ALL IN ONE
MATHEMATICS CHEAT SHEET
V2.6
Euler’s Identity:
iπ
e +1=0
CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL
AND UNIVERSITY MATHEMATICS
COMPILED FROM MANY SOURCES BY ALEX SPARTALIS
2009-2012
Page 1 of 286
REVISION HISTORY
2.1. 08/06/2012
UPDATED: Format
NEW: Multivariable Calculus
UPDATED: Convergence tests
UPDATED: Composite Functions
2.2. 10/07/2012
NEW: Three Phase – Delta & Y
NEW: Electrical Power
2.3. 14/08/2012
NEW: Factorial
NEW: Electromagnetics
NEW: Linear Algebra
NEW: Mathematical Symbols
NEW: Algebraic Identities
NEW: Graph Theory
UPDATED: Linear Algebra
UPDATED: Linear Transformations
2.4. 31/08/2012
NEW: Graphical Functions
NEW: Prime numbers
NEW: Power Series Expansion
NEW: Inner Products
UPDATED: Pi Formulas
UPDATED: General Trigonometric Functions Expansion
UPDATED: Linear Algebra
UPDATED: Matrix Inverse
2.5. 10/09/2012
NEW: Machin-Like Formulae
NEW: Infinite Summations To Pi
NEW: Classical Mechanics
NEW: Relativistic Formulae
NEW: Statistical Distributions
NEW: Logarithm Power Series
NEW: Spherical Triangle Identities
NEW: Bernoulli Expansion
UPDATED: Pi Formulas
UPDATED: Logarithm Identities
UPDATED: Riemann Zeta Function
UPDATED: Eigenvalues and Eigenvectors
2.6. 3/10/2012
NEW: QR Factorisation
NEW: Jordan Forms
NEW: Macroeconomics
NEW: Golden Ratio & Fibonacci Sequence
NEW: Complex Vectors and Matrices
NEW: Numerical Computations for Matrices
UPDATED: Prime Numbers
UPDATED: Errors within Matrix Formula
2.7. 2012
TO DO: USV Decomposition
Page 2 of 286
CONTENTS
REVISION HISTORY
2
CONTENTS
3
PART 1: PHYSICAL CONSTANTS
20
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
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23
24
25
SI PREFIXES:
SI BASE UNITS:
SI DERIVED UNITS:
UNIVERSAL CONSTANTS:
ELECTROMAGNETIC CONSTANTS:
ATOMIC AND NUCLEAR CONSTANTS:
PHYSICO-CHEMICAL CONSTANTS:
ADOPTED VALUES:
NATURAL UNITS:
PART 2: MATHEMTAICAL SYMBOLS
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
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BASIC MATH SYMBOLS
GEOMETRY SYMBOLS
ALGEBRA SYMBOLS
LINEAR ALGEBRA SYMBOLS
PROBABILITY AND STATISTICS SYMBOLS
COMBINATORICS SYMBOLS
SET THEORY SYMBOLS
LOGIC SYMBOLS
CALCULUS & ANALYSIS SYMBOLS
PART 3: AREA, VOLUME AND SURFACE AREA
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3.1 AREA
TRIANGLE:
RECTANGLE:
SQUARE:
PARALLELOGRAM:
RHOMBUS:
TRAPEZIUM:
QUADRILATERAL:
RECTANGLE WITH ROUNDED CORNERS:
REGULAR HEXAGON:
REGULAR OCTAGON:
REGULAR POLYGON:
3.2 VOLUME
CUBE:
CUBOID:
PYRAMID:
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TETRAHEDRON:
OCTAHEDRON:
DODECAHEDRON:
ICOSAHEDRON:
3.3 SURFACE AREA:
CUBE:
CUBOIDS:
TETRAHEDRON:
OCTAHEDRON:
DODECAHEDRON:
ICOSAHEDRON:
CYLINDER:
3.4 MISELANIOUS
DIAGONAL OF A RECTANGLE
DIAGONAL OF A CUBOID
LONGEST DIAGONAL (EVEN SIDES)
LONGEST DIAGONAL (ODD SIDES)
TOTAL LENGTH OF EDGES (CUBE):
TOTAL LENGTH OF EDGES (CUBOID):
CIRCUMFERENCE
PERIMETER OF RECTANGLE
SEMI PERIMETER
EULER’S FORMULA
3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4)
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PART 4: ALGEBRA
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4.1 POLYNOMIAL FORMULA:
QUDARATIC:
CUBIC:
4.2 ALGEBRAIC EXPANSION:
BABYLONIAN IDENTITY:
COMMON PRODUCTS AND FACTORS:
BINOMIAL THEOREM:
BINOMIAL EXPANSION:
DIFFERENCE OF TWO SQUARES:
BRAHMAGUPTA–FIBONACCI IDENTITY:
DEGEN'S EIGHT-SQUARE IDENTITY:
4.3 LIMIT MANIPULATIONS:
4.4 SUMATION MANIPULATIONS:
4.5 COMMON FUNCTIONS:
CONSTANT FUNCTION:
LINE/LINEAR FUNCTION:
PARABOLA/QUADRATIC FUNCTION:
CIRCLE:
ELLIPSE:
HYPERBOLA:
4.6 LINEAR ALGEBRA:
VECTOR SPACE AXIOMS:
SUBSPACE:
COMMON SPACES:
ROWSPACE OF A SPANNING SET IN RN
COLUMNSPACE OF A SPANNING SET IN RN
NULLSPACE:
NULLITY:
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LINEAR DEPENDENCE:
BASIS:
STANDARD BASIS:
ORTHOGONAL COMPLEMENT:
ORTHONORMAL BASIS:
GRAM-SCHMIDT PROCESS:
COORDINATE VECTOR:
DIMENSION:
4.7 COMPLEX VECTOR SPACES:
FORM:
DOT PRODUCT:
INNER PRODUCT:
4.8 LINEAR TRANSITIONS & TRANSFORMATIONS:
TRANSITION MATRIX:
CHANGE OF BASIS TRANSITION MATRIX:
TRANSFORMATION MATRIX:
4.9 INNER PRODUCTS:
DEFINITION:
AXIOMS:
UNIT VECTOR:
CAVCHY-SCHUARZ INEQUALITY:
INNER PRODUCT SPACE:
ANGLE BETWEEN TWO VECTORS:
DISTANCE BETWEEN TWO VECTORS:
GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS:
4.10 PRIME NUMBERS:
DETERMINATE:
LIST OF PRIME NUMBERS:
PERFECT NUMBERS:
LIST OF PERFECT NUMBERS:
AMICABLE NUMBERS:
LIST OF AMICABLE NUMBERS:
SOCIABLE NUMBERS:
LIST OF SOCIABLE NUMBERS:
4.11 GOLDEN RATIO & FIBONACCI SEQUENCE:
RELATIONSHIP:
INFINITE SERIES:
CONTINUED FRACTIONS:
TRIGONOMETRIC EXPRESSIONS:
FIBONACCI SEQUENCE:
4.12 FERMAT’S LAST THEOREM:
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PART 5: COUNTING TECHNIQUES & PROBABILITY
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5.1 2D
TRIANGLE NUMBER
SQUARE NUMBER
PENTAGONAL NUMBER
5.2 3D
TETRAHEDRAL NUMBER
SQUARE PYRAMID NUMBER
5.3 PERMUTATIONS
PERMUTATIONS:
PERMUTATIONS (WITH REPEATS):
5.4 COMBINATIONS
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ORDERED COMBINATIONS:
UNORDERED COMBINATIONS:
ORDERED REPEATED COMBINATIONS:
UNORDERED REPEATED COMBINATIONS:
GROUPING:
5.5 MISCELLANEOUS:
TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE:
NUMBER OF INTERPRETERS:
MAX NUMBER OF PIZZA PIECES:
MAX PIECES OF A CRESCENT:
MAX PIECES OF CHEESE:
CARDS IN A CARD HOUSE:
DIFFERENT ARRANGEMENT OF DOMINOS:
UNIT FRACTIONS:
ANGLE BETWEEN TWO HANDS OF A CLOCK:
WINNING LINES IN NOUGHTS AND CROSSES:
BAD RESTAURANT SPREAD:
FIBONACCI SEQUENCE:
ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5)
5.6 FACTORIAL:
DEFINITION:
TABLE OF FACTORIALS:
APPROXIMATION:
5.7 THE DAY OF THE WEEK:
5.8 BASIC PROBABILITY:
5.9 VENN DIAGRAMS:
COMPLEMENTARY EVENTS:
TOTALITY:
CONDITIONAL PROBABILITY:
UNION :
INDEPENDENT EVENTS:
MUTUALLY EXCLUSIVE:
BAYE’S THEOREM:
5.11 BASIC STATISTICAL OPERATIONS:
VARIANCE:
MEAN:
STANDARDIZED SCORE:
5.12 DISCRETE RANDOM VARIABLES:
STANDARD DEVIATION:
EXPECTED VALUE:
VARIANCE:
PROBABILITY MASS FUNCTION:
CUMULATIVE DISTRIBUTION FUNCTION:
5.13 COMMON DRVS:
BERNOULLI TRIAL:
BINOMIAL TRIAL:
GEOMETRIC TRIAL:
NEGATIVE BINOMIAL TRIAL:
5.14 CONTINUOUS RANDOM VARIABLES:
PROBABILITY DENSITY FUNCTION:
CUMULATIVE DISTRIBUTION FUNCTION:
INTERVAL PROBABILITY:
EXPECTED VALUE:
VARIANCE:
5.15 COMMON CRVS:
UNIFORM DISTRIBUTION:
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EXPONENTIAL DISTRIBUTION:
NORMAL DISTRIBUTION:
5.16 MULTIVARIABLE DISCRETE:
PROBABILITY:
MARGINAL DISTRIBUTION:
EXPECTED VALUE:
INDEPENDENCE:
COVARIANCE:
5.17 MULTIVARIABLE CONTINUOUS:
PROBABILITY:
MARGINAL DISTRIBUTION:
EXPECTED VALUE:
INDEPENDENCE:
COVARIANCE:
CORRELATION COEFFICIENT:
ABBREVIATIONS
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PART 6: FINANCIAL
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6.1 GENERAL FORMUALS:
PROFIT:
PROFIT MARGIN:
SIMPLE INTEREST:
COMPOUND INTEREST:
CONTINUOUS INTEREST:
ABBREVIATIONS (6.1):
6.2 MACROECONOMICS:
GDP:
RGDP:
NGDP:
GROWTH:
NET EXPORTS:
WORKING AGE POPULATION:
LABOR FORCE:
UNEMPLOYMENT:
NATURAL UNEMPLOYMENT:
UNEMPLOYMENT RATE:
EMPLOYMENT RATE:
PARTICIPATION RATE:
CPI:
INFLATION RATE:
ABBREVIATIONS (6.2)
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PART 7: PI
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7.1 AREA:
CIRCLE:
CYCLIC QUADRILATERAL:
AREA OF A SECTOR (DEGREES)
AREA OF A SECTOR (RADIANS)
AREA OF A SEGMENT (DEGREES)
AREA OF AN ANNULUS:
ELLIPSE :
7.2 VOLUME:
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SPHERE:
CAP OF A SPHERE:
CONE:
ICE-CREAM & CONE:
DOUGHNUT:
SAUSAGE:
ELLIPSOID:
7.3 SURFACE AREA:
SPHERE:
HEMISPHERE:
DOUGHNUT:
SAUSAGE:
CONE:
7.4 MISELANIOUS:
LENGTH OF ARC (DEGREES)
LENGTH OF CHORD (DEGREES)
PERIMETER OF AN ELLIPSE
7.6 PI:
JOHN WALLIS:
ISAAC NEWTON:
JAMES GREGORY:
LEONARD EULER:
JOZEF HOENE-WRONSKI:
FRANCISCUS VIETA:
INTEGRALS:
INFINITE SERIES:
CONTINUED FRACTIONS:
7.7 CIRCLE GEOMETRY:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR RECTANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR SQUARES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR TRIANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR QUADRILATERALS:
RADIUS OF INSCRIBED CIRCLE FOR SQUARES:
RADIUS OF INSCRIBED CIRCLE FOR TRIANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE:
RADIUS OF INSCRIBED CIRCLE:
7.8 ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7):
7.9 CRESCENT GEOMETRY:
AREA OF A LUNAR CRESCENT:
AREA OF AN ECLIPSE CRESCENT:
7.10 ABBREVIATIONS (7.9):
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PART 8: PHYSICS
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8.1 MOVEMENT:
STOPPING DISTANCE:
CENTRIPETAL ACCELERATION:
CENTRIPETAL FORCE:
DROPPING TIME :
FORCE:
KINETIC ENERGY:
MAXIMUM HEIGHT OF A CANNON:
PENDULUM SWING TIME:
POTENTIAL ENERGY:
RANGE OF A CANNON:
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TIME IN FLIGHT OF A CANNON:
UNIVERSAL GRAVITATION:
ABBREVIATIONS (8.1):
8.2 CLASSICAL MECHANICS:
NEWTON’S LAWS:
INERTIA:
MOMENTS OF INERTIA:
VELOCITY AND SPEED:
ACCELERATION:
TRAJECTORY (DISPLACEMENT):
KINETIC ENERGY:
CENTRIPETAL FORCE:
CIRCULAR MOTION:
ANGULAR MOMENTUM:
TORQUE:
WORK:
LAWS OF CONSERVATION:
ABBREVIATIONS (8.2)
8.3 RELATIVISTIC EQUATIONS:
KINETIC ENERGY:
MOMENTUM:
TIME DILATION:
LENGTH CONTRACTION:
RELATIVISTIC MASS:
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PART 9: TRIGONOMETRY
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9.1 CONVERSIONS:
9.2 BASIC RULES:
SIN RULE:
COS RULE:
TAN RULE:
AUXILIARY ANGLE:
PYTHAGORAS THEOREM:
9.3 RECIPROCAL FUNCTIONS
9.4 BASIC IDENTITES:
9.5 IDENTITIES (SINΘ):
9.6 IDENTITIES (COSΘ):
9.7 IDENTITIES (TANΘ):
9.8 IDENTITIES (CSCΘ):
9.9 IDENTITIES (COTΘ):
9.10 ADDITION FORMULAE:
9.11 DOUBLE ANGLE FORMULAE:
9.12 TRIPLE ANGLE FORMULAE:
9.13 HALF ANGLE FORMULAE:
9.14 POWER REDUCTION:
9.15 PRODUCT TO SUM:
9.16 SUM TO PRODUCT:
9.17 HYPERBOLIC EXPRESSIONS:
9.18 HYPERBOLIC RELATIONS:
9.19 MACHIN-LIKE FORMULAE:
FORM:
FORMULAE:
IDENTITIES:
9.20 SPHERICAL TRIANGLE IDENTITIES:
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9.21
ABBREVIATIONS (9.1-9.19)
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PART 10: EXPONENTIALS & LOGARITHIMS
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10.1
10.2
10.3
10.4
10.5
10.6
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FUNDAMENTAL THEORY:
IDENTITIES:
CHANGE OF BASE:
LAWS FOR LOG TABLES:
COMPLEX NUMBERS:
LIMITS INVOLVING LOGARITHMIC TERMS
PART 11: COMPLEX NUMBERS
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11.1 GENERAL:
FUNDAMENTAL:
STANDARD FORM:
POLAR FORM:
ARGUMENT:
MODULUS:
CONJUGATE:
EXPONENTIAL:
DE MOIVRE’S FORMULA:
EULER’S IDENTITY:
11.2 OPERATIONS:
ADDITION:
SUBTRACTION:
MULTIPLICATION:
DIVISION:
SUM OF SQUARES:
11.3 IDENTITIES:
EXPONENTIAL:
LOGARITHMIC:
TRIGONOMETRIC:
HYPERBOLIC:
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PART 12: DIFFERENTIATION
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12.1 GENERAL RULES:
PLUS OR MINUS:
PRODUCT RULE:
QUOTIENT RULE:
POWER RULE:
CHAIN RULE:
BLOB RULE:
BASE A LOG:
NATURAL LOG:
EXPONENTIAL (X):
FIRST PRINCIPLES:
12.2 EXPONETIAL FUNCTIONS:
12.3 LOGARITHMIC FUNCTIONS:
12.4 TRIGONOMETRIC FUNCTIONS:
12.5 HYPERBOLIC FUNCTIONS:
12.5 PARTIAL DIFFERENTIATION:
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FIRST PRINCIPLES:
GRADIENT:
TOTAL DIFFERENTIAL:
CHAIN RULE:
IMPLICIT DIFFERENTIATION:
HIGHER ORDER DERIVATIVES:
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PART 13: INTEGRATION
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13.1 GENERAL RULES:
POWER RULE:
BY PARTS:
CONSTANTS:
13.2 RATIONAL FUNCTIONS:
13.3 TRIGONOMETRIC FUNCTIONS (SINE):
13.4 TRIGONOMETRIC FUNCTIONS (COSINE):
13.5 TRIGONOMETRIC FUNCTIONS (TANGENT):
13.6 TRIGONOMETRIC FUNCTIONS (SECANT):
13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT):
13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE):
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):
13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE):
13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT):
13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT):
13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT):
13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT):
13.19 EXPONETIAL FUNCTIONS
13.20 LOGARITHMIC FUNCTIONS
13.21 HYPERBOLIC FUNCTIONS
13.22 INVERSE HYPERBOLIC FUNCTIONS
13.23 ABSOLUTE VALUE FUNCTIONS
13.24 SUMMARY TABLE
13.25 SQUARE ROOT PROOFS
13.26 CARTESIAN APPLICATIONS
AREA UNDER THE CURVE:
VOLUME:
VOLUME ABOUT X AXIS:
VOLUME ABOUT Y AXIS:
SURFACE AREA ABOUT X AXIS:
LENGTH WRT X-ORDINATES:
LENGTH WRT Y-ORDINATES:
LENGTH PARAMETRICALLY:
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PART 14: FUNCTIONS
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14.1 COMPOSITE FUNCTIONS:
14.2 MULTIVARIABLE FUNCTIONS:
LIMIT:
DISCRIMINANT:
CRITICAL POINTS:
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14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS:
SEPARABLE
LINEAR
HOMOGENEOUS
EXACT
14.4 SECOND ORDER
HOMOGENEOUS
UNDETERMINED COEFFICIENTS
VARIATION OF PARAMETERS
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PART 15: MATRICIES
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15.1 BASIC PRINICPLES:
SIZE
15.2 BASIC OPERTAIONS:
ADDITION:
SUBTRACTION:
SCALAR MULTIPLE:
TRANSPOSE:
SCALAR PRODUCT:
SYMMETRY:
CRAMER’S RULE:
LEAST SQUARES SOLUTION
15.3 SQUARE MATRIX:
DIAGONAL:
LOWER TRIANGLE MATRIX:
UPPER TRIANGLE MATRIX:
15.4 DETERMINATE:
2X2
3X3
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NXN
RULES
15.5 INVERSE
2X2:
3X3:
MINOR:
COFACTOR:
ADJOINT METHOD FOR INVERSE:
LEFT INVERSE:
RIGHT INVERSE:
15.6 LINEAR TRANSFORMATION
AXIOMS FOR A LINEAR TRANSFORMATION:
TRANSITION MATRIX:
ZERO TRANSFORMATION:
IDENTITY TRANSFORMATION:
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:
EIGENVALUES:
EIGENVECTORS:
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CHARACTERISTIC POLYNOMIAL:
ALGEBRAIC MULTIPLICITY:
GEOMETRIC MULTIPLICITY:
TRANSFORMATION:
LINEARLY INDEPENDENCE:
DIGITALIZATION:
CAYLEY-HAMILTON THEOREM:
ORTHONORMAL SET:
QR FACTORISATION:
15.9 JORDAN FORMS
GENERALISED DIAGONLISATION:
JORDAN BLOCK:
JORDAN FORM:
ALGEBRAIC MULTIPLICITY:
GEOMETRIC MULTIPLICITY:
GENERALISED CHAIN:
POWERS:
15.10 COMPLEX MATRICIS:
CONJUGATE TRANSPOSE:
HERMITIAN MATRIX:
SKEW-HERMITIAN:
UNITARY MATRIX:
NORMAL MATRIX:
DIAGONALISATION:
SPECTRAL THEOREM:
15.11 NUMERICAL COMPUTATIONS:
RAYLEIGH QUOTIENT:
POWER METHOD:
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PART 16: VECTORS
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16.1 BASIC OPERATIONS:
ADDITION:
SUBTRACTION:
EQUALITY:
SCALAR MULTIPLICATION:
PARALLEL:
MAGNITUDE:
UNIT VECTOR:
ZERO VECTOR:
DOT PRODUCT:
ANGLE BETWEEN TWO VECTORS:
ANGLE OF A VECTOR IN 3D:
PERPENDICULAR TEST:
SCALAR PROJECTION:
VECTOR PROJECTION:
CROSS PRODUCT:
16.2 LINES
16.3 PLANES
16.4 CLOSEST APPROACH
TWO POINTS:
POINT AND LINE:
POINT AND PLANE:
TWO SKEW LINES:
16.5 GEOMETRY
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AREA OF A TRIANGLE:
AREA OF A PARALLELOGRAM:
AREA OF A PARALLELEPIPED:
16.6 SPACE CURVES
WHERE:
VELOCITY:
ACCELERATION:
DEFINITION OF “S”:
UNIT TANGENT:
CHAIN RULE:
NORMAL:
CURVATURE:
UNIT BINOMIAL:
TORTION:
16.7 VECTOR SPACE
16.8 ABBREVIATIONS
137
137
137
137
137
137
137
137
137
137
138
138
138
138
138
138
PART 17: SERIES
140
17.1 MISCELLANEOUS
GENERAL FORM:
INFINITE FORM:
PARTIAL SUM OF A SERIES:
17.2 TEST FOR CONVERGENCE AND DIVERGENCE
TEST FOR CONVERGENCE:
TEST FOR DIVERGENCE:
GEOMETRIC SERIES
P SERIES
THE SANDWICH THEOREM
THE INTEGRAL TEST
THE DIRECT COMPARISON TEST
THE LIMIT COMPARISON TEST
D’ALMBERT’S RATIO COMPARISON TEST
THE NTH ROOT TEST
NEGATIVE TERMS
ALTERNATING SERIES TEST
ALTERNATING SERIES ERROR
17.3 ARITHMETIC PROGRESSION:
DEFINITION:
NTH TERM:
SUM OF THE FIRST N TERMS:
17.4 GEOMETRIC PROGRESSION:
DEFINITION:
NTH TERM:
SUM OF THE FIRST N TERMS:
SUM TO INFINITY:
GEOMETRIC MEAN:
17.5 SUMMATION SERIES
LINEAR:
QUADRATIC:
CUBIC:
17.6 APPROXIMATION SERIES
TAYLOR SERIES
MACLAURUN SERIES
LINEAR APPROXIMATION:
140
140
140
140
140
140
140
140
140
140
140
140
141
141
141
141
141
141
142
142
142
142
142
142
142
142
142
142
142
142
142
142
142
142
142
143
Page 14 of 286
QUADRATIC APPROXIMATION:
CUBIC APPROXIMATION:
17.7 MONOTONE SERIES
STRICTLY INCREASING:
NON-DECREASING:
STRICTLY DECREASING:
NON-INCREASING:
CONVERGENCE:
17.8 RIEMANN ZETA FUNCTION
FORM:
EULER’S TABLE:
ALTERNATING SERIES:
PROOF FOR N=2:
17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS
17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS
17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS
17.12 INFINITE SUMMATIONS TO PI
17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS
ABBREVIATIONS
17.14 POWER SERIES EXPANSION
EXPONENTIAL:
TRIGONOMETRIC:
EXPONENTIAL AND LOGARITHM SERIES:
FOURIER SERIES:
17.15 BERNOULLI EXPANSION:
FUNDAMENTALLY:
EXPANSIONS:
LIST OF BERNOULLI NUMBERS:
143
143
143
143
143
143
143
143
143
143
143
144
144
145
145
146
148
148
148
148
148
149
151
152
152
152
153
153
PART 18: ELECTRICAL
155
18.1 FUNDAMENTAL THEORY
CHARGE:
CURRENT:
RESISTANCE:
OHM’S LAW:
POWER:
CONSERVATION OF POWER:
ELECTRICAL ENERGY:
KIRCHOFF’S VOLTAGE LAW:
KIRCHOFF’S CURRENT LAW:
AVERAGE CURRENT:
RMS CURRENT:
∆ TO Y CONVERSION:
18.2 COMPONENTS
RESISTANCE IN SERIES:
RESISTANCE IN PARALLEL:
INDUCTIVE IMPEDANCE:
CAPACITOR IMPEDANCE:
CAPACITANCE IN SERIES:
CAPACITANCE IN PARALLEL:
VOLTAGE, CURRENT & POWER SUMMARY:
18.3 THEVENIN’S THEOREM
THEVENIN’S THEOREM:
MAXIMUM POWER TRANSFER THEOREM:
155
155
155
155
155
155
155
155
155
155
155
155
155
156
156
156
156
156
156
156
156
156
156
157
Page 15 of 286
18.4 FIRST ORDER RC CIRCUIT
18.5 FIRST ORDER RL CIRCUIT
18.6 SECOND ORDER RLC SERIES CIRCUIT
CALCULATION USING KVL:
IMPORTANT VARIABLES
SOLVING:
MODE 1:
MODE 2:
MODE 3:
MODE 4:
CURRENT THROUGH INDUCTOR:
PLOTTING MODES:
18.7 SECOND ORDER RLC PARALLEL CIRCUIT
CALCULATION USING KCL:
IMPORTANT VARIABLES
SOLVING:
18.8 LAPLANCE TRANSFORMATIONS
IDENTITIES:
PROPERTIES:
18.9 THREE PHASE – Y
LINE VOLTAGE:
PHASE VOLTAGE:
LINE CURRENT:
PHASE CURRENT:
POWER:
18.10 THREE PHASE – DELTA
LINE VOLTAGE:
PHASE VOLTAGE:
LINE CURRENT:
PHASE CURRENT:
POWER:
18.11 POWER
INSTANTANEOUS:
AVERAGE:
MAXIMUM POWER:
TOTAL POWER:
COMPLEX POWER:
18.12 ELECTROMAGNETICS
DEFINITIONS:
PERMEABILITY OF FREE SPACE:
MAGNETIC FIELD INTENSITY:
RELUCTANCE:
OHM’S LAW:
MAGNETIC FORCE ON A CONDUCTOR:
ELECTROMAGNETIC INDUCTION:
MAGNETIC FLUX:
ELECTRIC FIELD:
MAGNETIC FORCE ON A PARTICLE:
157
157
157
157
157
158
158
158
159
159
160
160
161
161
161
162
162
162
163
164
164
164
164
164
164
164
164
164
164
164
164
164
164
165
165
165
165
165
165
165
165
165
165
165
165
165
165
165
PART 19: GRAPH THEORY
166
19.1 FUNDAMENTAL EXPLANATIONS
LIST OF VERTICES:
LIST OF EDGES:
SUBGAPHS:
166
166
166
166
Page 16 of 286
DEGREE OF VERTEX:
DISTANCE:
DIAMETER:
TOTAL EDGES IN A SIMPLE BIPARTITE GRAPH:
TOTAL EDGES IN K-REGULAR GRAPH:
19.2 FACTORISATION:
1 FACTORISATION:
1 FACTORS OF A K n ,n BIPARTITE GRAPH:
166
166
166
166
166
166
166
166
1 FACTORS OF A K 2 n GRAPH:
19.3 VERTEX COLOURING
CHROMATIC NUMBER:
UNION/INTERSECTION:
EDGE CONTRACTION:
COMMON CHROMATIC POLYNOMIALS:
19.4 EDGE COLOURING:
COMMON CHROMATIC POLYNOMIALS:
166
166
167
167
167
167
167
167
PART 98: LIST OF DISTRIBUTION FUNCTIONS
168
5.18 FINITE DISCRETE DISTRIBUTIONS
BERNOULLI DISTRIBUTION
RADEMACHER DISTRIBUTION
BINOMIAL DISTRIBUTION
BETA-BINOMIAL DISTRIBUTION
DEGENERATE DISTRIBUTION
DISCRETE UNIFORM DISTRIBUTION
HYPERGEOMETRIC DISTRIBUTION
POISSON BINOMIAL DISTRIBUTION
FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE)
FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE)
WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE)
WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE)
5.19 INFINITE DISCRETE DISTRIBUTIONS
BETA NEGATIVE BINOMIAL DISTRIBUTION
MAXWELL–BOLTZMANN DISTRIBUTION
GEOMETRIC DISTRIBUTION
LOGARITHMIC (SERIES) DISTRIBUTION
NEGATIVE BINOMIAL DISTRIBUTION
POISSON DISTRIBUTION
CONWAY–MAXWELL–POISSON DISTRIBUTION
SKELLAM DISTRIBUTION
YULE–SIMON DISTRIBUTION
