigonometric: Page 194 of 330 Page 195 of 330 Exponential and Logarithm Series: x −1 , y = x +1 Page 196 of 330 Fourier Series: ∞ = a0 + + fW x)( ∑ak cos( kx ) bk sin( kx ) 2 = k 1 a f x)( = 0 + a cos( x) + a cos( x)2 +... + a cos( nx ) + b sin( x) + b sin( x)2 +... + b sin( nx ) W 2 1 2 n 1 2 n π 1 2 a = xf )( cos( kx )dx k = 2,1,0 ,..., n k π ∫ 0 π 1 2 b = xf )( sin( kx )dx k = 2,1 ,..., n k π ∫ 0 17.15 Bernoulli Expansion: A polynomial in n(n +1) k odd Fundamentally: 1k + 2k + 3k +... + nk = (2n +1) x A polynomial in n(n +1) k even Page 197 of 330 Expansions: 1 1+ 2 + 3+... + n = (nn + )1 2 1 1 1+ 2 + 3+... + n = n 2 + n 2 2 2 2 + + + + = 1 2 + 1 2 3 ... n B0n 1nB 2 0 1 1 12 + 22 + 32 +... + n2 = ()2n +1 (nn + )1 6 1 1 1 12 + 22 + 32 +... + n2 = n3 + n2 + n 3 2 6 3 3 3 2 + 2 + 2 + + 2 = 1 3 + 2 + 1 2 3 ... n B0n 1nB B2n 30 1 2 13 + 23 + 33 +... + n3 = ()1+ 2 + 3+... + n 2 1 13 + 23 + 33 +... + n3 = ()(nn + )1 2 4 1 1 1 13 + 23 + 33 +... + n3 = n4 + n3 + n2 4 2 4 4 4 4 4 3 + 3 + 3 + + 3 = 1 4 + 3 + 2 + 1 2 3 ... n B0n 1nB B2n B3n 4 0 1 2 3 1 14 + 24 + 34 +... + n4 = ()2n +1 (nn + ()(3)1 nn + )1 −1 30 1 1 1 1 14 + 24 + 34 +... + n4 = n5 + n 4 + n3 − n 5 2 3 30 5 5 5 5 5 4 + 4 + 4 + + 4 = 1 5 + 4 + 3 + 2 + 1 2 3 ... n B0n 1nB B2n B3n B4n 5 0 1 2 3 4 k +1 k +1 k +1 k +1 k +1 k + k + k + + k = 1 k+1 + k −+ 11 + k −+ 21 + + 2 + 1 2 3 ... n B0n 1nB B2n ... Bk−1n Bk n k +1 0 1 2 k −1 k List of Bernoulli Numbers: n B(n) 0 1 1 1 − 2 1 2 6 3 0 Page 198 of 330 1 4 − 30 5 0 1 6 42 7 0 1 8 − 30 9 0 5 10 66 11 0 691 12 − 2730 13 0 7 14 6 15 0 3617 16 − 510 17 0 43867 18 798 19 0 174611 20 − 330 Page 199 of 330 PART 18: ELECTRICAL 18.1 FUNDAMENTAL THEORY Charge: q = 24.6 ×10 18 Coulombs dq Current: I = dt ρl Resistance: R = A Ohm’s Law: V = IR V 2 Power: P =VI = I 2 R = R = Conservation of Power: ∑ PCONSUMED ∑ PDELIVERED t Electrical Energy: W = P×t = I 2 × R×t = ∫ Pdt 0 Kirchoff’s Voltage Law: The sum of the volt drops around a close loop is equal to zero. ∑V = 0 Kirchoff’s Current Law: The sum of the currents entering any junction is equal to the sum of the currents leaving that junction. = ∑ I IN ∑ IOUT T = 1 Average Current: I AVE ∫ )( dttI T 0 1 I = × Area (under I(t)) AVE T T 1 2 RMS Current: ∫()tI )( dt T 0 to Y Conversion: Page 200 of 330 + + = R R21 R R32 R R31 RA R1 + + = R R21 R R32 R R31 RB R2 + + = R R21 R R32 R R31 RC R3 18.2 COMPONENTS = + + + Resistance in Series: RT R1 R2 R3 ... 1 1 1 1 Resistance in Parallel: = + + + ... RT R1 R2 R3 = ω = π Inductive Impedance: X L j L j2 fL 1 1 Capacitor Impedance: X −= j −= j C ωC 2πfC 1 1 1 1 Capacitance in Series: = + + + ... CT C1 C2 C3 = + + + Capacitance in Parallel: CT C1 C2 C3 ... Voltage, Current & Power Summary: 18.3 THEVENIN’S THEOREM Thevenin’s Theorem: VTH = Open Circuit Voltage between a & b Page 201 of 330 RTH = Short Circuit any voltage source and Open Circuit any current source and calculate RTH as the resistance = VTH from a & b. With dependant sources, SC terminals a & b and calculate the current in the wire ( I SC ). RTH I SC ( )2 = VTH = Maximum Power Transfer Theorem: PMAX , where RL RTH 4RTH 18.4 FIRST ORDER RC CIRCUIT 18.5 FIRST ORDER RL CIRCUIT 18.6 SECOND ORDER RLC SERIES CIRCUIT Calculation using KVL: − + + + = V S V R V L V C 0 + + = V R V L V C V S di Ri + L + V = V di C S Circuit current: dV i = i = C C C dt di d 2V ∴ = C C dt dt 2 dV d 2V ∴RC C + LC C +V = V dt dt 2 C S d 2V dV LC C + RC C +V = V dt 2 dt C S d 2V R dV 1 V C + C + V = S dt 2 L dt LC C LC Important Variables 2 + α +ω 2 = Standard Format: s 2 s 0 0 1 R Damping Factor: α = 2 L Page 202 of 330 dV Natural Frequency: s = c dt 1 Undamped Natural Frequency: ω = 0 LC ω = ω 2 −α 2 Damping Frequency: d 0 ∆ = α 2 −ω 2 Mode Delta: 0 ( ) = + VC: VC t TRANSIENT FINAL Solving: R 1 s2 + s + = 0 L LC −= α ± α 2 −ω 2 −= α ± ∆ s 0 Mode 1: If: ∆ > 0 , then : s −= α ± ∆ = + V C t )( TRANSIENT FINAL TRANSIENT = Ae 1ts + Be 2ts = ∝ = FINAL V C ( ) V S = 1ts + 2ts + V C t )( Ae Be V S Finding A & B: + = − = VC )0( VC )0( V0 ∴ + + = → + = − A B VS V0 A B V0 VS dV c = As e 1ts + Bs e 2ts dt 1 2 dV + )0( i + )0( i + )0( i − )0( I C = C = L = L = 0 = As + Bs dt C C C C 1 2 V −V = A + B 0 S ∴ I 0 = As + Bs C 1 2 Mode 2: If: ∆ = 0 , then : s = −α = + VC t)( TRANSIENT FINAL TRANSIENT = ()()A + Bt est = A + Bt e−αt = ∝ = FINAL VC )( VS = ()+ −αt + VC t)( A Bt e VS Finding A & B: Page 203 of 330 + = − = VC )0( VC )0( V0 ∴ + = → = − A VS V0 A V0 VS dV c = ()A + Bt se st + Be st dt dV + )0( i + )0( i + )0( i − )0( I C = C = L = L = 0 = As + B dt C C C C V −V = A 0 S ∴ I 0 = As + B C Mode 3: ∆ < ω = ω 2 −α 2 If: 0 , and letting d 0 , then : −= α ± ω s j d = + VC t)( TRANSIENT FINAL = ()ω + ω −αt TRANSIENT Acos( d t) Bsin( d t) e = ∝ = FINAL VC )( VS = ()ω + ω −αt + VC t)( Acos( d t) Bsin( d t) e VS Finding A & B: + = − = VC )0( VC )0( V0 ∴ + = → = − A VS V0 A V0 VS dV −α −α c = ()()− Aω sin( ω t) + Bω cos( ω t) e t −α Acos( ω t) + Bsin( ω t) e t dt d d d d d d dV + )0( i + )0( i + )0( i − )0( I C = C = L = L = 0 = Bω −αA dt C C C C d V −V = A 0 S ∴ I 0 = Bω −αA C d Mode 4: If: R = 0 , then : α = ω = ω ,0 d 0 ±= ω ±= ω s j d j 0 = + VC t)( TRANSIENT FINAL = ω + ω TRANSIENT Acos( d t) Bsin( d t) = ∝ = FINAL VC )( VS = ω + ω + VC t)( Acos( d t) Bsin( d t) VS Page 204 of 330 Finding A & B: + = − = VC )0( VC )0( V0 ∴ + = → = − A VS V0 A V0 VS dV c −= Aω sin( ω t) + Bω cos( ω t) dt d d d d dV + )0( i + )0( i + )0( i − )0( I C = C = L = L = 0 = Bω dt C C C C d V −V = A 0 S ∴ I 0 = Bω C d Current through Inductor: dV i = i = C C L C dt Plotting Modes: Mode 1: Over Damped Mode 2: Critically Damped Mode 3: Sinusoidal Damped Page 205 of 330 Mode 4: Not Damped (Oscillates indefinitely) 18.7 SECOND ORDER RLC PARALLEL CIRCUIT Calculation using KCL: = + + i S i R i L i C V dV i = + i + C S R L dt Node Voltage: di V = L L = V L dt dV 2id = L L dt dt 2 L di 2id ∴i = L + i + LC L S R dt L dt 2 2id L di LC L + L + i = i dt 2 R dt L S 2id 1 di 1 1 L + L + i = i dt 2 RC dt LC L LC S Important Variables 2 + α +ω 2 = Standard Format: s 2 s 0 0 1 1 Damping Factor: α = 2 RC 1 Undamped Natural Frequency: ω = 0 LC ω = ω 2 −α 2 Damping Frequency: d 0 ∆ = α 2 −ω 2 Mode Delta: 0 Page 206 of 330 Solving: 1 1 s 2 + s + = 0 RC LC −= α ± α 2 −ω 2 −= α ± ∆ s 0 18.8 LAPLANCE TRANSFORMATIONS Identities: Page 207 of 330 Properties: Page 208 of 330 18.9 THREE PHASE – Y = × Line Voltage: VLINE VPHASE 3 V Phase Voltage: V = LINE PHASE 3 = Line Current: I LINE I PHASE = Phase Current: I PHASE I LINE S = 3 ×V × I Power: LINE LINE = × × S 3 VPHASE I PHASE 18.10 THREE PHASE – DELTA = Line Voltage: VLINE VPHASE = Phase Voltage: VPHASE VLINE = × Line Current: I LINE I PHASE 3 I Phase Current: I = LINE PHASE 3 S = 3 ×V × I Power: LINE LINE = × × S 3 VPHASE I PHASE 18.11 POWER Instantaneous: tP )( = tV )( × tI )( Page 209 of 330 T = 1 = 1 ()()θ −θ = θ −θ Average: ∫ )( dttP VMAX I MAX cos V I VRMS I RMS cos V I T 0 2 V 2 = TH = Maximum Power: PMAX where Z L ZTH 8RTH = 2 Total Power: I RMS R Complex Power: = S VRMS I RMS = 2 S I RMS Z S = P + jQ where P = Average or Active Power (W) [positive = load, negative = generator] where Q = Reactive Power (VAr) [positive = inductive, negative = capacitive] 18.12 Electromagnetics Definitions: Magnetic Flux Φ Strength of magnetic field Wb Reluctance ℜ Relative difficulty for flux to establish A-t/Wb Permeability µ Relative ease for flux to establish H/m Magnetomotive Force ℑ Ability of coil to produce flux A-t Flux density B Flux per unit area Wb/m 2 or T Magnetic Field Intensity H MMF per unit length A-t/m µ = π × −7 −1 Permeability of free space: 0 4 10 Hm ℑ NI Magnetic Field Intensity: H = = l l 1 Reluctance: ℜ = µA ℑ Ohm’s Law: Φ = OR ℑ = NI ℜ Magnetic Force on a conductor: F = BI l Φ − Φ EMF −= N 2 1 Electromagnetic Induction: t EMF = Bv l Magnetic Flux: Φ = BA F V Electric Field: E = = q d Magnetic force on a particle: F = qvB Page 210 of 330 PART 19: GRAPH THEORY 19.1 FUNDAMENTAL EXPLANATIONS: List of vertices: = V {v1,v2 ,v3 ,...} List of edges: = E {e1, e2 , e3 ,...} Subgaphs: Any subgraph H such that V (H ) ⊂ V (G &) E(H ) ⊂ E(G) Tree: Any subgraph H where V (H ) = V (G) , there are no cycles and all verticis are connected. Degree of vertex: Number of edges leaving a vertex ∑ vd )( = 2 E(G) ∈ GVv )( Distance: d vu ),( =Shortest path between u & v Diameter: diam (G) = max {d vu ),( } & ∈ GVvu )( Total Edges in a simple bipartite graph: V (X ) V Y )( E(G) = 2 ∑d x)( = ∑d y)( ∈Xx ∈ Yy Total Edges in K-regular graph: (kk − )1 E(G) = 2 19.2 FACTORISATION: 1 Factorisation: A spanning union of 1 Factors and only exists if there are an even number of vertices. 1 Factors of a K ,nn bipartite graph: = F1 ,...]'33,'22,'11[ = F2 ,...]'34,'23,'12[ = F3 35,'24,'13[ '...] = Fn ... where all numbers are MOD(n) 1 Factors of a K 2n graph: = ∞ − + F0 ,1{( 2,3(),0,2(), n 2),..., ,( nn )}1 = ∞ + − + + − + Fi i,{( (), i 2,1 n 2 1),..., (i n ,1 i n} = F n−22 ... Where all numbers are MOD(2n-1) 19.3 VERTEX COLOURING: Page 211 of 330 Chromatic Number: χ(G) ≥ 3 if there are triangles or an odd cycle χ(G) ≥ 2 if is an even cycle χ ≥ (G) n if is K n is a subgraph of G Union/Intersection: = ∪ ∩ = If G G1 G2 and G1 G2 K m , then P(G λ P(), G λ), P G λ),( = 1 2 λ P(K m ), Edge Contraction: P G λ),( = P(G − e λ), − P eG λ),.( Common Chromatic Polynomials: λ = λ λ − n−1 P(Tn ), ( )1 λ = λ − n + − n λ − P(Cn ), ( )1 ( ()1 )1 λ = λ λ − λ − λ − + P(K n ), ( )(1 2)...( n )1 Where the highest power is the number of vertices Where the lowest power is the number of components Where the coefficient of the 2 nd highest power is the number of edges. 19.4 EDGE COLOURING: Common Chromatic Polynomials: χ'(G) ≥ ∆(G) χ = (' K ,nn ) n χ = (' C2n ) 2 χ = (' C n+12 ) 3 χ = − (' K 2n ) 2n 1 χ = + (' K n+12 ) 2n 1 Page 212 of 330