ZETA DISTRIBUTION
ZIPF'S LAW
ZIPF–MANDELBROT LAW
5.20 BOUNDED INFINITE DISTRIBUTIONS
ARCSINE DISTRIBUTION
BETA DISTRIBUTION
LOGITNORMAL DISTRIBUTION
CONTINUOUS UNIFORM DISTRIBUTION
IRWIN-HALL DISTRIBUTION
KUMARASWAMY DISTRIBUTION
RAISED COSINE DISTRIBUTION
TRIANGULAR DISTRIBUTION
168
168
168
169
170
171
172
174
175
175
176
176
177
177
177
178
179
181
182
183
184
185
185
187
188
189
189
189
191
193
194
195
196
197
198
Page 17 of 286
TRUNCATED NORMAL DISTRIBUTION
U-QUADRATIC DISTRIBUTION
VON MISES DISTRIBUTION
WIGNER SEMICIRCLE DISTRIBUTION
5.21 SEMI-BOUNDED CUMULATIVE DISTRIBUTIONS
BETA PRIME DISTRIBUTION
CHI DISTRIBUTION
NONCENTRAL CHI DISTRIBUTION
CHI-SQUARED DISTRIBUTION
INVERSE-CHI-SQUARED DISTRIBUTION
NONCENTRAL CHI-SQUARED DISTRIBUTION
SCALED-INVERSE-CHI-SQUARED DISTRIBUTION
DAGUM DISTRIBUTION
EXPONENTIAL DISTRIBUTION
FISHER'S Z-DISTRIBUTION
FOLDED NORMAL DISTRIBUTION
FRÉCHET DISTRIBUTION
GAMMA DISTRIBUTION
ERLANG DISTRIBUTION
INVERSE-GAMMA DISTRIBUTION
INVERSE GAUSSIAN/WALD DISTRIBUTION
LÉVY DISTRIBUTION
LOG-CAUCHY DISTRIBUTION
LOG-LOGISTIC DISTRIBUTION
LOG-NORMAL DISTRIBUTION
MITTAG–LEFFLER DISTRIBUTION
PARETO DISTRIBUTION
RAYLEIGH DISTRIBUTION
RICE DISTRIBUTION
TYPE-2 GUMBEL DISTRIBUTION
WEIBULL DISTRIBUTION
5.22 UNBOUNDED CUMULATIVE DISTRIBUTIONS
CAUCHY DISTRIBUTION
EXPONENTIALLY MODIFIED GAUSSIAN DISTRIBUTION
FISHER–TIPPETT/ GENERALIZED EXTREME VALUE DISTRIBUTION
GUMBEL DISTRIBUTION
FISHER'S Z-DISTRIBUTION
GENERALIZED NORMAL DISTRIBUTION
GEOMETRIC STABLE DISTRIBUTION
HOLTSMARK DISTRIBUTION
HYPERBOLIC DISTRIBUTION
HYPERBOLIC SECANT DISTRIBUTION
LAPLACE DISTRIBUTION
LÉVY SKEW ALPHA-STABLE DISTRIBUTION
LINNIK DISTRIBUTION
LOGISTIC DISTRIBUTION
NORMAL DISTRIBUTION
NORMAL-EXPONENTIAL-GAMMA DISTRIBUTION
SKEW NORMAL DISTRIBUTION
STUDENT'S T-DISTRIBUTION
NONCENTRAL T-DISTRIBUTION
VOIGT DISTRIBUTION
GENERALIZED PARETO DISTRIBUTION
TUKEY LAMBDA DISTRIBUTION
5.23 JOINT DISTRIBUTIONS
DIRICHLET DISTRIBUTION
Page 18 of 286
200
201
202
203
205
205
206
207
207
208
210
211
212
213
216
216
217
218
219
220
221
222
224
225
226
227
228
229
230
231
232
233
233
234
236
237
238
238
240
240
241
242
243
244
246
246
248
249
249
250
252
252
253
254
254
254
BALDING–NICHOLS MODEL
MULTINOMIAL DISTRIBUTION
MULTIVARIATE NORMAL DISTRIBUTION
NEGATIVE MULTINOMIAL DISTRIBUTION
WISHART DISTRIBUTION
INVERSE-WISHART DISTRIBUTION
MATRIX NORMAL DISTRIBUTION
MATRIX T-DISTRIBUTION
5.24 OTHER DISTRIBUTIONS
CATEGORICAL DISTRIBUTION
CANTOR DISTRIBUTION
PHASE-TYPE DISTRIBUTION
TRUNCATED DISTRIBUTION
255
256
256
257
258
258
258
259
259
259
260
261
261
PART 99: CONVERSIONS
263
99.1
99.2
99.3
99.4
99.5
99.6
99.7
99.8
99.9
99.10
99.11
99.12
99.13
99.14
99.15
99.16
99.17
99.18
99.19
99.20
99.21
99.22
99.23
99.24
99.25
99.26
99.27
99.28
99.29
99.30
99.31
99.32
99.33
99.34
99.35
99.36
99.37
99.38
99.39
263
265
266
270
270
270
272
273
274
274
275
276
276
277
278
278
280
281
281
281
281
282
282
282
283
283
283
283
283
283
284
284
285
285
285
285
285
285
286
LENGTH:
AREA:
VOLUME:
PLANE ANGLE:
SOLID ANGLE:
MASS:
DENSITY:
TIME:
FREQUENCY:
SPEED OR VELOCITY:
FLOW (VOLUME):
ACCELERATION:
FORCE:
PRESSURE OR MECHANICAL STRESS:
TORQUE OR MOMENT OF FORCE:
ENERGY, WORK, OR AMOUNT OF HEAT:
POWER OR HEAT FLOW RATE:
ACTION:
DYNAMIC VISCOSITY:
KINEMATIC VISCOSITY:
ELECTRIC CURRENT:
ELECTRIC CHARGE:
ELECTRIC DIPOLE:
ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE:
ELECTRICAL RESISTANCE:
CAPACITANCE:
MAGNETIC FLUX:
MAGNETIC FLUX DENSITY:
INDUCTANCE:
TEMPERATURE:
INFORMATION ENTROPY:
LUMINOUS INTENSITY:
LUMINANCE:
LUMINOUS FLUX:
ILLUMINANCE:
RADIATION - SOURCE ACTIVITY:
RADIATION – EXPOSURE:
RADIATION - ABSORBED DOSE:
RADIATION - EQUIVALENT DOSE:
Page 19 of 286
PART 1: PHYSICAL CONSTANTS
1.1
SI PREFIXES:
Prefix
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
1.2
Symbol
Y
Z
E
P
T
G
M
k
h
da
d
c
m
µ
n
p
f
a
z
y
1000m
10008
10007
10006
10005
10004
10003
10002
10001
⁄
10002 3
⁄
10001 3
10000
1000−1⁄3
1000−2⁄3
1000−1
1000−2
1000−3
1000−4
1000−5
1000−6
1000−7
1000−8
10n
1024
1021
1018
1015
1012
109
106
103
102
101
100
10−1
10−2
10−3
10−6
10−9
10−12
10−15
10−18
10−21
10−24
Decimal
Scale
1000000000000000000000000
1000000000000000000000
1000000000000000000
1000000000000000
1000000000000
1000000000
1000000
1000
100
10
1
0.1
0.01
0.001
0.000001
0.000000001
0.000000000001
0.000000000000001
0.000000000000000001
0.000000000000000000001
0.000000000000000000000001
Septillion
Sextillion
Quintillion
Quadrillion
Trillion
Billion
Million
Thousand
Hundred
Ten
One
Tenth
Hundredth
Thousandth
Millionth
Billionth
Trillionth
Quadrillionth
Quintillionth
Sextillionth
Septillionth
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 20 of 286
1.3
SI DERIVED UNITS:
Quantity
Unit
Symbol
angle, plane
angle, solid
radian*
steradian*
degree
Celsius
farad
rad
sr
coulomb
siemens
henry
C
S
H
Celsius temperature
electric capacitance
electric charge, quantity
of electricity
electric conductance
electric inductance
electric potential
difference, electromotive
force
electric resistance
energy, work, quantity of
heat
force
frequency (of a periodic
phenomenon)
illuminance
luminous flux
magnetic flux
magnetic flux density
power, radiant flux
pressure, stress
activity (referred to a
radionuclide)
absorbed dose, specific
energy imparted, kerma
dose equivalent, ambient
dose equivalent,
directional dose
equivalent, personal dose
equivalent, organ dose
equivalent
catalytic activity
°C
F
Expression in terms of other SI
units
m/m = 1
m2/m2 = 1
K
C/V
A·s
A/V
Wb/A
W/A
volt
ohm
V
Ω
joule
newton
J
N
hertz
lux
lumen
weber
tesla
watt
pascal
Hz
lx
lm
Wb
T
W
Pa
becquerel
Bq
V/A
N·m
kg·m/s2
1/s
lm/m2
cd·sr
V·s
Wb/m2
J/s
N/m2
1/s
J/kg
gray
Gy
J/kg
sievert
katal
Sv
kat
Page 21 of 286
mol/s
1.4
UNIVERSAL CONSTANTS:
Quantity
speed of light in
vacuum
Newtonian constant
of gravitation
Planck constant
reduced Planck
constant
1.5
Value
299 792 458 m·s−1
Relative Standard
Uncertainty
defined
6.67428(67)×10−11 m3·kg−1·s−2 1.0 × 10−4
6.626 068 96(33) × 10−34 J·s
5.0 × 10−8
1.054 571 628(53) × 10−34 J·s 5.0 × 10−8
ELECTROMAGNETIC CONSTANTS:
Quantity
magnetic constant
(vacuum permeability)
electric constant
(vacuum permittivity)
characteristic
impedance of vacuum
Coulomb's constant
elementary charge
Bohr magneton
conductance quantum
inverse conductance
quantum
Josephson constant
magnetic flux quantum
nuclear magneton
von Klitzing constant
1.6
Symbol
Symbol
Value (SI units)
Relative Standard
Uncertainty
4π × 10−7 N·A−2 = 1.256
defined
637 061... × 10−6 N·A−2
8.854 187 817... × 10−12
defined
F·m−1
376.730 313 461... Ω
8.987 551 787... × 109
N·m²·C−2
1.602 176 487(40) ×
10−19 C
927.400 915(23) × 10−26
J·T−1
7.748 091 7004(53) ×
10−5 S
defined
defined
2.5 × 10−8
2.5 × 10−8
6.8 × 10−10
12 906.403 7787(88) Ω 6.8 × 10−10
4.835 978 91(12) × 1014
2.5 × 10−8
−1
Hz·V
2.067 833 667(52) ×
2.5 × 10−8
10−15 Wb
5.050 783 43(43) ×
8.6 × 10−8
10−27 J·T−1
25 812.807 557(18) Ω
ATOMIC AND NUCLEAR CONSTANTS:
Page 22 of 286
6.8 × 10−10
Quantity
Symbol
Relative
Standard
Uncertainty
5.291 772
108(18) ×
3.3 × 10−9
−11
10 m
2.817 940
2894(58) ×
2.1 × 10−9
−15
10 m
9.109 382
15(45) × 10−31 5.0 × 10−8
kg
Bohr radius
classical
electron
radius
electron mass
Fermi
coupling
constant
fine-structure
constant
Hartree
energy
proton mass
quantum of
circulation
Rydberg
constant
Thomson
cross section
weak mixing
angle
1.7
Value (SI
units)
1.166 39(1) ×
10−5 GeV−2
8.6 × 10−6
7.297 352 537
6(50) × 10−3
4.359 744
17(75) × 10−18
J
1.672 621
637(83) ×
10−27 kg
3.636 947
550(24) × 10−4
m² s−1
10 973
731.568
525(73) m−1
6.652 458
73(13) × 10−29
m²
6.8 × 10−10
0.222 15(76)
3.4 × 10−3
1.7 × 10−7
5.0 × 10−8
6.7 × 10−9
6.6 × 10−12
2.0 × 10−8
PHYSICO-CHEMICAL CONSTANTS:
Quantity
Symbol
Value (SI units)
Relative
Standard
Uncertainty
atomic mass unit
(unified atomic mass
unit)
1.660 538 86(28) ×
10−27 kg
1.7 × 10−7
Avogadro's number
6.022 141 5(10) ×
1023 mol−1
1.7 × 10−7
Boltzmann constant
1.380 6504(24) ×
1.8 × 10−6
Page 23 of 286
10−23 J·K−1
96
8.6 × 10−8
485.3383(83)C·mol−1
3.741 771 18(19) ×
5.0 × 10−8
10−16 W·m²
Faraday constant
first
radiation
constant
for
spectral
radiance
at
T=273.15
Loschmidt
K and
constant
p=101.325
kPa
molar Planck constant
1.8
1.7 × 10−7
2.686 777 3(47) ×
1025 m−3
1.8 × 10−6
8.314 472(15)
1.7 × 10−6
J·K−1·mol−1
3.990 312 716(27) ×
6.7 × 10−9
10−10 J·s·mol−1
gas constant
at
T=273.15
K and
molar
p=100 kPa
volume of
an ideal at
T=273.15
gas
K and
p=101.325
kPa
at T=1 K
and p=100
Sackur- kPa
Tetrode at T=1 K
constant and
p=101.325
kPa
second radiation
constant
Stefan–Boltzmann
constant
Wien displacement
law constant
1.191 042 82(20) ×
10−16 W·m² sr−1
4.965 114 231...
2.2710 981(40) ×
10−2 m³·mol−1
1.7 × 10−6
2.2413 996(39) ×
10−2 m³·mol−1
1.7 × 10−6
−1.151 704 7(44)
3.8 × 10−6
−1.164 867 7(44)
3.8 × 10−6
1.438 775 2(25) ×
1.7 × 10−6
10−2 m·K
5.670 400(40) × 10−8
7.0 × 10−6
W·m−2·K−4
2.897 768 5(51) ×
1.7 × 10−6
10−3 m·K
ADOPTED VALUES:
Quantity
Symbol
conventional value of
Page 24 of 286
Relative
Standard
Uncertainty
4.835 979 × defined
Value (SI
units)
1014 Hz·V−1
25 812.807
defined
Ω
1 × 10−3
defined
kg·mol−1
1.2 × 10−2
defined
kg·mol−1
Josephson constant
conventional value of von
Klitzing constant
constant
molar mass
of carbon-12
standard acceleration of
gravity (gee, free-fall on
Earth)
standard atmosphere
1.9
9.806 65
m·s−2
defined
101 325 Pa
defined
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
Planck charge
Electric charge
(Q)
1.875 545 870(47) × 10−18
C
Planck
temperature
Temperature (Θ)
1.416 785(71) × 1032 K
Page 25 of 286
PART 2: MATHEMTAICAL SYMBOLS
2.1
BASIC MATH SYMBOLS
Symbol Name
equals sign
not equal sign
strict inequality
strict inequality
inequality
inequality
parentheses
brackets
plus sign
minus sign
plus - minus
equality
inequality
greater than
less than
greater than or equal to
less than or equal to
calculate expression inside first
calculate expression inside first
addition
subtraction
both plus and minus operations
5 = 2+3
5≠4
5>4
4<5
5≥4
4≤5
2 × (3+5) = 16
[(1+2)*(1+5)] = 18
1+1=2
2−1=1
3 ± 5 = 8 and -2
∓
*
×
·
÷
/
minus - plus
both minus and plus operations
asterisk
times sign
multiplication dot
division sign / obelus
division slash
multiplication
multiplication
multiplication
division
division
3 ∓ 5 = -2 and 8
2*3=6
2×3=6
2·3=6
6÷2=3
6/2=3
–
horizontal line
division / fraction
modulo
period
power
caret
remainder calculation
decimal point, decimal separator
exponent
exponent
7 mod 2 = 1
2.56 = 2+56/100
23 = 8
2 ^ 3= 8
square root
√a · √a = a
√9 = ±3
Symbol
=
≠
>
<
≥
≤
()
[]
+
−
±
mod
.
ab
a^b
√a
√a
4
√a
n
√a
%
‰
ppm
ppb
ppt
3
2.2
cube root
forth root
n-th root (radical)
percent
per-mille
per-million
per-billion
per-trillion
Meaning / definition
Example
√8 = 2
√16 = ±2
for n=3, n√8 = 2
10% × 30 = 3
10‰ × 30 = 0.3
10ppm × 30 = 0.0003
10ppb × 30 = 3×10-7
10ppb × 30 = 3×10-10
3
4
1% = 1/100
1‰ = 1/1000 = 0.1%
1ppm = 1/1000000
1ppb = 1/1000000000
1ppb = 10-12
GEOMETRY SYMBOLS
Symbol
Symbol Name
∠
angle
∡
measured angle
∢
∟
º
´
spherical angle
right angle
degree
arcminute
Meaning / definition
formed by two rays
Example
∠ABC = 30º
∡ABC
= 30º
∢AOB = 30º
= 90º
1 turn = 360º
1º = 60´
Page 26 of 286
α = 90º
α = 60º
α = 60º59'
´´
AB
α = 60º59'59''
arcsecond
line
1´ = 60´´
line from point A to point B
ray
line that start from point A
|
perpendicular
perpendicular lines (90º angle)
AC | BC
||
parallel
parallel lines
AB || CD
≅
congruent to
equivalence of geometric shapes and
∆ABC ≅ ∆XYZ
size
~
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
rad
grad
2.3
pi constant
is the ratio between the circumference c = π·d = 2·π·r
and diameter of a circle
radians
grads
radians angle unit
grads angle unit
360º = 2π rad
360º = 400 grad
ALGEBRA SYMBOLS
Symbol
Symbol Name
x
x variable
≡
equivalence
Meaning / definition
unknown value to find
identical to
Example
when 2x = 4, then x = 2
≜
equal by definition
equal by definition
:=
~
≈
equal by definition
equal by definition
approximately equal
approximately equal
weak approximation
approximation
11 ~ 10
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
parentheses
brackets
braces
calculate expression inside first
calculate expression inside first
set
1000000 ≫ 1
2 * (3+5) = 16
[(1+2)*(1+5)] = 18
⌊x⌋
floor brackets
rounds number to lower integer
⌊4.3⌋= 4
⌈x⌉
x!
|x|
f (x)
ceiling brackets
rounds number to upper integer
exclamation mark
single vertical bar
function of x
factorial
absolute value
maps values of x to f(x)
⌈4.3⌉= 5
4! = 1*2*3*4 = 24
| -5 | = 5
f (x) = 3x+5
(f ◦g)
function composition
(f ◦g) (x) = f (g(x))
f (x)=3x, 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}
change / difference
∆ = b2 - 4ac
summation - sum of all values in range of
series
x ∈ [2,6]
∆t = t1 - t0
∆
∆
delta
discriminant
∑
sigma
Page 27 of 286
∑ xi= x1+x2+...+xn
∑∑
∏
double summation
capital pi
product - product of all values in range of
series
∏ xi=x1·x2·...·xn
e = 2.718281828...
e = lim (1+1/x)x , x→∞
e constant / Euler's
number
Euler-Mascheroni
constant
golden ratio
e
γ
φ
2.4
sigma
γ = 0.527721566...
golden ratio constant
LINEAR ALGEBRA SYMBOLS
Symbol
·
×
A⊗B
Symbol Name
dot
cross
Meaning / definition
scalar product
vector product
Example
a·b
a×b
tensor product
tensor product of A and B
A⊗B
inner product
[]
()
|A|
det(A)
|| x ||
2.5
brackets
parentheses
determinant
determinant
double vertical bars
matrix of numbers
matrix of numbers
determinant of matrix A
determinant of matrix A
norm
AT
transpose
matrix transpose
(AT)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
PROBABILITY AND STATISTICS SYMBOLS
Symbol
P(A)
P(A ∩ B)
P(A ∪ B)
P(A | B)
f (x)
F(x)
µ
E(X)
E(X | Y)
Symbol Name
probability function
probability of events
intersection
probability of events
union
conditional probability
function
probability density
function (pdf)
cumulative distribution
function (cdf)
population mean
expectation value
Meaning / definition
probability of event A
P(A) = 0.5
Example
probability that of events A and B
P(A∩B) = 0.5
probability that of events A or B
P(A∪B) = 0.5
probability of event A given event B
occured
P(A | B) = 0.3
P(a ≤ x ≤ b) = ∫ f (x) dx
F(x) = P(X ≤ x)
mean of population values
expected value of random variable X
expected value of random variable X
conditional expectation
given Y
Page 28 of 286
µ = 10
E(X) = 10
E(X | Y=2) = 5
var(X)
σ2
variance
variance
std(X)
standard deviation
σX
standard deviation
variance of random variable X
variance of population values
standard deviation of random variable
X
standard deviation value of random
variable X
middle value of random variable x
median
covariance
corr(X,Y)
correlation
ρX,Y
correlation
∑
summation
summation - sum of all values in range
of series
∑∑
double summation
double summation
Mo
mode
value that occurs most frequently in
population
MR
mid-range
MR = (xmax+xmin)/2
Md
Q1
sample median
lower / first quartile
Q3
x
s2
s
zx
X~
N(µ,σ2)
U(a,b)
exp(λ)
gamma(c, λ)
χ 2(k)
standard score
zx = (x-x) / sx
distribution of X
normal distribution
uniform distribution
exponential distribution
distribution of random variable X
gaussian distribution
equal probability in range a,b
f (x) = λe-λx , x≥0
gamma distribution
f (x) = λ c xc-1e-λx / Γ(c), x≥0
chi-square distribution
f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) )
F distribution
Bin(n,p)
binomial distribution
f (k) = nCk pk(1-p)n-k
Poisson(λ)
Poisson distribution
f (k) = λke-λ / k!
geometric distribution
f (k) = p (1-p) k
HG(N,K,n)
Bern(p)
σX = 2
half the population is below this value
25% of population are below this value
50% of population are below this value
median / second quartile
= median of samples
upper / third quartile
75% of population are below this value
sample mean
average / arithmetic mean
x = (2+5+9) / 3 = 5.333
sample variance
population samples variance estimator s 2 = 4
sample standard
population samples standard deviation
s=2
deviation
estimator
F (k1, k2)
Geom(p)
std(X) = 2
covariance of random variables X and
cov(X,Y) = 4
Y
correlation of random variables X and
corr(X,Y) = 3
Y
correlation of random variables X and
ρX,Y = 3
Y
cov(X,Y)
Q2
var(X) = 4
σ2 = 4
hyper-geometric
distribution
Bernoulli distribution
Page 29 of 286
X ~ N(0,3)
X ~ N(0,3)
X ~ U(0,3)
2.6
COMBINATORICS SYMBOLS
Symbol
Symbol Name
n!
factorial
nPk
Meaning / definition
n! = 1·2·3·...·n
Example
5! = 1·2·3·4·5 = 120
permutation
5P3
= 5! / (5-3)! = 60
combination
5 C3
= 5!/[3!(5-3)!]=10
n Ck
2.7
SET THEORY SYMBOLS
Symbol
{}
set
A∩B
intersection
A∪B
union
A⊆B
subset
A⊂B
proper subset / strict
subset
Meaning / definition
Example
a collection of elements
A={3,7,9,14}, B={9,14,28}
objects that belong to set A and set
A ∩ B = {9,14}
B
objects that belong to set A or set
A ∪ B = {3,7,9,14,28}
B
subset has less elements or equal to
{9,14,28} ⊆ {9,14,28}
the set
subset has less elements than the
{9,14} ⊂ {9,14,28}
set
A⊄B
not subset
left set not a subset of right set
{9,66} ⊄ {9,14,28}
A⊇B
superset
set A has more elements or equal
to the set B
{9,14,28} ⊇ {9,14,28}
A⊃B
proper superset / strict
superset
set A has more elements than set B {9,14,28} ⊃ {9,14}
A⊅B
2A
not superset
set A is not a superset of set B
power set
all subsets of A
Ƅ (A)
power set
all subsets of A
A=B
equality
both sets have the same members
all the objects that do not belong to
set A
objects that belong to A and not to
B
objects that belong to A and not to
B
objects that belong to A or B but
not to their intersection
objects that belong to A or B but
not to their intersection
A={3,9,14}, B={3,9,14}, A=B
Ac
Symbol Name
complement
A\B
relative complement
A-B
relative complement
A∆B
symmetric difference
A⊖B
symmetric difference
{9,14,28} ⊅ {9,66}
A={3,9,14},
B={1,2,3}, A-B={9,14}
A={3,9,14},
B={1,2,3}, A-B={9,14}
A={3,9,14},
B={1,2,3}, A ∆ B={1,2,9,14}
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
(a,b)
not element of
no set membership
A={3,9,14}, 1 ∉ A
A×B
cartesian product
|A|
#A
ordered pair
cardinality
cardinality
collection of 2 elements
set of all ordered pairs from A and
B
the number of elements of set A
A={3,9,14}, |A|=3
the number of elements of set A
A={3,9,14}, #A=3
Page 30 of 286
‫א‬
Ø
U
infinite cardinality
Ø={}
set of all possible values
C = {Ø}
ℕ0 = {0,1,2,3,4,...}
0 ∈ ℕ0
ℕ1
aleph
empty set
universal set
natural numbers set (with
zero)
natural numbers set
(without zero)
ℕ1 = {1,2,3,4,5,...}
6 ∈ ℕ1
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
complex numbers set
ℂ = {z | z=a+bi, -∞<a<∞,
∞<b<∞}
ℕ0
2.8
-
6+2i ∈ ℂ
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
∀
for all
∃
there exists
∄
there does not exists
∴
therefore
∵
because / since
if and only if
Page 31 of 286
2.9
CALCULUS & ANALYSIS SYMBOLS
Symbol
Symbol Name
Meaning / definition
limit
ε
e
y'
y ''
y(n)
limit value of a function
e constant / Euler's number
derivative
second derivative
nth derivative
represents a very small number, near
zero
e = 2.718281828...
derivative - Leibniz's notation
derivative of derivative
n times derivation
e = lim (1+1/x)x , x→∞
(3x3)' = 9x2
(3x3)'' = 18x
(3x3)(3) = 18
derivative
derivative - Lagrange's notation
d(3x3)/dx = 9x2
second derivative
derivative of derivative
d2(3x3)/dx2 = 18x
nth derivative
n times derivation
time derivative
derivative by time - Newton notation
time second derivative
derivative of derivative
epsilon
∫
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
∰
[a,b]
(a,b)
i
z*
z
closed volume integral
closed interval
open interval
imaginary unit
complex conjugate
complex conjugate
[a,b] = {x | a ≤ x ≤ b}
(a,b) = {x | a < x < b}
i ≡ √-1
z = a+bi → z*=a-bi
z = a+bi → z = a-bi
z = 3 + 2i
z* = 3 + 2i
z = 3 + 2i
nabla / del
gradient / divergence operator
∇f (x,y,z)
vector
unit vector
x*y
ε→0
∂(x2+y2)/∂x = 2x
partial derivative
∇
Example
convolution
y(t) = x(t) * h(t)
ℒ
Laplace transform
F(s) = ℒ{f (t)}
ℱ
δ
Fourier transform
X(ω) = ℱ{f (t)}
delta function
Page 32 of 286
PART 3: AREA, VOLUME AND SURFACE AREA
3.1
AREA
Triangle:
Rectangle:
Square:
Parallelogram:
Rhombus:
Trapezium:
Quadrilateral:
1
1
a 2 sin B sin C
bh = ab sin C =
= s (s − a )(s − b )(s − c )
2
2
2 sin A
A = lw
A = a2
A = bh = ab sin A
A = a 2 sin A
a+b
A = h

 s 
A=
(s − a )(s − b )(s − c )(s − d ) − abcd × cos 2  ∠AB + ∠CD 
A=

d1 d 2 sin I
2
Rectangle with rounded corners: A = lw − r 2 (4 − π )
A=
Regular Hexagon:
Regular Octagon:
Regular Polygon:
3.2
(
A=
)
2
na
 180 
4 tan 

 n 
VOLUME
Cube:
Cuboid:
Pyramid:
Tetrahedron:
Octahedron:
Dodecahedron:
Icosahedron:
3.3
3 3 × a2
2
A = 2 1+ 2 × a2
A=
V = a3
V = abc
1
V = × A(b ) × h
3
2
V=
× a3
12
2
V=
× a3
3
15 + 7 5
V=
× a3
4
53+ 5
V=
× a3
12
(
)
SURFACE AREA:
Cube:
Cuboids:
SA = 6a 2
SA = 2(ab + bc + ca )
Tetrahedron:
SA = 3 × a 2
Page 33 of 286
2

Octahedron:
SA = 2 × 3 × a 2
Dodecahedron:
SA = 3 × 25 + 10 5 × a 2
Icosahedron:
Cylinder:
SA = 5 × 3 × a 2
SA = 2πr (h + r )
3.4
MISELANIOUS
Diagonal of a Rectangle
d = l 2 + w2
d = a2 + b2 + c2
a
Longest Diagonal (Even Sides)
=
 180 
sin 

 n 
a
Longest Diagonal (Odd Sides)
=
 90 
2 sin  
 n 
Total Length of Edges (Cube):
= 12a
Total Length of Edges (Cuboid): = 4(a + b + c )
Diagonal of a Cuboid
Circumference
Perimeter of rectangle
Semi perimeter
Euler’s Formula
3.5
C = 2πr = πd
P = 2(a + b )
P
s=
2
Faces + Verticies = Edges + 2
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
r1=radius 1
Page 34 of 286
s=semi-perimeter
SA=Surface Area
V=Volume
w=width
Page 35 of 286
PART 4: ALGEBRA
4.1
POLYNOMIAL FORMULA:
Qudaratic:
Where ax 2 + bx + c = 0 ,
− b ± b 2 − 4ac
2a
3
2
Where ax + bx + cx + d = 0 ,
b
Let, x = y −
3a
3
2
b 
b 
b 



∴ a y −  + b y −  + c y −  + d = 0
3a 
3a 
3a 





b2 
bc 
2b 3
ay 3 +  c −  y +  d +
−  = 0
2
3a 
3a 
27a


x=
Cubic:


b2 
2b 3
bc 
 c − 
 d +
− 
2
3a 
3a 
27a
y3 + 
y+
=0
a
a


b2 
bc 
2b 3
 c − 
 d +
− 
2
3a 
3a 
27a
y3 + 
y = −
a
a
2

b 
 c − 
3a 
Let, A = 
= 3st...(1)
a

2b 3
bc 
 d +

−
27 a 2 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 ) + 3st (s − t ) = s 3 − t 3
3
(s
3
− 3s 2 t + 3st 2 − t 3 ) + (3s 2 t − 3st 2 ) = s 3 − t 3
Solving (1) for s and substituting into (2) yields:
Let, u = t 3
A3
∴ u + Bu −
=0
27
2
Page 36 of 286
ie : αu 2 + βu + γ = 0
α =1
β =B
A3
γ =−
27
− β ± β 2 − 4αγ
u=
2α
u=
− B ± B2 +
4 A3
27
2
4 A3
−B± B +
27
2
2
∴t = 3 u =
3
Substituting into (2) yields:

3 
 − B ± B2 + 4A 
3
27 
s3 = B + t 3 = B + 

2







3 
 − B ± B2 + 4A 
3
27 
∴s = 3 B + 

2






Now, y = s − t
3
3
3

3 
4 A3
 − B ± B2 + 4A 
2
−
B
±
B
+
3
27  3
27
∴y = 3 B+
 −
2
2






b
Now, x = y −
3a
3



3 
3 

4
A
4
A
 − B ± B2 +

− B ± B2 +


3
3


27
27  b

x = 3 B+
−
 −

 3a
2
2












2
3


b 
2b
bc 
 c − 
 d +
− 
2
3a 
3a 
27 a
Where, A = 
&B = −
a
a
Page 37 of 286
4.2
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 38 of 286
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:
and,
Page 39 of 286
4.3
LIMIT MANIPULATIONS:
n→∞
(
n→∞
n→∞
)(
)
lim(an ± bn ) = lim(an ) ± lim(bn )
( )
lim(a b ) = (lim(a ))(lim(b ))
lim( f (a )) = f (lim(a ))
n→∞
n→∞
lim(kan ) = k lim(an )
n→∞
n→∞
4.4
n n
n
n→∞
n
n→∞
n→∞
n
n
SUMATION MANIPULATIONS:
, where C is a constant
4.5
COMMON FUNCTIONS:
Constant Function:
y=a or f (x)=a
Page 40 of 286
Graph is a horizontal line passing through the point (0,a)
x=a
Graph is a vertical line passing through the point (a,0)
Line/Linear Function:
y = mx + c
Graph is a line with point (0,c) and slope m.
Where the gradient is between any two points ( x1 , y1 ) & ( x2 , y 2 )
rise y 2 − y1
m=
=
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 = a ( x − 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
 − b  − b 
has a 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
  − b   − b 
has a 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
a2
b2
=1
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.
Page 41 of 286
Hyperbola:
(x − h )2 − ( y − k )2
=1
a2
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 center with slope ±
b
.
a
( y − k )2 − (x − h )2
=1
b2
a2
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 through center with slope ±
4.6
b
.
a
LINEAR ALGEBRA:
Vector Space Axioms:
Let V be a set on which addition and scalar multiplication are defined (this means that if u
and v are objects in V and c is a scalar then we’ve defined
and cu in
some way). If the following axioms are true for all objects u, v, and w in V and all
scalars c and k then V is called a vector space and the objects in V are called vectors.
(a)
is in V This is called closed under addition.
This is called closed under scalar multiplication.
(b) cu is in V
(c)
(d)
(e) There is a special object in V, denoted 0 and called the zero vector, such that
for all u in V we have
.
(f) For every u in V there is another object in V, denoted
and called the
negative of u, such that
.
(g)
(h)
(i)
(j)
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 42 of 286
Real Numbers
Complex Numbers:
Polynomials
All continuous functions
ℜ, ℜ 2 , ℜ 3 ,..., ℜ n (n denotes dimension)
C, C 2 , C 3 ,..., C n (n denotes dimension)
P1 , P2 , P3 ,..., Pn (n denotes the highest order of x)
C [a, b](a & b denote the interval) (This is never a
vector space as it has infinite dimensions)
Rowspace of a spanning set in Rn
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 Rn
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 A x = 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:
c1r1 + c2 r2 + ... + cn rn = 0
Then, c1 = c 2 = c n = 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 , u 2 , u3 ,..., u n }
Page 43 of 286
x 
y
The general vector within the vector space is: w =  
z 
 
...
w = c1u1 + c2 u 2 + c3u3 + ... + cn u n
u11 u 21 u31 ... u n1   c1 
u
u 22 u32 ... u n 2  c2 
12

Therefore,
[w] = u13 u 23 u33 ... u n3  c3 

 
 ... ... ... ... ...   ... 
u1n u 2 n u 3n ... u nn  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:
Real Numbers
 1   0   0   0  
        
 0   1   0   0  


n
S (ℜ ) =  0 ,  0 ,  1 ,...,  0  
... ... ... ... 
        
 0   0   0   1  
Polynomials
S ( Pn ) = 1, x, x 2 , x 3 ,..., x n
{
}
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
< vi , v j >= 
1 i = j
In an orthonormal basis:
u =< u , v1 > v1 + < u , v2 > v2 + < u , v3 > v3 + ...+ < u , vn > vn )
u = c1v1 + c2 v2 + c3v3 + ... + cn vn )
Page 44 of 286
Gram-Schmidt Process:
This finds an orthonormal basis recursively.
In a basis
B = {u1 , u 2 , u3 ,..., u n }
q1 = u1
^
v1 = q1 =
q1
q1
Next vector needs to be orthogonal to v1 ,
q2 = u 2 − < u 2 , v1 > v1
Similarly
q3 = u3 − < u3 , v1 > v1 − < u3 , v2 > v2
qn = u n − < u3 , v1 > v1 − < u 3 , v2 > v2 − ...− < u3 , vn > vn
^
vn = q n =
qn
qn
Coordinate Vector:
If
v = c1e1 + c2 e2 + ... + cn en
 c1 
c 
vB =  2 
 ... 
 
c n 
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 Rn and then translated back into the general vector
space.
Dimension:
Real Numbers
Polynomials
dim(ℜ n ) = n
dim( Pn ) = n + 1
Matricis
dim( M p ,q ) = 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.
4.7
COMPLEX VECTOR SPACES:
Form:
 a1 + ib1 
 a + ib 
2
Cn =  2
 ... 


a n + ibn 
Dot Product:
_
_
_
u • v = u1 v1 + u 2 v 2 + ... + u n v n
Where:
Page 45 of 286
u •v = v•u ≠ v•u
(u + v) • w = u • w + v • w
su • v = s (u • v), s ∈ C
u •u ≥ 0
u • u = 0 iff u = 0
Inner Product:
u = u •u =
u1 + u 2 + ... + u n
2
2
2
d (u , v) = u − v
Orthogonal if u • v = 0
Parallel if u = sv, s ∈ C
4.8
LINEAR TRANSITIONS & TRANSFORMATIONS:
Transition Matrix:
From 1 vector space to another vector space
T (u ) = T (c1u1 + c2 u 2 + c3u3 + ... + cn u n )
T (u ) = c1T (u1 ) + c2T (u 2 ) + c3T (u3 ) + ... + cnT (u n )
Nullity(T)+Rank(T)=Dim(V)=Columns(T)
Change of Basis Transition Matrix:
−1
vB ' = M B ' M B vB
v B ' = C BB 'v B
For a general vector space with the standard basis:
M B = [(v1 ) S | ... | (vn ) S ]
S = {s1 , s 2 ,..., s n }
M B ' = [(u1 ) S | ... | (u m ) S ]
Transformation Matrix:
From 1 basis to another basis
V = span({v1 , v2 , v3 ,..., vn })
B1 = {v1 , v2 , v3 ,..., vn }
U = span({u1 , u 2 , u3 ,..., u m })
B2 = {u1 , u 2 , u3 ,..., u m }
A = [(T (v1 ) )B 2 | (T (v2 ) )B 2 | ... | (T (vn ) )B 2 ]
−1
4.9
A' = C B 'B AC B 'B
INNER PRODUCTS:
Definition:
Axioms:
An extension of the dot product into a general vector space.
1. < u , v >=< v, u >
2. < u , v + w >=< u, v > + < u , w >
Page 46 of 286
3. < ku , v >= k < u, v >
< u , u >≥ 0
4.
< u , u >= 0 iff u = 0
^
u
Unit Vector: u =
u
< u , v > 2 ≤< u , u > × < v, v >
Cavchy-Schuarz Inequality:
Inner Product Space:
1
u =< u, u > 2 = < u, u >
u =< u, u >
2
2
 < u, v > 
 ≤ 1 ⇒ −1 ≤ < u , v > ≤ 1
< u , v > ≤ u × v ⇒ 

u v
 u v 
u ≥ 0, u = 0 iff u = 0
2
2
2
ku = k u
u+v = u + v
Angle between two vectors:
As defined by the inner product,
< u, v >
cos(θ ) =
u v
Orthogonal if: < u , v >= 0
Distance between two vectors:
As defined by the inner product,
d (u , v) = u − v
Generalised Pythagoras for orthogonal vectors:
2
2
2
u+v = u + v
4.10
PRIME NUMBERS:



Determinate: ∆( N ) =  
 
 
1 +

List of Prime Numbers:



if N is odd and prime
 = 1


N +1
 0 if N is odd and composite

2 
 2k + 1  N   
×
∑


 2k + 1  
k =1  N
3
1+  
N
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
Page 47 of 286
811
821
823
827
829
839
853
857
859
863
877
881
883
887
907
911
919
929
937
941
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
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 2p−1(2p−1), where (2p−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
1
2
3
4
2
3
5
7
6
28
496
8128
1
2
3
4
Known to the Greeks
Known to the Greeks
Known to the Greeks
Known to the Greeks
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
13
17
19
31
61
89
107
127
521
607
1279
2203
2281
3217
4253
4423
9689
9941
11213
33550336
8589869056
8
10
12
19
37
54
65
77
314
366
770
1327
1373
1937
2561
2663
5834
5985
6751
1456
1588
1588
1772
1883
1911
1914
1876
1952
1952
1952
1952
1952
1957
1961
1961
1963
1963
1963
Page 48 of 286
Discoverer
First seen in the medieval manuscript,
Codex Lat. Monac.
Cataldi
Cataldi
Euler
Pervushin
Powers
Powers
Lucas
Robinson
Robinson
Robinson
Robinson
Robinson
Riesel
Hurwitz
Hurwitz
Gillies
Gillies
Gillies
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
19937
21701
23209
44497
86243
110503
132049
216091
756839
859433
1257787
1398269
2976221
3021377
6972593
13466917
20996011
24036583
25964951
12003
13066
13973
26790
51924
66530
79502
130100
455663
517430
757263
841842
1791864
1819050
4197919
8107892
12640858
14471465
15632458
1971
1978
1979
1979
1982
1988
1983
1985
1992
1994
1996
1996
1997
1998
1999
2001
2003
2004
2005
43
30402457
18304103
2005
44
45
46
47
32582657
37156667
42643801
43112609
19616714
22370543
25674127
25956377
2006
2008
2009
2008
Tuckerman
Noll & Nickel
Noll
Nelson & Slowinski
Slowinski
Colquitt & Welsh
Slowinski
Slowinski
Slowinski & Gage
Slowinski & Gage
Slowinski & Gage
Armengaud, Woltman, et al.
Spence, Woltman, et al.
Clarkson, Woltman, Kurowski, et al.
Hajratwala, Woltman, Kurowski, et al.
Cameron, Woltman, Kurowski, et al.
Shafer, Woltman, Kurowski, et al.
Findley, Woltman, Kurowski, et al.
Nowak, Woltman, Kurowski, et al.
Cooper, Boone, Woltman, Kurowski, et
al.
Cooper, Boone, Woltman, Kurowski, et
al.
Elvenich, Woltman, Kurowski, et al.
Strindmo, Woltman, Kurowski, et al.
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
220
1,184
2,620
5,020
6,232
10,744
12,285
17,296
63,020
66,928
67,095
69,615
79,750
100,485
122,265
122,368
141,664
142,310
171,856
176,272
284
1,210
2,924
5,564
6,368
10,856
14,595
18,416
76,084
66,992
71,145
87,633
88,730
124,155
139,815
123,152
153,176
168,730
176,336
180,848
Amicable Pairs
1,328,470
1,358,595
1,392,368
1,466,150
1,468,324
1,511,930
1,669,910
1,798,875
2,082,464
2,236,570
2,652,728
2,723,792
2,728,726
2,739,704
2,802,416
2,803,580
3,276,856
3,606,850
3,786,904
3,805,264
Amicable Pairs
1,483,850 8,619,765
1,486,845 8,666,860
1,464,592 8,754,130
1,747,930 8,826,070
1,749,212 9,071,685
1,598,470 9,199,496
2,062,570 9,206,925
1,870,245 9,339,704
2,090,656 9,363,584
2,429,030 9,478,910
2,941,672 9,491,625
2,874,064 9,660,950
3,077,354 9,773,505
2,928,136 10,254,970
2,947,216 10,533,296
3,716,164 10,572,550
3,721,544 10,596,368
3,892,670 10,634,085
4,300,136 10,992,735
4,006,736 11,173,460
Page 49 of 286
9,627,915
10,638,356
10,893,230
10,043,690
9,498,555
9,592,504
10,791,795
9,892,936
9,437,056
11,049,730
10,950,615
10,025,290
11,791,935
10,273,670
10,949,704
10,854,650
11,199,112
14,084,763
12,070,305
13,212,076
185,368
196,724
280,540
308,620
319,550
356,408
437,456
469,028
503,056
522,405
600,392
609,928
624,184
635,624
643,336
667,964
726,104
802,725
879,712
898,216
947,835
998,104
1,077,890
1,154,450
1,156,870
1,175,265
1,185,376
1,280,565
203,432
202,444
365,084
389,924
430,402
399,592
455,344
486,178
514,736
525,915
669,688
686,072
691,256
712,216
652,664
783,556
796,696
863,835
901,424
980,984
1,125,765
1,043,096
1,099,390
1,189,150
1,292,570
1,438,983
1,286,744
1,340,235
4,238,984
4,246,130
4,259,750
4,482,765
4,532,710
4,604,776
5,123,090
5,147,032
5,232,010
5,357,625
5,385,310
5,459,176
5,726,072
5,730,615
5,864,660
6,329,416
6,377,175
6,955,216
6,993,610
7,275,532
7,288,930
7,489,112
7,577,350
7,677,248
7,800,544
7,850,512
8,262,136
4,314,616
4,488,910
4,445,050
5,120,595
6,135,962
5,162,744
5,504,110
5,843,048
5,799,542
5,684,679
5,812,130
5,495,264
6,369,928
6,088,905
7,489,324
6,371,384
6,680,025
7,418,864
7,158,710
7,471,508
8,221,598
7,674,088
8,493,050
7,684,672
7,916,696
8,052,488
8,369,864
11,252,648
11,498,355
11,545,616
11,693,290
11,905,504
12,397,552
12,707,704
13,671,735
13,813,150
13,921,528
14,311,688
14,426,230
14,443,730
14,654,150
15,002,464
15,363,832
15,938,055
16,137,628
16,871,582
17,041,010
17,257,695
17,754,165
17,844,255
17,908,064
18,056,312
18,194,715
18,655,744
12,101,272
12,024,045
12,247,504
12,361,622
13,337,336
13,136,528
14,236,136
15,877,065
14,310,050
13,985,672
14,718,712
18,087,818
15,882,670
16,817,050
15,334,304
16,517,768
17,308,665
16,150,628
19,325,698
19,150,222
17,578,785
19,985,355
19,895,265
18,017,056
18,166,888
22,240,485
19,154,336
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 50 of 286
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 51 of 286
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 52 of 286
358336
418904
366556
274924
275444
243760
376736
381028
285778
152990
122410
97946
48976
45946
22976
22744
19916
17716
2^6*11*509
2^3*52363
2^2*91639
2^2*13*17*311
2^2*13*5297
2^4*5*11*277
2^5*61*193
2^2*95257
2*43*3323
2*5*15299
2*5*12241
2*48973
2^4*3061
2*22973
2^6*359
2^3*2843
2^2*13*383
2^2*43*103
This list is exhaustive for known social numbers where
C>4
4.11
GOLDEN RATIO & FIBONACCI SEQUENCE:
Relationship:
Infinite Series:
Continued Fractions:
Page 53 of 286
Trigonometric Expressions:
Fibonacci Sequence:
F ( n) =
ϕ n − (1 − ϕ ) n
5
=
ϕ n − (−ϕ ) − n
5
n
1   1 + 5   1 − 5  

 −

F ( n) =
5   2   2  


n
4.12
FERMAT’S LAST THEOREM:
a n + b n ≠ c n for integers a, b & c and n > 2
Proposed by Fermat in 1637 and proved by Andrew Wiles in 1994. The proof is too long
to be written here. See: http://www.cs.berkeley.edu/~anindya/fermat.pdf
Page 54 of 286
PART 5: COUNTING TECHNIQUES & PROBABILITY
5.1
2D
n(n + 1)
2
2
n = Tn + Tn −1
Triangle Number
Tn =
Square Number
Tn = n 2
n(3n − 1)
Tn =
2
Pentagonal Number
5.2
3D
Tetrahedral Number
Square Pyramid Number
n 3 + 3n 2 + 2n
6
3
2n + 3n 2 + n
Tn =
6
Tn =
5.3
PERMUTATIONS
Permutations:
= n!
Permutations (with repeats):
=
5.4
n!
(groupA)!×(groupB )!×...
COMBINATIONS
Ordered Combinations:
Unordered Combinations:
n!
(n − p )!
n
n!
=   =
 p  p!(n − p )!
=
Ordered Repeated Combinations: = n p
Unordered Repeated Combinations: =
Grouping:
( p + n − 1)!
p!×(n − 1)!
 n  n − n1  n − n1 − n2 
n!

... =
=  
n3
n1!n2 !n3!...nr !
 n1  n2 

5.5
MISCELLANEOUS:
Total Number of Rectangles and Squares from a a x b rectangle:
∑ = Ta × Tb
Number of Interpreters:
Max number of pizza pieces:
Max pieces of a crescent:
Max pieces of cheese:
= TL −1
c(c + 1)
=
+1
2
c(c + 3)
=
+1
2
c 3 + 5c
=
+1
6
Page 55 of 286
l (3l + 1)
2
n−d
Different arrangement of dominos: = 2 × n!
=
Cards in a card house:
b 
a − MOD  
a
1
a
Unit Fractions:
=
+
b
b 


b
INT   + 1 b INT   + 1
a 
a  

Angle between two hands of a clock: θ = 5.5m − 30h
Winning Lines in Noughts and Crosses: = 2(a + 1)
Bad Restaurant Spread:
=
P
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= nth 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!
2!
3!
4!
5!
6!
7!
1 (by definition)
1
2
6
24
120
720
5040
11!
12!
13!
14!
15!
16!
17!
39916800
479001600
6227020800
87178291200
1307674368000
20922789888000
355687428096000
Page 56 of 286
8!
9!
10!
Approximation:
5.7
40320
362880
3628800
n!= 2π × n
18!
19!
20!
n+
1
2
6402373705728000
121645100408832000
2432902008176640000
× e −n
(within 1% for n>10)
THE DAY OF THE WEEK:
This only works after 1753

 31m   y   y   y  
= MOD7 d + 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:
∑P =1
5.9
VENN DIAGRAMS:
Complementary Events:
()
1 − P ( A) = P A
m
Totality:
P( A) = ∑ P( A | Bi ) P( Bi )
i =1
Conditional Probability:
Union :
Independent Events:
Mutually Exclusive:
P( A) = P( A ∩ B ) + P( A ∩ B ' )
P( A ∩ B )
P( A | B ) =
P (B )
P( A ∩ B ) = P(B ) ⋅ P( A | B )
P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B )
P ( A ∩ B ) = P ( A) ⋅ P ( B )
P ( A ∪ B ) = P ( A) + P ( B ) − P ( A) ⋅ P ( B )
P (B | A) = P (B )
P( A ∩ B ) = 0
P( A ∩ B ') = P( A)
P ( A ∪ B ) = P ( A) + P ( B )
P ( A ∪ B ') = P ( B ')
Baye’s Theorem:
Page 57 of 286
P( B | A) =
P( A | B ) P( B)
P( A | B ) P( B)
=
P ( A)
P ( A | B) P ( B) + P( A | B' ) P( B' )
m
P( A) = ∑ P( A ∩ Bi )
Event’s Space:
i =1
5.11 BASIC STATISTICAL OPERATIONS:
Variance:
v =σ2
∑ xi
Mean:
µ=
ns
x −µ
Standardized Score:
z= i
σ
Confidence Interval:
5.12
DISCRETE RANDOM VARIABLES:
Standard Deviation:
σ=
∑ (x
i
−x
)
2
ns
Expected Value:
i
E[ X ] = ∑ P ( xi ) × xi
1
E[aX + b] = aE[ X ] + b
Variance:
∑ (x
v=
i
−x
)
2
ns
v = (E [x − E[ x]])
2
v = E[ x 2 ] − (E[ x])
2
var[aX + b] = a 2 var[ X ]
Probability Mass Function: P( x) = f ( x) = P( X = x)
Cumulative Distribution Function: F ( x ) = P( X ≤ x)
5.13
COMMON DRVs:
Bernoulli Trial:
Definition:
Outcomes:
Probability:
Expected Value:
Variance:
1 trial, 1 probability that is either fail or success
S X = {0,1}
x =1
 p
PX ( x) = 
1 − p x = 0
E[ X ] = p
Var[ X ] = p − p 2 = p(1 − p )
Binomial Trial:
Definition:
Outcomes:
Repeated Bernoulli Trials
S X = {0,1,2,3,...n}
Page 58 of 286
n
x
n− x
PX ( x ) =   ⋅ ( p ) ⋅ (1 − p )
x
 
E[ X ] = np
Var[ X ] = np(1 − p )
Probability:
Expected Value:
Variance:
n=number to choose from
p=probability of x occurring
x=number of favorable results
Geometric Trial:
Number of Bernoulli Trials to get 1st Success.
Definition:
Outcomes:
S X = {0,1,2,3,...}
PX (x ) = p (1 − p )
Probability:
x −1
Negative Binomial Trial:
Definition:
Number to 1st get to n success.
Probability:
 x − 1 x
 p (1 − p )n− x
PX ( x ) = 
 n − 1
5.14 CONTINUOUS RANDOM VARIABLES:
Probability Density Function: = f (x )
∞
If
∫ f ( x)dx = 1 & f ( x) ≥ 0 for − ∞ ≤ x ≤ ∞
−∞
x
Cumulative Distribution Function: = F ( x ) = P ( X ≤ x ) =
∫ f ( x ) dx
−∞
b
P(a ≤ X ≤ b) = F (b) − F (a) = ∫ f ( x)dx
Interval Probability:
a
E ( x) =
Expected Value:
∞
∫ x × f ( x)dx
−∞
E ( g ( x)) =
∞
∫ g ( x) × f ( x)dx
−∞
Var ( X ) = E ( X 2 ) − ( E ( X )) 2
Variance:
5.15
COMMON CRVs:
Uniform Distribution:
Declaration:
PDF:
X ~ Uniform(a, b)
 1

a≤ x≤b
f ( x) =  b − a
 0
otherwise
Page 59 of 286
 0
x−a
F ( x ) = ∫ f ( x ) dx = 
−∞
b − a
 1
x
CDF:
Expected Value:
Variance:
a+b
2
(b − a )2
=
12
=
Exponential Distribution:
Declaration:
PDF:
X ~ Exponential (λ )
x<0
 0
f ( x ) =  −λx
x≥0
λe
Page 60 of 286
x<a
a≤ x≤b
x>b

x
F ( x) =
CDF:
0
∫ f ( x)dx = 1 − e λ
− x
−∞
Expected Value:
=
Variance:
=
x<0
x≥0
1
λ
1
λ2
Normal Distribution:
Declaration:
Standardized Z Score:
σ
−1 x − µ 

σ 
2

1
f ( x) =
e2
σ 2π
PDF:
CDF:
Expected Value:
=µ
Variance:
5.16
X ~ Normal ( µ , σ 2 )
x−µ
Z=
=
−1
z2
1
e2
σ 2π
Φ(Z ) (The integration is provided within statistic tables)
=σ
2
MULTIVARIABLE DISCRETE:
Probability:
Marginal Distribution:
P ( X = x, Y = y ) = f ( x, y )
P( X ≤ x, Y ≤ y ) = ∑ f ( x, y ) over all values of x & y
f X ( x ) = ∑ f ( xi , y )
y
f Y ( y ) = ∑ f ( x, yi )
x
Expected Value:
E[ X ] = ∑ x × f X ( x)
x
E[Y ] = ∑ y × f Y ( y )
y
E[ X , Y ] = ∑∑ x × y × f X ,Y ( x, y )
x
Independence:
y
f ( x, y ) = f X ( x ) × f Y ( y )
Page 61 of 286
Covariance:
5.17
Cov = E[ X , Y ] − E[ X ] × E[Y ]
MULTIVARIABLE CONTINUOUS:
Probability:
y x
P ( X ≤ x, Y ≤ y ) =
∫ ∫ f ( x, y)dxdy
−∞ −∞
y
P(Y < y ) = P(−∞ < X < ∞, Y < y ) =
∫f
Y
( y )dy
−∞
Marginal Distribution:
b
f X ( x ) = ∫ f ( x, y )dy where a & b are bounds of y
a
b
f Y ( y ) = ∫ f ( x, y )dx where a & b are bounds of x
a
Expected Value:
∞
E[ X ] = ∫ x × f X ( x)dx
−∞
E[Y ] =
∞
∫ y× f
Y
( y )dy
−∞
∞ ∞
E[ X , Y ] =
∫ ∫ x× y× f
X ,Y
( x, y )dxdy
− ∞−∞
Independence:
Covariance:
Correlation Coefficient:
f ( x, y ) = f X ( x ) × f Y ( y )
Cov = E[ X , Y ] − E[ X ] × E[Y ]
Cov( X , Y )
ρ X ,Y =
σ XσY
ABBREVIATIONS
σ = Standard Deviation
µ = mean
ns = number of scores
p = probability of favourable result
v = variance
xi = Individual x score
x = mean of the x scores
z = Standardized Score
Page 62 of 286
Page 63 of 286
PART 6: FINANCIAL
6.1
GENERAL FORMUALS:
p = s−c
p
m = × 100
c
= P(1 + tr )
Profit:
Profit margin:
Simple Interest:
Compound Interest:
= P(1 + r )
Continuous Interest:
= Pe rt
t
ABBREVIATIONS (6.1):
c=cost
I=interest
m=profit margin (%)
p=profit
P=premium
r=rate
s=sale price
t=time
6.2
MACROECONOMICS:
GDP:
RGDP:
NGDP:
y = AE = AD = C + I + G + NX
y = Summation of all product quantities multiplied by cost
RGDP = Summation of all product quantities multiplied by base year cost
NGDP = Summation of all product quantities multiplied by current year cost
Growth:
Growth =
Net Exports:
NX = X - M
RGDPCURRENT − RGDPBASE
×100
RGDPBASE
Working Age Population:
Labor Force:
Unemployment:
Natural Unemployment:
WAP = Labor Force + Not in Labor Force
LF = Employed + Unemployed
UE = Frictional + Structural + Cyclical
NUE = Frictional + Structural
Unemployment Rate:
∆UE% =
UE
×100
LF
E
× 100
LF
LF
UE + E
Participation Rate: ∆P% =
×100 =
×100
WAP
WAP
CPI:
CPI = Indexed Average Price of all Goods and Services
CPI CURRENT − CPI BASE
Inflation Rate:
Inflation Rate =
× 100
CPI BASE
Employment Rate: ∆E% =
ABBREVIATIONS (6.2)
AD=Aggregate Demand
Page 64 of 286
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 65 of 286
PART 7: PI
7.1
AREA:
πd 2
Cd
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
r2  Q

Area of a segment (degrees) A = 
× π − sin Q 
2  180

Circle:
A = πr =
Area of an annulus:
A = π r2 − r1
Ellipse :
A=
2
(
π
4
2
2
=
)
 w
= π 
2
lw = πr1 r2
7.2
VOLUME:
Cylinder:
V = πr 2 h
4
Sphere:
V = πr 3
3
1
2
Cap of a Sphere: V = πh 3r1 + h 2
6
1 2
Cone:
V = πr h
3
1
Ice-cream & Cone: V = πr 2 (h + 2r )
3
(
Doughnut:
V = 2π 2 r2 r1 =
2
)
π2
4
πw  w 
V=
l − 
4 
3
4
V = πr1 r2 r3
3
(b + a )(b − a )2
2
Sausage:
Ellipsoid:
7.3
SURFACE AREA:
Sphere:
SA = 4πr 2
Hemisphere: SA = 3πr 2
Doughnut:
SA = 4π 2 r2 r1 = π 2 b 2 − a 2
Sausage:
SA = πwl
(
Cone:
(
SA = πr r + r 2 + h 2
)
)
Page 66 of 286
2
7.4
MISELANIOUS:
Q
Q
×C =
× πr
360
180
Q
l = 2r × sin   = 2 r 2 − h 2
2
2

 1 + 3(r1 − r2 )

(r1 + r2 )2
P ≈ π (r1 + r2 )
2

3(r1 − r2 )
10
+
4
−

(r1 + r2 )2

l=
Length of arc (degrees)
Length of chord (degrees)
Perimeter of an ellipse
7.6







PI:
John Wallis:
Isaac Newton:
James Gregory:
Leonard Euler:
π ≈ 3.14159265358979323846264338327950288...
C
π=
d
∞
4n 2
π 2 2 4 4 6 6 8 8
= × × × × × × × × ... = ∏ 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 
= + 
+

+

 + ...
6 2 2  3 × 23  2 × 4  5 × 25  2 × 4 × 6  7 × 2 7 
π
1 1 1 1 1 1 1
= 1 − + − + − + − ...
4
3 5 7 9 11 13 15
2
π
1
1
1
1
= 2 + 2 + 2 + 2 + ...
6 1
2
3
4
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.
`π
= 1+
1 1 1 1 1 1 1 1 1 1 1 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
i
n→ ∞
Franciscus Vieta:
2
π
=
2+ 2+ 2
2
2+ 2
×
×
× ...
2
2
2
Integrals:
Page 67 of 286
Infinite Series:
n
1 ∞ (− 1) 
25
1
28
26
22
22
1 

 =π
−
−
+
−
−
−
+
6 ∑
10 n 
2 n=0 2  4n + 1 4n + 3 10n + 1 10n + 3 10n + 5 10n + 7 10n + 9 
See also: Zeta Function within Part 17
Continued Fractions:
Page 68 of 286
7.7
CIRCLE GEOMETRY:
Radius of Circumscribed Circle for Rectangles: r =
Radius of Circumscribed Circle for Squares:
r=
a2 + b2
2
a
2
a
Radius of Circumscribed Circle for Triangles: r =
2 sin 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
Radius of Circumscribed Circle:
Radius of Inscribed Circle:
7.8
a
 180 
2 sin 

 n 
a
r=
 180 
2 tan 

 n 
r=
ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7):
A=Angle ‘A’
A=Area
Page 69 of 286
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




−1  w + l − b 
−1  w + l − b







2π  cos 
  sin 2 cos 

2 wl
2wl





2
A = w π −
+
360
2





 w2 + l 2 − b 2

2π  cos −1 

2 wl

− b 2 π − 
360



7.10


 w2 + l 2 − b 2
  sin 2 cos −1 


2wl



+
2
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 70 of 286
  
 

 




  
 

 




PART 8: PHYSICS
8.1
MOVEMENT:
Stopping distance:
v2
s=
− 2a
Centripetal acceleration:
a=
Centripetal force:
FC = ma =
Dropping time :
t=
Force:
F=
Kinetic Energy:
 v2  2
1 − 2 
 c 
1
E k = mv 2
2
2
(
u sin θ )
h=
g
Maximum height of a cannon:
v2
r
mv 2
r
2h
g
ma
3
Potential Energy:
l
g
E p = mgh
Range of a cannon:
s = t (u cos θ ) =
Time in flight of a cannon:
t=
Pendulum swing time:
Universal Gravitation:
t = 2π
2u sin θ
g
mm
F = G 12 2
r
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 = 6.67 × 10 −11
h=height
l=length of a pendulum
m=mass
m1=mass 1
m2=mass 2
Page 71 of 286
2u sin θ
× (u cos θ )
g
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 72 of 286
Moments of Inertia:
Description
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)
Diagram
Formulae
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 73 of 286
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
r2, length h and mass m
or when defining the normalized thickness tn = t/r and letting
r = r2,
then
Sphere (hollow) of
radius r and mass m
Ball (solid) of radius r
and mass m
Page 74 of 286
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 75 of 286
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:
a AVE =
∆V
∆t
Trajectory (Displacement):
Page 76 of 286
Kinetic Energy:
Centripetal Force:
Circular Motion:
, or
,
Angular Momentum:
Page 77 of 286
Torque:
Work:
Laws of Conservation:
Momentum:
Energy:
Force:
∑E
∑F
IN
= ∑ EOUT
NET
= 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 78 of 286
Momentum:
Time Dilation:
Length Contraction:
Relativistic Mass:
.
Page 79 of 286
PART 9: TRIGONOMETRY
9.1
CONVERSIONS:
60°
Degrees 30°
120°
150°
210°
240°
300°
330°
Radians
Grads
33⅓
grad
66⅔
grad
133⅓
grad
166⅔
grad
233⅓
grad
266⅔
grad
333⅓
grad
366⅔
grad
Degrees
45°
90°
135°
180°
225°
270°
315°
360°
Radians
Grads 50 grad
9.2
100
grad
150 grad 200 grad 250 grad 300 grad 350 grad 400 grad
BASIC RULES:
tan θ =
Sin Rule:
Cos Rule:
sin θ
cos θ
a
b
c
sin A sin B sin C
or
=
=
=
=
sin A sin B sin C
a
b
c
2
2
2
b +c −a
or a 2 = b 2 + c 2 − 2bc cos A
cos A =
2bc
Tan Rule:
Auxiliary Angle:
Pythagoras Theorem:
a 2 + b2 = c2
Page 80 of 286
9.3
RECIPROCAL FUNCTIONS
1
secθ =
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 (SINΘ):
•
•
•
•
•
Page 81 of 286
•
9.6
IDENTITIES (COSΘ):
•
•
•
•
•
•
9.7
IDENTITIES (TANΘ):
•
•
•
•
•
•
9.8
IDENTITIES (CSCΘ):
•
•
•
•
•
•
9.9
IDENTITIES (COTΘ):
•
Page 82 of 286
•
•
•
•
•
9.10
ADDITION FORMULAE:
Sine:
Cosine:
Tangent:
Arcsine:
Arccosine:
Arctangent:
9.11
DOUBLE ANGLE FORMULAE:
Sine:
Generally,
n
n
1

sin (nx ) = ∑   cos k ( x )sin n−k ( x )sin  (n − k )π 
2

k =0  k 
Cosine:
Generally,
n
n
1

cos(nx ) = ∑   cos k ( x )sin n−k ( x )cos (n − k )π 
2

k =0  k 
Tangent:
Page 83 of 286
Generally.
n
1

  cos ( x ) sin ( x ) sin  (n − k )π 
∑
sin (nx )
2

k 
tan (nx ) =
=
n
k
cos(nx )
k =0
 n

1
∑  k  cos (x )sin (x ) cos 2 (n − k )π 
n
k
k =0
 
Cot:
9.12
TRIPLE ANGLE FORMULAE:
Sine:
Cosine:
Tangent:
Cot:
9.13
n −k
HALF ANGLE FORMULAE:
Sine:
Cosine:
Tangent:
Cot:
Page 84 of 286
n− k
9.14
POWER REDUCTION:
Sine:
If n is even:
If n is odd:
Cosine:
If n is even:
If n is odd:
Page 85 of 286
Sine & Cosine:
9.15
PRODUCT TO SUM:
9.16
SUM TO PRODUCT:
9.17
HYPERBOLIC EXPRESSIONS:
Hyperbolic sine:
Hyperbolic cosine:
Hyperbolic tangent:
Hyperbolic cotangent:
Page 86 of 286
Hyperbolic secant:
Hyperbolic cosecant:
9.18
HYPERBOLIC RELATIONS:
9.19
MACHIN-LIKE FORMULAE:
Form:
Formulae:
Page 87 of 286
Identities:
for
for
for
for
9.20
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 
9.21
ABBREVIATIONS (9.1-9.19)
A=Angle ‘A’
Page 88 of 286
,
,
,
.
a=side ‘a’
B=Angle ‘B’
b=side ‘b’
B=Angle ‘B’
c=side ‘c’
Page 89 of 286
PART 10: EXPONENTIALS & LOGARITHIMS
10.1
FUNDAMENTAL THEORY:
10.2
IDENTITIES:
10.3
CHANGE OF BASE:
10.4
LAWS FOR LOG TABLES:
Page 90 of 286
10.5
COMPLEX NUMBERS:
10.6
LIMITS INVOLVING LOGARITHMIC TERMS
Page 91 of 286
PART 11: COMPLEX NUMBERS
11.1
GENERAL:
Fundamental:
Standard Form:
Polar Form:
i 2 = −1
z = a + bi
z = rcisθ = r (cos θ + i sin θ )
Argument:
arg( z ) = θ , where tan θ =
Modulus:
mod( z ) = r = z = a + bi = a 2 + b 2
b
a
Conjugate:
z = a − bi
Exponential:
z = r ⋅ e iθ
De Moivre’s Formula:
z = rcisθ
Euler’s Identity:
 θ + 2kπ  , k=0,1,…,(n-1)
z = r cis

 n 
(Special Case when n=2)
e iπ + 1 = 0
1
n
1
n
n −1
2 iπk
n
∑e
=0
(Generally)
k =0
11.2
OPERATIONS:
Addition:
Subtraction:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i.
Multiplication:
(a + bi)(c + di) = ac + bci + adi + bdi 2 = (ac - bd) + (bc + ad)i.
Division:
(a + bi) (a + bi)(c - di) ac + bci - adi + bd  ac + bd   bc - ad 
=
= 2
+ 2
i.
=
2 
2 
(c + di) (c + di)(c - di)
(c + di)(c - di)
c +d  c +d 
Sum of Squares:
11.3
IDENTITIES:
Exponential:
Logarithmic:
Trigonometric:
Page 92 of 286
Hyperbolic:
Page 93 of 286
PART 12: DIFFERENTIATION
For Differential Equations, see Functions
12.1
GENERAL RULES:
Plus Or Minus:
y = f ( x ) ± g ( x ) ± h( x ) ...
y ' = f ' ( x ) ± g ' ( x ) ± h' ( x ) ...
Product Rule:
y = uv
y ' = u ' v + uv'
Quotient Rule:
u
v
u ' v − uv'
y' =
v2
y=
Power Rule:
y = ( f (x) )
n
y ' = n( f ( x ) )
n −1
× f '(x )
Chain Rule:
dy dy du dv
=
×
×
dx du dv dx
Blob Rule:
y=e
f( x )
y ' = f '( x ) ×e
Base A Log:
y = log a f ( x )
y' =
Natural Log:
f( x )
f '(x )
f ( x ) × ln (a )
y = a ln ( f ( x ) )
y' = a ×
f '(x )
f (x )
Exponential (X):
y = kx
y ' = ln k × k x
First Principles:
 f (x+h ) − f ( x ) 


lim
h
h → 0 

EXPONETIAL FUNCTIONS:
f '(x ) =
12.2
Page 94 of 286
12.3
LOGARITHMIC FUNCTIONS:
12.4
TRIGONOMETRIC FUNCTIONS:
12.5
HYPERBOLIC FUNCTIONS:
Page 95 of 286
12.5
PARTIAL DIFFERENTIATION:
First Principles:
ie:
Gradient:
Total Differential:
Chain Rule:
Page 96 of 286
Implicit Differentiation:
Page 97 of 286
Therefore,
Higher Order Derivatives:
Page 98 of 286
PART 13: INTEGRATION
13.1
GENERAL RULES:
[ f( ) ]
∫ f '( ) [ f( ) ] dx = n + 1
n +1
n
Power Rule:
x
x
x
[f ]
a ∫ f '( ) [ f ( ) ] dx = a ( )
n +1
n +1
n
x
By Parts:
+C
x
x
∫ udv = uv − ∫ vdu
f ( x)
Constants:
∫ kdy = kf ( x)
0
13.2
RATIONAL FUNCTIONS:
Page 99 of 286
+C
For
||
13.3
TRIGONOMETRIC FUNCTIONS (SINE):
Page 100 of 286
13.4
TRIGONOMETRIC FUNCTIONS (COSINE):
Page 101 of 286
13.5
TRIGONOMETRIC FUNCTIONS (TANGENT):
13.6
TRIGONOMETRIC FUNCTIONS (SECANT):
Page 102 of 286
13.7
TRIGONOMETRIC FUNCTIONS (COTANGENT):
13.8
TRIGONOMETRIC FUNCTIONS (SINE & COSINE):
Page 103 of 286
also:
also:
also:
Page 104 of 286
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 105 of 286
13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE):
13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT):
13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT):
Page 106 of 286
13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT):
13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT):
13.19 EXPONETIAL FUNCTIONS
for
Page 107 of 286
(erf is the Error function)
where
where
when b≠0, λ≠0 and
Page 108 of 286
for
, which is the logarithmic mean
(!! is the double factorial)
(I0 is the modified Bessel function of the first kind)
13.20 LOGARITHMIC FUNCTIONS
Page 109 of 286
Page 110 of 286
13.21 HYPERBOLIC FUNCTIONS
Page 111 of 286
13.22 INVERSE HYPERBOLIC FUNCTIONS
Page 112 of 286
13.23 ABSOLUTE VALUE FUNCTIONS
13.24 SUMMARY TABLE
Page 113 of 286
13.25 SQUARE ROOT PROOFS
Page 114 of 286
∫
a 2 + x 2 dx
Let x = a tan θ ∴ dx = a sec 2 θdθ → tan θ =
x
a
= ∫ a 2 + (a tan θ ) 2 × a sec 2 θdθ
= ∫ a 2 + a 2 tan 2 θ × a sec 2 θdθ
= ∫ a 2 + a 2 (sec 2 θ − 1) × a sec 2 θdθ
= ∫ a 2 + a 2 sec 2 θ − a 2 × a sec 2 θdθ
= ∫ a 2 sec 2 θ × a sec 2 θdθ
= ∫ a secθ × a sec 2 θdθ
= ∫ a 2 sec 3 θ dθ
= a 2 ∫ secθ × sec 2 θ dθ
u = secθ , dv = sec 2 θdθ
du = secθ tan θdθ , v = tan θ
∴ a 2 ∫ sec 3 θ dθ = secθ × tan θ − ∫ tan θ × secθ tan θdθ
a 2 ∫ sec 3 θ dθ = secθ tan θ − ∫ tan 2 θ secθdθ
a 2 ∫ sec 3 θ dθ = secθ tan θ − ∫ (sec 2 θ − 1) secθdθ
a 2 ∫ sec 3 θ dθ = secθ tan θ − ∫ sec 3 θ − secθdθ
a 2 ∫ sec 3 θ dθ = secθ tan θ − ∫ sec 3 θdθ + ∫ secθdθ
2a 2 ∫ sec 3 θ dθ = secθ tan θ + ∫ secθdθ
(
)
1
secθ tan θ + ∫ secθdθ
2a 2
1
3
∫ sec θ dθ = 2a 2 (secθ tan θ + ln secθ + tan θ ) + C
1  a2 + x2 s
a 2 + x 2 s 
∴ ∫ a 2 + x 2 dx = 2 
× + ln
+
+C
2a 
a
a
a
a


∫ sec
3
θ dθ =
Page 115 of 286
∫
a 2 − x 2 dx
Let x = a sin θ ∴ dx = a cosθdθ → sin θ =
x
a
= ∫ a 2 − (a sin θ ) × a cosθdθ
2
= ∫ a 2 − a 2 sin 2 θ × a cosθdθ
(
)
= ∫ a 2 − a 2 1 − cos 2 θ × a cosθdθ
= ∫ a 2 − a 2 + a 2 cos 2 θ × a cosθdθ
= ∫ a 2 cos 2 θ × a cosθdθ
= ∫ a cosθ × a cosθdθ
= ∫ a 2 cos 2 θdθ
= a 2 ∫ cos 2 θdθ
= a2 ∫
a2
2
a2
=
2
=
1 + cos(2θ )
dθ
2
∫1 + cos(2θ )dθ
sin (2θ ) 

θ + 2  + C
a2 
2 sin θ cosθ 
=
θ+

 + C
2 
2
a2
[θ + sin θ cosθ ] + C
2
a2 
a2 − x2
x x
=
arcsin  + ×
2 
a
a a
=

+C

Page 116 of 286
∫
x 2 − a 2 dx
Let x = a secθ ∴ dx = a secθ tan θdθ → secθ =
=∫
x
a
(a secθ )2 − a 2 × a secθ tan θdθ
= ∫ a 2 sec 2 θ − a 2 × a secθ tan θdθ
(
)
= ∫ a 2 1 + tan 2 θ − a 2 × a secθ tan θdθ
= ∫ a 2 + a 2 tan 2 θ − a 2 × a secθ tan θdθ
= ∫ a 2 tan 2 θ × a secθ tan θdθ
= ∫ a tan θ × a secθ tan θdθ
= ∫ a 2 tan 2 θ secθdθ
= a 2 ∫ tan 2 θ secθdθ
(
)
= a 2 ∫ sec 2 θ − 1 secθdθ
= a 2 ∫ sec 3 θ − secθdθ
(
= a 2 ∫ sec 3 θdθ − ∫ secθdθ
)
 1


= a 2   2 (secθ tan θ + ln secθ + tan θ ) − (ln secθ + tan θ ) + C

  2a

1
= (secθ tan θ + ln secθ + tan θ ) − a 2 (ln secθ + tan θ ) + C
2
1 x
x2 − a2
x
x 2 − a 2  2  x
x 2 − a 2 
=  ×
+ ln +
− a ln +
+C

 a

2a
a
a
a
a




=
1 x × x2 − a2 1 x
x2 − a2
x
x2 − a2
+
ln
+
−
ln
+
+C
2
a2
2 a
a
a
a
1 x × x2 − a2  1
x
x2 − a2
2
=
+  − a  ln +
+C
2
a
a2
2
 a
13.26 CARTESIAN APPLICATIONS
b
Area under the curve:
A = ∫ f ( x ) dx
a
b
Volume:
V =∫A
a
b
Volume about x axis:
b
[ ]
Vx = π ∫ [ y ] dx = π ∫ f ( x ) dx
2
a
2
a
Page 117 of 286
d
Volume about y axis:
V y = π ∫ [x ] dy
2
c
Surface Area about x axis: SA = 2π ∫ f ( x ) 1 + ( f '( x ) ) dx
b
2
a
b
Length wrt x-ordinates:
L=∫
a
d
Length wrt y-ordinates:
L=∫
c
t2
Length parametrically:
L=
∫
t1
2
 dy 
1 +   dx
 dx 
2
 dx 
1 +   dy (Where the function is continually increasing)
 dy 
2
2
 dx   dy 
  +   dt
 dt   dt 
Page 118 of 286
PART 14: FUNCTIONS
14.1
COMPOSITE FUNCTIONS:
Odd ± Odd = Odd
Odd ± Even = Neither
Even ± Even = Even
Odd x Odd = Even
Odd / Odd = Odd
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
a
∫ f ( x)dx = 0
If f(x) is odd:
−a
a
∫
If f(x) is even:
−a
14.2
a
f ( x)dx = 2∫ f ( x)dx
0
MULTIVARIABLE FUNCTIONS:
Limit:
lim
( x , y )→( 0 , 0 )
(f )=
( x, y )
lim
( x , mx ) →( 0 , 0 )
Discriminant:
D( x0 , y0 ) = z xx z yy − (z xy )
Critical Points:
z = f ( x, y)
(f
( x , mx )
)=
lim ( f ( x ,mx ) )
( x )→( 0 )
2
z x = 0

z y = 0
 If the critical point (x0,y0) is a local maximum, then
Solve for:
D(x0,y0) >= 0
fxx(x0,y0) <= 0
and fyy(x0,y0) <= 0
 If D(x0,y0) > 0, and either
fxx(x0,y0) < 0
or fyy(x0,y0) < 0
then the critical point (x0,y0) is a local maximum.
 If the critical point (x0,y0) is a local minimum, then
D(x0,y0) >= 0
fxx(x0,y0) >= 0
and fyy(x0,y0) >= 0
 If D(x0,y0) > 0, and either
fxx(x0,y0) > 0
or fyy(x0,y0) > 0
then the critical point (x0,y0) is a local minimum.
Page 119 of 286
 If the critical point (x0,y0) is a saddle point, then
D(x0,y0) <= 0
 If
D(x0,y0) < 0,
then the critical point (x0,y0) is a saddle point.
14.3
FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS:
Separable:
dy f ( x)
=
dx g ( y )
g ( y )dy = f ( x)dx
∫ g ( y )dy = ∫ f ( x)dx
Linear:
dy
+ P ( x ) × y = Q ( x)
dx
P ( x ) dx
I ( x) = e ∫
y=
Homogeneous:
1
I ( x)
(∫ I ( x) × Q( x)dx )
f ( λx , λ y ) = f ( x , y )
dy
 y
= f ( x, y ) = F  
dx
 x
y dy
dv
Let v ( x ) = ,∴
=v+ x
x dx
dx
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
= f ( x, y ) → M ( x, y )dx + N ( x, y )dy = 0
dx
If:
M y = Nx
When:
FX = M & FY = N
Page 120 of 286
Therefore,
F = ∫ M ( x , y ) dx =Φ ( x , y ) + g ( y )
∂
(Φ + g ( y ) ) = Φ Y + g ' ( y ) = N
∂y
∴ g ( y ) = ...
FY =
So:
14.4
F ( x , y ) = Φ ( x, y ) + g ( y ) = C
SECOND ORDER
d2y
dy
+ b + cy = f ( x)
2
Where dx
dx
ay ' '+by'+cy = f ( x)
a
Homogeneous:
ay' '+by'+cy = 0
⇒ am 2 + bm + c = 0
m=
− b ± b 2 − 4ac
2a
There are three possible outcomes:
1)
m1 , m2 where m1 ≠ m2
2)
m1 ,m2 where m1 = m2
3)
m1, 2 = α ± β j
⇒ yh = Ae m1x + Be m2 x
⇒ y h = ( A + Bx )e m1x
⇒ y h = eαx ( A cos(βx ) + B sin (βx ))
Undetermined Coefficients
Where f ( x) is in the form of
1) A polynomial
2)
α sin (kx )
3)
αe kt
⇒ y p = An x n + An−1 x n−1 + A1 x + A0
⇒ y p = A sin(kx) + B cos(kx)
⇒ y p = Ae kt
NB: Multiplication is OK: eg:
f ( x) = 3 x 3 e x
( )(
= (e )(Bx
y p = Ae x Bx 3 + Cx 2 + Dx + E
yp
x
3
+ Cx + Dx + E
2
)
)
NB: If yp is part of yc, you multiply yp by x
To determine the unknown variables, substitute back into the original equation with
y p , y ' p , y ' ' p and compare the coefficients.
Then,
y = y h + y p1 + y p 2 + y p 3 + ...
Variation of Parameters
Page 121 of 286
y h = c1u1 ( x) + c2 u 2 ( x)
y p = v1 ( x)u1 ( x) + v2 ( x)u 2 ( x)
Where,
v1 ' =
u u2
− u 2 ( x) f ( x)
u ( x ) f ( x)
, v2 ' = 1
,∆ = 1
= u1u 2 '−u 2u1 '
u1 ' u 2 '
∆
∆
∴ v1 = ∫
− u 2 ( x) f ( x)
u ( x) f ( x )
dx, v2 = ∫ 1
dx
u1u 2 '−u 2u1 '
u1u 2 '−u 2 u1 '
Page 122 of 286
PART 15: MATRICIES
15.1
BASIC PRINICPLES:
Size = i × j , i=row, j=column
A = aij
[ ]
15.2
BASIC OPERTAIONS:
[
A − B = [a
kA = [ka ]
[A ] = A
]
−b ]
A + B = aij + bij
Addition:
Subtraction:
ij
Scalar Multiple:
ij
ij
T
Transpose:
ij
ji
eg:
( A + B + C + ...)T = AT + B T
( ABCD...)T = ...D T C T B T AT
Scalar Product:
a • b = [a1 a2
Symmetry:
AT = A
a3
+ C T + ...
 b1 
b 
...] 2 
b3 
 
 ... 
Cramer’s Rule:
Ax = B
det( Ai ) where Ai = column i replaced by B
xi =
det( A)
Least Squares Solution
In the form
(
Ax = b ,
For a linear approximation:
For a quadratic approximation:
Etc.
15.3
)
−1
x = AT A AT b
r0 + r1 x = b
r0 + r1 x + r2 x 2 = b
SQUARE MATRIX:
Page 123 of 286
Diagonal:
Lower Triangle Matrix:
Upper Triangle Matrix:
15.4
DETERMINATE:
2x2:
3x3:
det (A) = ad − bc
det (A) = aei + bfg + cdh − afh − bdi − ceg
nxn:
det (A) = a11C11 + a12 C12 + a1n C1n = ∑ a1 j C ij = ∑ a1 j Mi1 j × (− 1)(1+ j )
n
n
j =1
j =1
Rules:
Page 124 of 286
Page 125 of 286
15.5
INVERSE
a b 
c d 


2x2:
−1
=
1  d − b
ad − bc − c a 
3x3:
a
d

 g
Minor:
b
e
h
−1
c
 ei − fh ch − bi bf − ce 
1

 fg − di ai − cg cd − af 
f =

aei − afh − bdi + bfg + cdh − ceg 
dh − eg bg − ah ae − bd 
i 
Mij = Determinate of Sub matrix which has deleted row i and column j
Page 126 of 286
a
A = d
 g
h
c
f 
i 
b
M 21 = 
h
c
f 
b
e
Cij = Mij × (− 1)
Cofactor:
(i + j )
Adjoint Method for Inverse:
adj ( A) = C T
A−1 =
Left Inverse:
AC = I
(
C = AT A
)
−1
1
adj ( A)
det( A)
AT
(when rows(A)>columns(A))
Right Inverse:
CA = I
(
C = AT AAT
)
−1
(when rows(A)<columns(A))
15.6
LINEAR TRANSFORMATION
Axioms for a linear transformation:
If
F (u + v ) = F (u ) + F (v)
And F (λ u ) = λF (u )
[Preserves Addition]
[Preserves Scalar Multiplication]
Transition Matrix:
The matrix that represents the linear transformation
Page 127 of 286
T (v) = c1T (v1 ) + c2T (v2 ) + ... + cnT (vn )
T ( x) = Ax
A = [T (e1 ) | T (e 2 ) | ... | T (e 3 )] (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:
Eigenvalues:
Eigenvectors:
Characteristic Polynomial:
All solutions of Ax = λx
All solutions of λ of det(A-λI)=0
General solution of [A-λI][X]=0 (ie: the nullspace)
The function p (λ ) = det( A − λI )
Algebraic Multiplicity:
The number of times a root is repeated for a given
eigenvalue.
∑ of all algebraic multiplicity = degree of the
characteristic polynomial.
The number of linearly independent eigenvectors
you get from a given eigenvalue.
Geometric Multiplicity:
Page 128 of 286
Transformation:
Linearly Independence:
Digitalization:
T :V → V
T ( x) = λx
The same process for an ordinary matrix is used.
The set of eigenvectors for distinct eigenvalues is
linearly independent.
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.
λ1 0
0 λ
2


0
0
D=

 ... ...
0 0

Cayley-Hamilton Theorem:
Orthonormal Set:
... 0 
... 0 
λ3 ... 0 

... ... 0 
0 0 λn 
0
0
Every matrix satisfies its own polynomial:
P(λ ) = an λn + an−1λn−1 ... + a1λ + a 0 = 0
P(λ ) = an A n + an −1 A n−1... + a1 A + a 0 = 0
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 
Pn 
 P1 P2
QR Factorisation:
A = [u1 | u 2 | ... | u n ] = QR
dim( A) = n × k , k ≤ n
All columns are Linearly Independent
Q = [v1 | v2 | ... | vn ] by the Gram-Schmidt Process
 q1

 0
 0
R=
 0
 ...

 0
T
T
u 2 v1 u3 v1
T
q2
u 3 v2
0
q3
0
0
...
...
0
0
Page 129 of 286
T
u 4 v1
T
u 4 v2
T
u 4 v3
q4
...
0
...
...
...
...
...
...
T
u k v1 

T
u k v2 
T
u k v3 

T
u k v4 
... 

q k 
u 2 • v1 u 3 • v1
q2
u3 • v2
0
q3
0
0
...
...
0
0
 q1

 0
 0
R=
 0
 ...

 0
15.9
u 4 • v1
u 4 • v2
u 4 • v3
q4
...
0
...
...
...
...
...
...
u k • v1 

u k • v2 
u k • v3 

u k • v4 
... 

qk 
JORDAN FORMS
Generalised Diagonlisation:
P −1 AP = J
A = PJP −1
Jordan Block:
Jordan Form:
Algebraic Multiplicity:
Geometric Multiplicity:
Generalised Chain:
λ
0

0
JB = 
...
0

 0
1
λ
0
...
0
0
0 ... 0
1 ... 0
λ ... 0
... ... ...
0 ... λ
0 ... 0
0
0 
0

...
1

λ 
 J1 0 ... 0 
 0 J ... 0 
2

J =
 ... ... ... ... 


 0 0 ... J n 
The number of times λ appears on main diagonal
The number of times λ appears on main diagonal
without a 1 directly above it
= {u m , u m−1 ,..., u 2 , u1 } , where u1 is an eigenvector
u k = ( A − λI )u k +1
u k +1 = [ A − λI | u k ]
P = [P1 | P2 | ... | Pm | ...] , for every eigenvector of A
Page 130 of 286
Powers:
A k = PJ k P −1
 J1
0
k
J =
 ...

0
JB
k
 k
λ


=0

0
 ...

0
J2
...
0
k
 J 1k
0

... 0 
0
=

... ... 
...


... J n 
 0
...
 k  k −1
 λ
1
λk
 k  k −2
 λ
 2
 k  k −1
 λ
1
0
λk
...
...
0
J2
...
0
0 

... 0 
... ... 
k
... J n 
...
k

...


...

...
...
15.10 COMPLEX MATRICIS:
Conjugate Transpose:
A∗ = AT
∗
A∗ = A
( A + B )∗ = A∗ + B ∗
(zA)∗ = z A∗
( 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.
If A is Skew-Hermitian, iA is normal as: (iA) = i A = (− 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
Page 131 of 286
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.
λ1 0
0 λ
2

D=0 0

 ... ...
0 0

0
0 ... 0 
λ3 ... 0 

... ... 0 
0 0 λn 
−1
∗
If A is Hermitian, D = P AP = P AP as P are an orthonormal set of vectors.
0
...
Spectral Theorem:
For a nxn Normal matrix and eigenvectors form an orthonormal set
P = [P1 | P2 | ... | Pn ]
A = λ1P1 P1 + λ2 P2 P2 + ... + λn Pn Pn
*
*
*
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
15.11 NUMERICAL COMPUTATIONS:
Rayleigh Quotient:
if (λ;v) is an eigenvalue/eigenvector pair of A, then
Page 132 of 286
Power method:
If A is a nxn matrix with Linearly Independent Eigenvectors, and distinct eigenvectors
arranged such that:
λ1 ≥ λ2 ≥ ... ≥ λn
and the set of eigenvectors are:
{v1 , v2 ,..., vn }
Any vector “w” can be written as:
w0 = c1v1 + c2 v2 + ... + cn vn
w1 = Aw0 = c1 Av1 + c2 Av2 + ... + cn Avn = c1λ1v1 + c2 λ2v2 + ... + cn λn vn
Page 133 of 286
s
s

 λ2 
 λn  

ws = Aws −1 = c1λ1 v1 + c2 λ2 v2 + ... + cn λn vn = λ1 c1v1 + c2   v2 + ... + cn   vn

 λ1  
 λ1 

  λ s 
λi
As
< 1 , lim  i   = 0
s →∞  λ

λ1
 1  
s
∴ ws → c1λ1 v1
s
s
s
Appling this with the Rayleigh Quotient:
w
ws = A s −1
 ws −1

, λ = R( ws ), w0 can be any vector usually


Page 134 of 286
1
0
 
...
PART 16: VECTORS
16.1
Basic Operations:
 a1 + b1 
a + b = a 2 + b2 
 a3 + b3 
 a1 − b1 
a − b = a 2 − b2 
 a3 − b3 
Addition:
Subtraction:
a = b ⇔ a1 = b1 , a 2 = b2 , a3 = b3
Equality:
k a + lb = λ a + µb ⇒ k = λ, l = µ
 ka1 
k a =  ka 2 
 ka 3 
Scalar Multiplication:
Parallel:
a = kb ⇔ a b
Magnitude:
a =
^
(a1 )2 + (a 2 )2 + (a 3 )2
a
a
Unit Vector:
a=
Zero Vector:
A vector with no magnitude and no specific direction
Dot Product:
a • b = a ⋅ b ⋅ cos θ
a • b = a 1 b1 + a 2 b 2 + a 3 b 3
Angle Between two Vectors:
cos θ =
cos θ =
Angle of a vector in 3D:
Perpendicular Test:
a•b
a⋅b
 a 2
 1
a 1 b1 + a 2 b 2 + a 3 b 3
2
2
2
2
2
+ a 2 + a 3  ⋅  b1 + b 2 + b 3 
 

 a1 
 
 a   cos(α ) 
^
a 
a =  2  = cos( β ) 
a
   cos(γ ) 
a
 3
a
 
a•b = 0
Page 135 of 286
^
Scalar Projection:
a onto b: P = a • b
Vector Projection:
1
 ^^
a onto b: P =  a • b  b = 2 (a • b )b


b
Cross Product:
a × b = a 2 b3 − a 3 b 2 , a 3 b1 − a1 b3 , a1 b 2 − a 2 b1
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 )
j k
i
 a

a × b = det a1 a2 a3  = i det  2
 b2
b1 b2 b3 
16.2
x = a1 + λb1
where a is a point on the line, and b is
a vector parallel to the line
y = a 2 + λb2
z = a3 + λb3
λ=
x − a1 y − a 2 z − a3
=
=
b1
b2
b3
Planes
n • AR = 0
n•r = n•a
n•r = k
Where: n = a, b, c & r = x, y, z : ax + by + cz = k
16.4
 a a    a a  
j  det  1 3   + k  det  1 2  
 b1 b3    b1 b2  
Lines
r = a + λb ,
16.3
a3  
−
b3  
Closest Approach
Two Points:
d = PQ
Point and Line:
d = PQ × a
Point and Plane:
d = PQ • n
^
^
Page 136 of 286
d = PQ • n = PQ • (a × b )
^
Two Skew Lines:
Solving for t:
16.5
[r b (t ) − r a (t )] • [v b − v a ] = 0
[ a r b (t )] • [ a v b ] = 0
Geometry
A=
Area of a Triangle:
AB × AC
Area of a Parallelogram:
2
A = AB × AC
Area of a Parallelepiped:
A = AD • ( AB × AC )
16.6
Space Curves
Where:
r (t ) = x(t )i + y (t ) j + z (t )k
Velocity:
v (t ) = r ' (t ) = x ' (t )i + y ' (t ) j + z ' (t )k
Acceleration:
a(t ) = v ' (t ) = r ' ' (t ) = x' ' (t )i + y ' ' (t ) j + z ' ' (t )k
Definition of “s”:
The length of the curve from r to r+∆r
Unit Tangent:
T=
dr r ' (t )
=
ds r ' (t )
T =1
Chain Rule:
dr dr ds
=
×
dt ds dt
Page 137 of 286
As
Normal:
dr
dr
ds
=speed
= 1,
=
ds
dt
dt
T •T =1
d
(T • T ) = 0
ds
dT
dT
•T + T •
=0
ds
ds
dT
2T •
=0
ds
dT
=0
T•
ds
As T is tangent to the curve,
dT
is normal
ds
 dT 


 ds 
N=
dT
ds
Curvature:
dT dT
=
N = κN
ds
ds
r ' (t ) × r ' ' (t ) v(t ) × a (t )
dT
∴κ =
=
=
3
3
ds
r ' (t )
v(t )
Unit Binomial:
B =T × N
Tortion:
τ=
16.7
Vector Space
16.8
ABBREVIATIONS
dB
ds
λ = a scalar value
µ = 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 138 of 286
Page 139 of 286
PART 17: SERIES
17.1
MISCELLANEOUS
n
S n = a1 + a2 + a2 + a4 + ... + an = ∑ an
General Form:
n =1
∞
S ∞ = a1 + a2 + a2 + a4 + ... = ∑ an
Infinite Form:
n =1
i
Si = a1 + a2 + a2 + a4 + ... + ai = ∑ an
Partial Sum of a Series:
n =1
17.2
TEST FOR CONVERGENCE AND DIVERGENCE
lim (S ) = L , if L exists, it is convergent
lim (a ) ≠ 0
Test For Convergence:
n→ ∞
Test For Divergence:
n →∞
n
n
Geometric Series
∞
∑ ar
n −1
n=1
 Divergent , r ≥ 1

Convergent, r < 1
P Series
∞
1  Divergent , p ≤ 1
p
Convergent, p > 1
∑x
n=1
The Sandwich Theorem
an ≤ bn ≤ cn
If there is a positive series so that
If
lim (a ) = lim (c ) = L , then, lim (b ) = L
n
n→∞
Hence, if
n→∞
n
n→∞
n
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
= f ( n) = = f ( x )
n
x
∞
∞
1
∞
∴ ∫ f ( x ) dx = ∫ dx = [ln x ]1 = D.N .E.
x
1
1
an =
Eg:
∴ 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
Page 140 of 286
If
bn is convergent and an ≤ bn , then an is also convergent
The Limit Comparison Test
∞
 an 


<
∞
,
then
an converges
∑
lim
 
n→∞  c n 
n =1
n =1
∞
∞
a 
If there is a divergent series ∑ d n , then if lim  n  > 0 , then ∑ an diverges
n→∞  d n 
n=1
n =1
∞
If there is a convergent series
∑ cn , then if
D’almbert’s Ratio Comparison Test
FOR POSITIVE TERMS:
 an+1 
 <1
an 
lim 
Converges:
n→∞
 an+1 
 >1
an 
lim 
Diverges:
n→∞
 an+1 
 =1
an 
lim 
Not enough information:
n→∞
The nth Root Test
∞
For
∑a
n =1
where an
n
≥ 0 , then if
lim
n
n→∞
Converges:
an ,
lim
n
an < 1
lim
n
an > 1
lim
n
an = 1
n→∞
Diverges:
n→∞
Not enough information:
n→∞
Abel’s Test:
∞
If
∑a
n =1
∞
n
is positive and decreasing, and
n =1
∞
Then
∑a
n =1
∑c
n
n
is a convergent series.
× cn Converges
Negative Terms
∞
If
∑ an converges, then
n =1
∞
∑a
n =1
n
is said to be absolutely convergent
Alternating Series Test
This is the only test for an alternating series in the form
Let
∞
∞
n=1
n=1
∑ an = ∑ (−1) n × bn
bn be the sequence of positive numbers. If bn+1 < bn and
lim b
n→∞
n
= 0 , then the
series is convergent.
Alternating Series Error
Rn = S − sn ≤ bn+1 , where Rn is the error of the partial sum to the nth term.
Page 141 of 286
17.3
ARITHMETIC PROGRESSION:
a, a + d , a + 2d , a + 3d ,...
= a + d (n − 1)
n
n
∑a=1a = 2 (2a + d (n − 1))
Definition:
Nth Term:
Sum Of The First N Terms:
17.4
GEOMETRIC PROGRESSION:
a, ar , ar 2 , ar 3 ,...
= ar n−1
n
a 1− rn
Sn = ∑ a =
1− r
a =1
Definition:
Nth Term:
Sum Of The First N Terms:
)
(
)  =
 a 1− rn
S ∞ = lim 
n →∞  1 − r
P, A, Q,...
Sum To Infinity:


a
(given r < 1 )
1− r
A
Q
= r, = r
P
A
A Q
∴ = ⇒ A 2 = PQ ⇒ A = PQ
P A
Geometric Mean:
17.5
(
SUMMATION SERIES
n(n + 1)
2
a =1
n
n(n + 1)(2n + 1)
∑a =a1 2 =
6
n
Linear:
1+2+3+4+…
Quadratic:
12+22+32+42+…
3
3
3
1 +2 +3 +4 +…
Cubic:
17.6
3
∑a =
 n(n + 1) 
∑a=a1 =  2 
n
2
3
APPROXIMATION SERIES
Taylor Series
∞
∞
n=0
n=0
f ( x ) = ∑ a n ( x − x0 ) n = ∑
f ( n) ( x0 )
( x − x0 ) n = a 0 + a1 ( x − x0 ) + a 2 ( x − x0 ) 2 + a3 ( x − x0 ) 3 + ...
n!
f ( n ) ( x0 )
where, a n =
n!
Maclaurun Series
Special case of the Taylor Series where
x0 = 0
Page 142 of 286
Linear Approximation:
1
1
f ( x ) ≈ L( x ) = ∑ an ( x − x0 ) n = ∑
n =0
f (n)( x 0 )
( x − x0 ) n = a0 + a1 ( x − x0 )
n!
n =0
Quadratic Approximation:
2
2
f ( x ) ≈ Q( x ) = ∑ an ( x − x0 ) n = ∑
n =0
f (n)( x 0 )
( x − x0 ) n = a0 + a1 ( x − x0 ) + a2 ( x − x0 ) 2
n!
n=0
Cubic Approximation:
3
3
f ( x ) ≈ C( x ) = ∑ an ( x − x0 ) = ∑
n
n =0
17.7
f ( n ) ( x0 )
( x − x0 ) n = a0 + a1 ( x − x0 ) + a2 ( x − x0 ) 2 + a3 ( x − x0 ) 3
n!
n =0
MONOTONE SERIES
Strictly Increasing:
an+1 > an
Non-Decreasing:
an+1 ≥ an
Strictly Decreasing:
an+1 < an
Non-Increasing:
Convergence:
an+1 ≤ an
17.8
an+1
>1
an
an+1
<1
an
A monotone sequence is convergent if it is bounded, and hence the
limit exists when an → ∞
RIEMANN ZETA FUNCTION
∞
ζ (n ) = ∑
1
n
k =1 k
Form:
Euler’s Table:
n=2
1
1 1
π2
=
1
+
+
+
...
=
2
4 9
6
k =1 k
∞
ζ (2 ) = ∑
1
1 1
1
π4
=
1
+
+
+
+
...
=
4
16 81 256
90
k =1 k
∞
n=4
ζ (4 ) = ∑
n=6
ζ (6 ) = ∑
n=8
ζ (8) =
n=10
1
1
1
1
π6
=
1
+
+
+
+
...
=
6
64 729 4096
945
k =1 k
∞
π8
9450
ζ (10) =
π 10
93555
Page 143 of 286
n=12
n=14
n=16
n=18
n=20
n=22
n=24
n=26
691π 12
638512875
2π 14
ζ (14) =
18243225
3617π 16
ζ (16) =
325641566250
43867π 18
ζ (18) =
38979295480125
174611π 20
ζ (20 ) =
1531329465290625
155366π 22
ζ (22 ) =
13447856940643125
236364091π 24
ζ (24) =
201919571963756521875
1315862π 26
ζ (26) =
11094481976030578125
ζ (12) =
Alternating Series:
Proof for n=2:
Taylor Series Expansion:
Polynomial Expansion:
x3 x5 x7
+ − + ...
3! 5! 7!
sin( x) = x( x − π )( x + π )( x − 2π )( x + 2π )...
sin( x) = x −
(
)(
)(
)
sin( x) = x x 2 − π 2 x 2 − 4π 2 x 2 − 9π 2 ...

x 
x 
x2 
sin( x) = Ax1 − 2 1 − 2 2 1 − 2 2 ...
 π  2 π  3 π 
2
Page 144 of 286
2
Comparing the Coefficient of x3:
17.9
 sin( x) 
lim
 =1= A
x →0
 x 

x3 x5 x7
x 2 
x 2 
x2 
+
−
+ ... = x1 − 2 1 − 2 2 1 − 2 2 ..
x−
3! 5! 7!
 π  2 π  3 π 
1
1
1
1
1
− = − 2 − 2 2 − 2 2 − 2 2 ...
3!
2 π
3π
4 π
π
2
1 1
1
π
= 1 + 2 + 2 + 2 ...
6
2
3 4
SUMMATIONS OF POLYNOMIAL EXPRESSIONS
(Harmonic number)
where
Bernoulli number
17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS
Where
x ≠1
(m < n)
Page 145 of 286
denotes a
(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 146 of 286
[
Page 147 of 286
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 148 of 286
Trigonometric:
Page 149 of 286
Page 150 of 286
Exponential and Logarithm Series:
,
Page 151 of 286
y=
x −1
x +1
Fourier Series:
a0 ∞
+ ∑ a k cos(kx) + bk sin(kx)
2 k =1
a
fW ( x) = 0 + a1 cos( x) + a2 cos(2 x) + ... + a n cos(nx) + b1 sin( x) + b2 sin(2 x) + ... + bn sin(nx)
2
fW ( x ) =
ak =
bk =
1
π
1
π
2π
∫ f ( x) cos(kx)dx
k = 0,1,2,..., n
0
2π
∫ f ( x) sin(kx)dx
k = 1,2,..., n
0
17.15 Bernoulli Expansion:
Fundamentally:
A polynomial in n(n + 1)

1k + 2 k + 3k + ... + n k = 
(2n + 1) x A polynomial in n(n + 1)
Page 152 of 286
k odd
k even
Expansions:
1
1 + 2 + 3 + ... + n = n(n + 1)
2
1
1
1 + 2 + 3 + ... + n = n 2 + n
2
2

 2
1  2
1 + 2 + 3 + ... + n =    B0 n 2 +   B1n 
2 0
1

1
12 + 2 2 + 32 + ... + n 2 = (2n + 1) n(n + 1)
6
1
1
1
12 + 2 2 + 32 + ... + n 2 = n 3 + n 2 + n
3
2
6

 3
 3
1  3
12 + 2 2 + 32 + ... + n 2 =    B0 n 3 +   B1n 2 +   B2 n 
3 0
1
 2

13 + 2 3 + 33 + ... + n 3 = (1 + 2 + 3 + ... + n )
2
1
(n(n + 1))2
4
1
1
1
13 + 2 3 + 33 + ... + n 3 = n 4 + n 3 + n 2
4
2
4

4
 4
 4
1  4
13 + 2 3 + 33 + ... + n 3 =    B0 n 4 +   B1n 3 +   B2 n 2 +   B3 n 
4  0
1
 2
 3

13 + 2 3 + 33 + ... + n 3 =
14 + 2 4 + 34 + ... + n 4 = (2n + 1)
1
n(n + 1)(3n(n + 1) − 1)
30
1
1
1
1
14 + 2 4 + 34 + ... + n 4 = n 5 + n 4 + n 3 − n
5
2
3
30

 5
5
5
5
1  5
14 + 2 4 + 34 + ... + n 4 =    B0 n 5 +   B1n 4 +   B2 n 3 +   B3 n 2 +   B4 n 
5  0
1
 2
 3
 4


 k + 1 k +1−1  k + 1
 k + 1
 k + 1
1   k + 1
 
 B0 n k +1 + 
 B1n
 B2 n k +1−2 + ... + 
 Bk −1n 2 + 
 Bk n 
+ 

k +1 0 
 1 
 2 
 k − 1
 k 

1k + 2 k + 3k + ... + n k =
List of Bernoulli Numbers:
n
0
B(n)
1
1
−
2
3
1
2
1
6
0
Page 153 of 286
−
4
1
30
5
0
6
1
42
7
0
−
8
1
30
9
0
10
5
66
11
0
12
691
−
2730
13
0
14
7
6
15
0
16
−
3617
510
17
0
18
43867
798
19
0
20
−
174611
330
Page 154 of 286
PART 18: ELECTRICAL
18.1
FUNDAMENTAL THEORY
Conservation of Power:
q = 6.24 ×1018 Coulombs
dq
I=
dt
ρl
R=
A
V = IR
V2
P = VI = I 2 R =
R
∑ PCONSUMED = ∑ PDELIVERED
Electrical Energy:
W = P × t = I 2 × R × t = ∫ Pdt
Charge:
Current:
Resistance:
Ohm’s Law:
Power:
t
0
Kirchoff’s Voltage Law:
The sum of the volt drops around a close loop is equal to zero.
Kirchoff’s Current Law:
The sum of the currents entering any junction is equal to the sum of the
currents leaving that junction.
∑V = 0
∑I
= ∑ I OUT
IN
T
Average Current:
1
= ∫ I (t )dt
T0
I AVE
I AVE =
1
× Area (under I(t))
T
T
RMS Current:
1
(I (t ))2 dt
T ∫0
∆ to Y Conversion:
Page 155 of 286
18.2
RA =
R1 R2 + R2 R3 + R1 R3
R1
RB =
R1 R2 + R2 R3 + R1 R3
R2
RC =
R1 R2 + R2 R3 + R1 R3
R3
COMPONENTS
Resistance in Series:
Resistance in Parallel:
Inductive Impedance:
Capacitor Impedance:
Capacitance in Series:
Capacitance in Parallel:
RT = R1 + R2 + R3 + ...
1
1
1
1
=
+
+
+ ...
RT R1 R2 R3
X L = jωL = j 2πfL
1
1
=−j
XC = − j
ωC
2πfC
1
1
1
1
=
+
+
+ ...
CT C1 C2 C3
CT = C1 + C2 + C3 + ...
Voltage, Current & Power Summary:
18.3
THEVENIN’S THEOREM
Thevenin’s Theorem:
VTH = Open Circuit Voltage between a & b
Page 156 of 286
RTH = Short Circuit any voltage source and Open Circuit any current source and calculate RTH as the
resistance from a & b. With dependant sources, SC terminals a & b and calculate the current in the wire
( I SC ). RTH
=
VTH
I SC
Maximum Power Transfer Theorem:
PMAX =
(VTH )2 , where R
4 RTH
18.4
FIRST ORDER RC CIRCUIT
18.5
FIRST ORDER RL CIRCUIT
18.6
SECOND ORDER RLC SERIES CIRCUIT
Calculation using KVL:
− VS + VR + VL + VC = 0
V R + V L + VC = V S
Ri + L
di
+ VC = V S
di
Circuit current:
dVC
dt
d 2VC
di
∴ =C
dt
dt 2
dV
d 2VC
∴ RC C + LC
+ VC = VS
dt
dt 2
d 2VC
dV
+ RC C + VC = VS
LC
2
dt
dt
2
d VC R dVC
1
V
+
+
VC = S
2
dt
L dt
LC
LC
Important Variables
2
Standard Format:
s 2 + 2αs + ω0 = 0
i = iC = C
Page 157 of 286
L
= RTH
1R
2 L
dV
s= c
dt
α=  
Damping Factor:
Natural Frequency:
 1 

 LC 
Undamped Natural Frequency:
ω0 = 
Damping Frequency:
ωd = ω0 2 − α 2
Mode Delta:
∆ = α 2 − ω0
2
VC (t ) = TRANSIENT +FINAL
VC:
Solving:
s2 +
1
R
s+
=0
L
LC
s = −α ± α 2 − ω0 = −α ± ∆
Mode 1:
If: ∆ > 0 , then :
s = −α ± ∆
2
V C ( t ) = TRANSIENT
TRANSIENT
+ FINAL
= Ae s1t + Be s 2 t
FINAL = V C ( ∝ ) = V S
V C ( t ) = Ae s1t + Be s 2 t + V S
Finding A & B:
VC (0 + ) = VC (0 − ) = V0
∴ A + B + VS = V0 → A + B = V0 − VS
dVc
= As1e s1t + Bs2 e s2t
dt
dVC (0 + ) iC (0 + ) iL (0 + ) iL (0 − ) I 0
=
=
=
= = As1 + Bs2
dt
C
C
C
C
V0 − VS = A + B 

∴ I0
= As1 + Bs2 
C

Mode 2:
If:
∆ = 0 , then :
s = −α
VC (t ) = TRANSIENT + FINAL
TRANSIENT = ( A + Bt )e st = ( A + Bt )e −αt
FINAL = VC (∝) = VS
VC (t ) = ( A + Bt )e −αt + VS
Finding A & B:
Page 158 of 286
VC (0 + ) = VC (0 − ) = V0
∴ A + VS = V0 → A = V0 − VS
dVc
= ( A + Bt )se st + Be st
dt
dVC (0 + ) iC (0 + ) iL (0 + ) iL (0 − ) I 0
=
=
=
= = As + B
dt
C
C
C
C
V0 − VS = A 

∴ I0
= As + B 
C

Mode 3:
If:
∆ < 0 , and letting ωd = ω0 − α 2 , then :
2
s = −α ± jωd
VC (t ) = TRANSIENT + FINAL
TRANSIENT = ( A cos(ω d t ) + B sin(ω d t ) )e −αt
FINAL = VC (∝) = VS
VC (t ) = ( A cos(ω d t ) + B sin(ω d t ) )e −αt + VS
Finding A & B:
VC (0 + ) = VC (0 − ) = V0
∴ A + VS = V0 → A = V0 − VS
dVc
= (− Aωd sin(ωd t ) + Bωd cos(ωd t ) )e −αt − α ( A cos(ωd t ) + B sin(ωd t ) )e −αt
dt
dVC (0 + ) iC (0 + ) iL (0 + ) iL (0 − ) I 0
=
=
=
= = Bωd − αA
dt
C
C
C
C
V0 − VS = A 

∴ I0
= Bωd − αA
C

Mode 4:
If:
R = 0 , then :
α = 0, ωd = ω0
s = ± jωd = ± jω0
VC (t ) = TRANSIENT + FINAL
TRANSIENT = A cos(ωd t ) + B sin(ωd t )
FINAL = VC (∝) = VS
VC (t ) = A cos(ωd t ) + B sin(ωd t ) + VS
Page 159 of 286
Finding A & B:
VC (0 + ) = VC (0 − ) = V0
∴ A + VS = V0 → A = V0 − VS
dVc
= − Aωd sin(ωd t ) + Bωd cos(ωd t )
dt
dVC (0 + ) iC (0 + ) iL (0 + ) iL (0 − ) I 0
=
=
=
= = Bωd
dt
C
C
C
C
V0 − VS = A

∴ I0
= Bωd 
C

Current through Inductor:
dV
iL = iC = C C
dt
Plotting Modes:
Mode 1: Over Damped
Mode 2: Critically Damped
Mode 3: Sinusoidal Damped
Page 160 of 286
Mode 4: Not Damped
(Oscillates indefinitely)
18.7
SECOND ORDER RLC PARALLEL CIRCUIT
Calculation using KCL:
i S = i R + i L + iC
iS =
V
dV
+ iL + C
R
dt
Node Voltage:
diL
=V
dt
dV
d 2i
= L 2L
dt
dt
L diL
d 2i
∴ iS =
+ iL + LC 2L
R dt
dt
2
d i
L diL
LC 2L +
+ iL = iS
dt
R dt
d 2 iL
1 diL
1
1
+
+
iL =
iS
2
dt
RC dt LC
LC
VL = L
Important Variables
Standard Format:
s 2 + 2αs + ω0 = 0
Damping Factor:
α= 
Undamped Natural Frequency:
ω0 = 
Damping Frequency:
ωd = ω0 2 − α 2
Mode Delta:
∆ = α 2 − ω0
2
1 1 

2  RC 
 1 

 LC 
2
Page 161 of 286
Solving:
s2 +
1
1
s+
=0
RC
LC
s = −α ± α 2 − ω 0 = −α ± ∆
2
18.8
LAPLANCE TRANSFORMATIONS
Identities:
Page 162 of 286
Properties:
Page 163 of 286
18.9
THREE PHASE – Y
Line Voltage: VLINE = VPHASE × 3
V
Phase Voltage:
VPHASE = LINE
3
Line Current:
I LINE = I PHASE
Phase Current:
I PHASE = I LINE
Power:
S = 3 × VLINE × I 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
I PHASE = LINE
3
Phase Current:
Power:
S = 3 × VLINE × I LINE
S = 3 × VPHASE × I PHASE
18.11 POWER
Instantaneous:
P(t ) = V (t ) × I (t )
Page 164 of 286
Average:
T
=
1
1
P
(
t
)
dt
=
VMAX I MAX cos(θV − θ I ) = VRMS I RMS cos(θV − θ I )
T ∫0
2
2
V
= TH where Z L = Z TH
8RTH
Maximum Power:
PMAX
Total Power:
Complex Power:
= I RMS R
2
S = VRMS I RMS
S = I RMS Z
2
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
Reluctance
Permeability
Φ
ℜ
µ
Magnetomotive Force
ℑ
Flux density
B
Magnetic Field Intensity H
Permeability of free space:
Magnetic Field Intensity:
Reluctance:
Ohm’s Law:
Magnetic Force on a conductor:
Electromagnetic Induction:
Magnetic Flux:
Electric Field:
Magnetic force on a particle:
Strength of magnetic field
Relative difficulty for flux to establish
Relative ease for flux to establish
Wb
A-t/Wb
H/m
Ability of coil to produce flux
Flux per unit area
MMF per unit length
A-t
Wb/m2 or T
A-t/m
µ 0 = 4π × 10 −7 Hm −1
ℑ NI
=
l
l
1
ℜ=
µA
ℑ
OR ℑ = NI
Φ=
ℜ
F = BIl
Φ − Φ1
EMF = − N 2
t
EMF = Bvl
Φ = BA
F V
E= =
q d
F = qvB
H=
Page 165 of 286
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 )
Any subgraph H where V ( H ) = V (G ) , there are no cycles
and all verticies are connected.
Degree of vertex:
Number of edges leaving a vertex
∑ d (v) = 2 E (G )
v∈V ( G )
Distance:
Diameter:
d (u , v) =Shortest path between u & v
diam(G ) = max {d (u , v)}
u & v∈V ( G )
Total Edges in a simple bipartite graph:
E (G ) =
V ( X ) V (Y )
2
∑ d ( x) = ∑ d ( y )
x∈X
Total Edges in K-regular graph:
E (G ) =
19.2
y∈Y
k (k − 1)
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 n ,n bipartite graph:
F1 = [11' ,22' ,33' ,...]
F2 = [12' ,23' ,34' ,...]
F3 = [13' ,24' ,35'...]
Fn = ...
where all numbers are MOD(n)
1 Factors of a K 2 n graph:
F0 = {(1, ∞), (2,0), (3,2n − 2),..., (n, n + 1)}
Fi = {(i, ∞), (i + 1,2n − 2 + 1),..., (i + n − 1, i + n}
F2 n− 2 = ...
Where all numbers are MOD(2n-1)
19.3
Vertex Colouring
Page 166 of 286
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:
G = G1 ∪ G2 and G1 ∩ G2 = K m , then
P(G1 , λ ) P(G2 , λ )
P(G, λ ) =
P( K m , λ )
If
Edge Contraction:
P(G , λ ) = P(G − e, λ ) − P (G.e, λ )
Common Chromatic Polynomials:
P(Tn , λ ) = λ (λ − 1) n−1
P(C n , λ ) = (λ − 1) n + (−1) n (λ − 1)
P( K n , λ ) = λ (λ − 1)(λ − 2)...(λ − n + 1)
19.4
Where the highest power is the number of vertices
Where the lowest power is the number of
components
Where the the coefficient of the 2nd highest power is
the number of edges.
Edge Colouring:
Common Chromatic Polynomials:
χ ' (G ) ≥ ∆(G )
χ ' ( K n ,n ) = n
χ ' (C2 n ) = 2
χ ' (C2 n+1 ) = 3
χ ' ( K 2 n ) = 2n − 1
χ ' ( K 2 n +1 ) = 2n + 1
Page 167 of 286
PART 99: CONVERSIONS
99.1
LENGTH:
Name of unit
ångström
Symbol
Å
Definition
−10
≡ 1 × 10
m
Relation to SI units
= 0.1 nm
astronomical unit AU
≈ Distance from Earth to Sun
≈ 149 597 871 464 m
barleycorn (H)
≡ ⅓ in (see note above about
rounding)
= 8.46 × 10−3 m
bohr, atomic unit
a0
of length
≡ Bohr radius of hydrogen
≈ 5.291 772
0859 × 10−11 ±
3.6 × 10−20 m
cable length
(Imperial)
≡ 608 ft
= 185.3184 m
cable length
(International)
≡ 1/10 nmi
= 185.2 m
cable length
(U.S.)
≡ 720 ft
= 219.456 m
≡ 66 ft ≡ 4 rods
= 20.1168 m
chain (Gunter's;
Surveyor's)
ch
≡ Distance from fingers to elbow
≈ 0.5 m
≈ 18in
cubit (H)
ell (H)
ell
≡ 45 in
fathom
fm
≡ 6 ft
fm
≡ 1 × 10
fermi
= 1.143 m
= 1.8288 m
−15
m
= 1 × 10−15 m
finger
≡ 7/8 in
= 0.022 225 m
finger (cloth)
≡ 4½ in
= 0.1143 m
foot (Benoît) (H) ft (Ben)
≈ 0.304 799 735 m
foot (Clarke's;
Cape) (H)
ft (Cla)
≈ 0.304 797 2654 m
foot (Indian) (H)
ft Ind
≈ 0.304 799 514 m
foot
(International)
ft
foot (Sear's) (H)
ft (Sear)
foot (U.S.
Survey)
ft (US)
≡ 1 200/3 937 m
≈ 0.304 800 610 m
french; charriere
F
≡ ⅓ mm
= 3.3 × 10−4 m
furlong
fur
≡ 10 chains = 660 ft = 220 yd
= 201.168 m
≡ 4 in
= 0.1016 m
hand
≡ ⅓ yd = 12 inches
= 0.3048 m
≈ 0.304 799 47 m
inch
in
≡ 1/36 yd = 1/12 ft
= 0.0254 m
league (land)
lea
≡ 3 US Statute miles
= 4 828.032 m
≡ 24 light-hours
= 2.590 206 837
light-day
Page 263 of 286
12 × 1013 m
light-hour
≡ 60 light-minutes
= 1.079 252
8488 × 1012 m
light-minute
≡ 60 light-seconds
= 1.798 754
748 × 1010 m
light-second
≡ Distance light travels in one
second in vacuum
= 299 792 458 m
light-year
l.y.
≡ Distance light travels in
vacuum in 365.25 days
= 9.460 730 472
5808 × 1015 m
line
ln
≡ 1/12 in
= 0.002 116 m
link (Gunter's;
Surveyor's)
lnk
≡ 1/100 ch
= 0.201 168 m
link (Ramsden's;
Engineer's)
lnk
≡ 1 ft
= 0.3048 m
metre (SI base
unit)
m
≡ Distance light travels in 1/299
792 458 of a second in vacuum.
=1m
≡ 1/200 in
= 1.27 × 10−4 m
mickey
micron
µ
mil; thou
mil
≡ 1 × 10−6 m
≡ 1 × 10−3 in
= 2.54 × 10−5 m
mil (Sweden and
mil
Norway)
≡ 10 km
= 10 000 m
mile
≡ 1 760 yd = 5 280 ft = 80 chains = 1 609.344 m
mi
mile
(geographical)
(H)
≡ 6 082 ft
= 1 853.7936 m
mile (telegraph)
(H)
mi
≡ 6 087 ft
= 1 855.3176 m
mile (U.S.
Survey)
mi
≡ 5 280 ft (US Survey feet)
= 5 280 × 1 200/3 937
m ≈ 1 609.347 219 m
≡ 2¼ in
= 0.057 15 m
≡ 3 nmi
= 5 556 m
nail (cloth)
nautical league
NL; nl
nautical mile
(Admiralty)
NM (Adm);
≡ 6 080 ft
nmi (Adm)
nautical mile
(international)
NM; nmi
≡ 1 853.184 m
≡ 1 852 m
= 1 852 m
pace
≡ 2.5 ft
= 0.762 m
palm
≡ 3 in
= 0.0762 m
Distance of star with parallax
shift of one arc second from a
base of one astronomical unit
≈ 3.085 677 82 × 1016
± 6 × 106 m
≡ 12 points
Dependent on point
measures.
parsec
pica
pc
Page 264 of 286
point (American,
pt
English)
point (Didot;
European)
point (PostScript)
pt
≡ 1/72.272 in
≈ 0.000 351 450 m
≡ 1/12 × 1/72 of pied du roi;
≈ 0.000 375 97 m;
After 1878:
≡ 5/133 cm
After 1878:
≈ 0.000 375 939 85 m
[11]
pt
≡ 1/72 in
= 0.000 352 7 m
point (TeX)
pt
≡ 1/72.27 in
= 0.000 351 4598 m
≡ ¼ yd
= 0.2286 m
quarter
rod; pole; perch
(H)
rd
≡ 16½ ft
= 5.0292 m
rope (H)
rope
≡ 20 ft
= 6.096 m
≡ 9 in
= 0.2286 m
span (H)
≡ 1 × 1012 m
spat
≡ 2 in
stick (H)
stigma; bicron
(picometre)
pm
twip
twp
x unit; siegbahn
xu
yard
(International)
yd
99.2
= 0.0508 m
≡ 1 × 10−12 m
= 1.7638 × 10−5 m
≡ 1/1 440 in
≈ 1.0021 × 10−13 m
≡ 0.9144 m ≡ 3 ft ≡ 36 in
≡ 0.9144 m
Symbol
Relation to SI units
AREA:
Name of unit
Definition
acre (international)
ac
≡ 1 ch × 10 ch = 4 840
= 4 046.856 4224 m2
sq yd
acre (U. S. survey)
ac
≡ 10 sq ch = 4 840 sq
= 4 046.873 m2 [15]
yd
are
a
≡ 100 m2
= 100 m2
barn
b
≡ 10−28 m2
= 10−28 m2
≡ 4 000 ac
= 1.618 742 568
96 × 107 m2
≡ 1 in × 1 ft
= 7.741 92 × 10−3 m2
barony
board
bd
boiler horsepower equivalent bhp
direct radiation
EDR
≡ (1 ft2) (1 bhp) / (240
≈ 12.958 174 m2
BTUIT/h)
circular inch
circ in
≡ π/4 sq in
circular mil; circular thou
circ mil ≡ π/4 mil2
≈ 5.067 075 × 10−4 m2
≈ 5.067 075 × 10−10 m2
cord
≡ 192 bd
= 1.486 448 64 m2
dunam
≡ 1 000 m2
= 1 000 m2
Guntha
≡ 33 ft x 33 ft[citation
≈ 101.17 m2
Page 265 of 286
needed]
≡ 10 000 m2
= 10 000 m2
≈ 120 ac (variable)
≈ 5 × 105 m2
≡ ¼ ac
= 1 011.714 1056 m2
shed
≡ 10−52 m2
= 10−52 m2
square (roofing)
≡ 10 ft × 10 ft
= 9.290 304 m2
hectare
ha
hide
rood
ro
square chain (international)
sq ch
≡ 66 ft × 66 ft = 1/10
ac
= 404.685 642 24 m2
square chain (U.S. Survey)
sq ch
≡ 66 ft(US) ×
66 ft(US) = 1/10 ac
= 404.687 3 m2
square foot
sq ft
≡ 1 ft × 1 ft
= 9.290 304 × 10−2 m2
square foot (U.S. Survey)
sq ft
≡ 1 ft (US) × 1 ft (US)
≈ 9.290 341 161 327
49 × 10−2 m2
square inch
sq in
≡ 1 in × 1 in
= 6.4516 × 10−4 m2
square kilometre
km2
≡ 1 km × 1 km
= 106 m2
square link
sq lnk
≡ 1 lnk × 1 lnk
= 4.046 856 4224 × 10−2
m2
square metre (SI unit)
m2
≡1m×1m
= 1 m2
square mil; square thou
sq mil
≡ 1 mil × 1 mil
= 6.4516 × 10−10 m2
square mile; section
sq mi
≡ 1 mi × 1 mi
= 2.589 988 110
336 × 106 m2
square mile (U.S. Survey)
sq mi
≡ 1 mi (US) × 1 mi
(US)
≈ 2.589 998 × 106 m2
square rod/pole/perch
sq rd
≡ 1 rd × 1 rd
= 25.292 852 64 m2
square yard
sq yd
≡ 1 yd × 1 yd
= 0.836 127 36 m2
stremma
≡ 1 000 m2
= 1 000 m2
township
≡ 36 sq mi (US)
≈ 9.323 994 × 107 m2
yardland
≈ 30 ac
≈ 1.2 × 105 m2
99.3
VOLUME:
Name of unit
acre-foot
acre-inch
barrel (Imperial)
barrel (petroleum)
barrel (U.S. dry)
barrel (U.S. fluid)
board-foot
Symbol
Definition
≡ 1 ac x 1 ft = 43 560
ac ft
ft3
≡ 1 ac × 1 in
bl (Imp) ≡ 36 gal (Imp)
bl; bbl
≡ 42 gal (US)
≡ 105 qt (US) = 105/32
bl (US)
bu (US lvl)
fl bl (US) ≡ 31½ gal (US)
fbm
≡ 144 cu in
Page 266 of 286
Relation to SI units
= 1 233.481 837 547 52
m3
= 102.790 153 128 96 m3
= 0.163 659 24 m3
= 0.158 987 294 928 m3
= 0.115 628 198 985 075
m3
= 0.119 240 471 196 m3
= 2.359 737 216 × 10−3
bucket (Imperial)
bushel (Imperial)
bushel (U.S. dry heaped)
bushel (U.S. dry level)
bkt
bu (Imp)
bu (US)
bu (US
lvl)
≡ 4 gal (Imp)
≡ 8 gal (Imp)
≡ 1 ¼ bu (US lvl)
≡ 2 150.42 cu in
m3
= 0.018 184 36 m3
= 0.036 368 72 m3
= 0.044 048 837 7086 m3
= 0.035 239 070 166 88
m3
= 0.476 961 884 784 m3
= 0.145 474 88 m3
= 3.624 556 363 776 m3
= 0.453 069 545 472 m3
= 6.116 438 863 872 m3
= 0.028 316 846 592 m3
= 16.387 064 × 10−6 m3
= 1 m3
= 4 168 181 825.440 579
584 m3
= 0.764 554 857 984 m3
= 284.130 625 × 10−6 m3
= 227.3045 × 10−6 m3
= 250.0 × 10−6 m3
butt, pipe
coomb
cord (firewood)
cord-foot
cubic fathom
cubic foot
cubic inch
cubic metre (SI unit)
cu fm
cu ft
cu in
m3
≡ 126 gal (wine)
≡ 4 bu (Imp)
≡ 8 ft × 4 ft × 4 ft
≡ 16 cu ft
≡ 1 fm × 1 fm × 1 fm
≡ 1 ft × 1 ft × 1 ft
≡ 1 in × 1 in × 1 in
≡1m×1m×1m
cubic mile
cu mi
≡ 1 mi × 1 mi × 1 mi
cubic yard
cup (breakfast)
cup (Canadian)
cup (metric)
cu yd
cup (U.S. customary)
c (US)
≡ 27 cu ft
≡ 10 fl oz (Imp)
≡ 8 fl oz (Imp)
≡ 250.0 × 10−6 m3
≡ 8 US fl oz ≡ 1/16 gal
= 236.588 2365 × 10−6 m3
(US)
cup (U.S. food nutrition
labeling)
c (US)
c (CA)
c
dash (Imperial)
dash (U.S.)
≡ 240 mL[16]
= 2.4×10−4 m3
≡ 1/384 gi (Imp) = ½
pinch (Imp)
≡ 1/96 US fl oz = ½ US
pinch
= 369.961 751 302 08
3 × 10−9 m3
= 308.057 599 609
375 × 10−9 m3
= 11.838 776 0416 × 10−6
m3
= 98.656 467 013
8 × 10−9 m3
≈ 77.886 684 × 10−9 m3
= 83.03 × 10−9 m3
= 50.0 × 10−9 m3
= 82.148 693
22916 × 10−9 m3
≈ 64.854 231 × 10−9 m3
= 757.082 3568 × 10−6 m3
= 0.034 068 706 056 m3
= 3.551 632 8125 × 10−6
m3
≡ 1/12 gi (Imp)
dessertspoon (Imperial)
drop (Imperial)
gtt
≡ 1/288 fl oz (Imp)
drop (Imperial) (alt)
drop (medical)
drop (metric)
gtt
≡ 1/1 824 gi (Imp)
≡ 1/12 ml
≡ 1/20 mL
drop (U.S.)
gtt
≡ 1/360 US fl oz
drop (U.S.) (alt)
fifth
firkin
gtt
≡ 1/456 US fl oz
≡ 1/5 US gal
≡ 9 gal (US)
fluid drachm (Imperial)
fl dr
≡ ⅛ fl oz (Imp)
Page 267 of 286
fluid dram (U.S.); U.S.
fluidram
fl dr
≡ ⅛ US fl oz
= 3.696 691 195
3125 × 10−6 m3
fluid ounce (Imperial)
fl oz
(Imp)
≡ 1/160 gal (Imp)
= 28.413 0625 × 10−6 m3
US fl oz
≡ 1/128 gal (US)
= 29.573 529 5625 × 10−6
m3
US fl oz
≡ 30 mL[16]
= 3×10−5 m3
fluid scruple (Imperial)
fl s
≡ 1/24 fl oz (Imp)
gallon (beer)
beer gal
≡ 282 cu in
gallon (Imperial)
gal (Imp) ≡ 4.546 09 L
gallon (U.S. dry)
gal (US)
≡ ⅛ bu (US lvl)
gallon (U.S. fluid; Wine)
gal (US)
≡ 231 cu in
gill (Imperial); Noggin
gi (Imp);
≡ 5 fl oz (Imp)
nog
gill (U.S.)
gi (US)
hogshead (Imperial)
hogshead (U.S.)
jigger (bartending)
kilderkin
lambda
last
litre
load
peck (Imperial)
hhd (Imp) ≡ 2 bl (Imp)
hhd (US) ≡ 2 fl bl (US)
≡ 1½ US fl oz
≡ 18 gal (Imp)
λ
≡ 1 mm3
≡ 80 bu (Imp)
L
≡ 1 dm3 [17]
≡ 50 cu ft
≡ 1/480 fl oz (Imp) =
min
1/60 fl dr (Imp)
≡ 1/480 US fl oz = 1/60
min
US fl dr
pk
≡ 2 gal (Imp)
peck (U.S. dry)
pk
≡ ¼ US lvl bu
perch
per
≡ 16½ ft × 1½ ft × 1 ft
≡ 1/192 gi (Imp) = ⅛
tsp (Imp)
≡ 1/48 US fl oz = ⅛ US
tsp
≡ ⅛ gal (Imp)
≡ 1/64 bu (US lvl) ≡ ⅛
fluid ounce (U.S.
customary)
fluid ounce (U.S. food
nutrition labeling)
minim (Imperial)
minim (U.S.)
pinch (Imperial)
pinch (U.S.)
pint (Imperial)
pint (U.S. dry)
pt (Imp)
pt (US
≡ 4 US fl oz
Page 268 of 286
= 1.183 877 60416 × 10−6
m3
= 4.621 152 048 × 10−3
m3
= 4.546 09 × 10−3 m3
= 4.404 883 770
86 × 10−3 m3
= 3.785 411 784 × 10−3
m3
= 142.065 3125 × 10−6 m3
= 118.294 118 25 × 10−6
m3
= 0.327 318 48 m3
= 0.238 480 942 392 m3
≈ 44.36 × 10−6 m3
= 0.081 829 62 m3
= 1 × 10−9 m3
= 2.909 4976 m3
= 0.001 m3
= 1.415 842 3296 m3
= 59.193 880 208
3 × 10−9 m3
= 61.611 519 921
875 × 10−9 m3
= 9.092 18 × 10−3 m3
= 8.809 767 541
72 × 10−3 m3
= 0.700 841 953 152 m3
= 739.923 502
60416 × 10−9 m3
= 616.115 199 218
75 × 10−9 m3
= 568.261 25 × 10−6 m3
= 550.610 471
pint (U.S. fluid)
dry)
gal (US dry)
pt (US fl) ≡ ⅛ gal (US)
≡ 3/4 US fl oz
pony
pottle; quartern
quart (Imperial)
qt (Imp)
quart (U.S. dry)
qt (US)
quart (U.S. fluid)
quarter; pail
register ton
qt (US)
≡ ½ gal (Imp) = 80 fl
oz (Imp)
≡ ¼ gal (Imp)
≡ 1/32 bu (US lvl) = ¼
gal (US dry)
≡ ¼ gal (US fl)
≡ 8 bu (Imp)
≡ 100 cu ft
sack (Imperial); bag
≡ 3 bu (Imp)
sack (U.S.)
≡ 3 bu (US lvl)
seam
≡ 8 bu (US lvl)
shot
strike (Imperial)
≡ 1 US fl oz
≡ 2 bu (Imp)
strike (U.S.)
≡ 2 bu (US lvl)
tablespoon (Canadian)
tbsp
≡ ½ fl oz (Imp)
tablespoon (Imperial)
tbsp
≡ 5/8 fl oz (Imp)
tbsp
≡ ½ US fl oz
tbsp
≡ 15 mL[16]
teaspoon (Canadian)
tsp
≡ 1/6 fl oz (Imp)
teaspoon (Imperial)
tsp
≡ 1/24 gi (Imp)
tablespoon (metric)
tablespoon (U.S.
customary)
tablespoon (U.S. food
nutrition labeling)
3575 × 10−6 m3
= 473.176 473 × 10−6 m3
= 22.180 147 171
875 × 10−6 m3
= 2.273 045 × 10−3 m3
= 1.136 5225 × 10−3 m3
= 1.101 220 942
715 × 10−3 m3
= 946.352 946 × 10−6 m3
= 0.290 949 76 m3
= 2.831 684 6592 m3
= 0.109 106 16 m3[citation
needed]
= 0.105 717 210 500 64
m3
= 0.281 912 561 335 04
m3[citation needed]
≈ 29.57 × 10−6 m3
= 0.072 737 44 m3
= 0.070 478 140 333 76
m3
= 14.206 531 25 × 10−6
m3
= 17.758 164 0625 × 10−6
m3
≡ 15.0 × 10−6 m3
= 14.786 764 7825 × 10−6
m3
= 1.5×10−5 m3
= 4.735 510 416 × 10−6
m3
= 5.919 388 02083 × 10−6
m3
= 5.0 × 10−6 m3
= 4.928 921 595 × 10−6
m3
teaspoon (metric)
≡ 5.0 × 10−6 m3
teaspoon (U.S. customary) tsp
≡ 1/6 US fl oz
teaspoon (U.S. food
nutrition labeling)
timber foot
ton (displacement)
ton (freight)
≡ 5 mL[16]
= 5×10−6 m3
≡ 1 cu ft
≡ 35 cu ft
≡ 40 cu ft
= 0.028 316 846 592 m3
= 0.991 089 630 72 m3
= 1.132 673 863 68 m3
tsp
Page 269 of 286
≡ 28 bu (Imp)
≡ 252 gal (wine)
≡ 40 bu (US lvl)
ton (water)
tun
wey (U.S.)
99.4
PLANE ANGLE:
Name of unit Symbol
µ
≡ 2π/6400 rad
arcminute
'
≡ 1°/60
arcsecond
"
≡ 1°/3600
'
≡ 1 grad/100
"
≡ 1 grad/(10 000)
°
≡ π/180 rad = 1/360 of a revolution
grad
≡ 2π/400 rad = 0.9°
centesimal
minute of arc
centesimal
second of arc
degree (of
arc)
grad; gradian;
gon
octant
≡ 45°
quadrant
≡ 90°
radian (SI
unit)
rad
sextant
sign
The angle subtended at the center of a circle by
an arc whose length is equal to the circle's radius. = 1 rad
One full revolution encompasses 2π radians.
≈ 1.047 198
≡ 60°
rad
≈ 0.523 599
≡ 30°
rad
SOLID ANGLE:
Name of
Symbol
unit
steradian
(SI unit)
99.6
Relation to
SI units
≈ 0.981
748 × 10−3 rad
≈ 0.290
888 × 10−3 rad
≈ 4.848
137 × 10−6 rad
≈ 0.157
080 × 10−3 rad
≈ 1.570
796 × 10−6 rad
≈ 17.453
293 × 10−3 rad
≈ 15.707
963 × 10−3 rad
≈ 0.785 398
rad
≈ 1.570 796
rad
Definition
angular mil
99.5
= 1.018 324 16 m3
= 0.953 923 769 568 m3
= 1.409 562 806 6752 m3
sr
Relation to
SI units
Definition
The solid angle subtended at the center of a sphere of
radius r by a portion of the surface of the sphere
having an area r2. A sphere encompasses 4π sr.[14]
= 1 sr
MASS:
Name of unit
Symbol
Definition
Page 270 of 286
Relation to SI units
atomic mass unit, unified u; AMU
atomic unit of mass,
electron rest mass
bag (coffee)
bag (Portland cement)
barge
carat
carat (metric)
clove
crith
me
kt
ct
≡ 60 kg
≡ 94 lb av
≡ 22½ sh tn
≡ 3 1/6 gr
≡ 200 mg
≡ 8 lb av
dalton
Da
dram (apothecary; troy)
dram (avoirdupois)
dr t
dr av
electronvolt
eV
gamma
grain
γ
gr
long cwt or
≡ 112 lb av
cwt
hundredweight (long)
hundredweight (short);
cental
sh cwt
hyl (MKS unit)
kilogram, grave
kip
mark
mite
mite (metric)
ounce (apothecary; troy)
ounce (avoirdupois)
ounce (U.S. food nutrition
labeling)
pennyweight
point
pound (avoirdupois)
pound (metric)
pound (troy)
≡ 60 gr
≡ 27 11/32 gr
≡ 1 eV (energy unit)
= 1.7826 × 10−36 kg
/ c2
≡ 1 µg
= 1 µg
≡ 64.798 91 mg
= 64.798 91 mg
≡ 100 lb av
≡ 1 gee × 1 g × 1
s2/m
≡ 1 gee × 1 kg × 1
s2/m
hyl (CGS unit)
≈ 1.660 538 73 × 10−27 ±
1.3 × 10−36 kg
≈ 9.109 382 15 × 10−31 ±
45 × 10−39 kg [18]
= 60 kg
= 42.637 682 78 kg
= 20 411.656 65 kg
≈ 205.196 548 333 mg
= 200 mg
= 3.628 738 96 kg
≈ 89.9349 mg
≈ 1.660 902 10 × 10−27 ±
1.3 × 10−36 kg
= 3.887 9346 g
= 1.771 845 195 3125 g
= 50.802 345 44 kg
= 45.359 237 kg
= 9.806 65 g
= 9.806 65 kg
oz t
oz av
≡ 1 000 lb av
≡ 8 oz t
≡ 1/20 gr
≡ 1/20 g
≡ 1/12 lb t
≡ 1/16 lb
(SI base unit)[8]
= 453.592 37 kg
= 248.827 8144 g
= 3.239 9455 mg
= 50 mg
= 31.103 4768 g
= 28.349 523 125 g
oz
≡ 28 g[16]
= 28 g
dwt; pwt
≡ 1/20 oz t
≡ 1/100 ct
≡ 7 000 grains
≡ 500 g
≡ 5 760 grains
= 1.555 173 84 g
= 2 mg
= 0.453 592 37 kg
= 500 g
= 0.373 241 7216 kg
kg; G
kip
lb av
lb t
Page 271 of 286
≡ 1/4 long cwt = 2 st
= 28 lb av
≡ ¼ short tn
≡ ¼ long tn
≡ 100 kg
≡ 20 gr
≡ 1/700 lb av
≡ 1 gee × 1 lb av × 1
s2/ft
≡ 14 lb av
≡ 1 mg × 1 long tn ÷
1 oz t
≡ 1 mg × 1 sh tn ÷ 1
oz t
quarter (Imperial)
quarter (informal)
quarter, long (informal)
quintal (metric)
scruple (apothecary)
sheet
q
s ap
slug; geepound
slug
stone
st
ton, assay (long)
AT
ton, assay (short)
AT
ton, long
ton, short
tonne (mts unit)
long tn or
ton
sh tn
t
Zentner
99.7
Ztr.
= 226.796 185 kg
= 254.011 7272 kg
= 100 kg
= 1.295 9782 g
= 647.9891 mg
≈ 14.593 903 kg
= 6.350 293 18 kg
≈ 32.666 667 g
≈ 29.166 667 g
≡ 2 240 lb
= 1 016.046 9088 kg
≡ 2 000 lb
≡ 1 000 kg
= 907.184 74 kg
= 1 000 kg
= 114.305 277 24 kg
(variants exist)
≡ 252 lb = 18 st
wey
= 12.700 586 36 kg
Definitions vary; see
[19]
and.[14]
DENSITY:
Name of unit
gram per millilitre
kilogram per cubic metre (SI unit)
kilogram per litre
ounce (avoirdupois) per cubic foot
Symbol
g/mL
kg/m3
kg/L
oz/ft3
Definition
≡ g/mL
≡ kg/m3
≡ kg/L
≡ oz/ft3
ounce (avoirdupois) per cubic inch
oz/in3
≡ oz/in3
ounce (avoirdupois) per gallon (Imperial) oz/gal
ounce (avoirdupois) per gallon (U.S.
oz/gal
fluid)
pound (avoirdupois) per cubic foot
lb/ft3
≡ oz/gal
Relation to SI units
= 1,000 kg/m3
= 1 kg/m3
= 1,000 kg/m3
≈ 1.001153961 kg/m3
≈
1.729994044×103 kg/m3
≈ 6.236023291 kg/m3
≡ oz/gal
≈ 7.489151707 kg/m3
≡ lb/ft3
lb/in3
≡ lb/in3
≈ 16.01846337 kg/m3
≈
2.767990471×104 kg/m3
≈ 99.77637266 kg/m3
pound (avoirdupois) per cubic inch
pound (avoirdupois) per gallon (Imperial) lb/gal ≡ lb/gal
pound (avoirdupois) per gallon (U.S.
lb/gal ≡ lb/gal
≈ 119.8264273 kg/m3
fluid)
slug per cubic foot
slug/ft3 ≡ slug/ft3 ≈ 515.3788184 kg/m3
Page 272 of 286
99.8
TIME:
Name of unit
atomic unit of
time
Symbol
au
Definition
≡ a0/(α·c)
fortnight
helek
≡ 441 mo (hollow) + 499 mo (full) = 76
a of 365.25 d
≡ 100 a (see below for definition of year
length)
= 24 h
≡ Time needed for the Earth to rotate
once around its axis, determined from
successive transits of a very distant
astronomical object across an observer's
meridian (International Celestial
Reference Frame)
≡ 10 a (see below for definition of year
length)
≡ 2 wk
≡ 1/1 080 h
Hipparchic cycle
≡ 4 Callippic cycles - 1 d
Callippic cycle
century
day
d
day (sidereal)
d
decade
hour
jiffy
jiffy (alternate)
h
≡ 60 min
≡ 1/60 s
≡ 1/100 s
ke (quarter of an
hour)
≡ ¼ h = 1/96 d
ke (traditional)
≡ 1/100 d
lustre; lustrum
Metonic cycle;
enneadecaeteris
≡ 5 a of 365 d
≡ 110 mo (hollow) + 125 mo (full) =
6940 d ≈ 19 a
≡ 1 000 a (see below for definition of
year length)
millennium
milliday
md
≡ 1/1 000 d
minute
moment
month (full)
min
≡ 60 s
≡ 90 s
≡ 30 d[20]
mo
Page 273 of 286
Relation to SI
units
≈ 2.418 884
254 × 10−17 s
= 2.398 3776 × 109
s
= 100 × year
= 86400 s
≈ 86 164.1 s
= 10 × year
= 1 209 600 s
= 3.3 s
= 9.593 424 × 109
s
= 3 600 s
= .016 s
= 10 ms
= 60 × 60 / 4 s =
900 s = 60 / 4 min
= 15 min
= 24 × 60 × 60 /
100 s = 864 s = 24
* 60 / 100 min =
14.4 min
= 1.5768 × 108 s
= 5.996 16 × 108 s
= 1000 × year
= 24 × 60 × 60 / 1
000 s = 86.4 s
= 60 s
= 90 s
= 2 592 000 s
Average Gregorian month = 365.2425/12
d = 30.436875 d
≡ 29 d[20]
Cycle time of moon phases ≈ 29.530589
days (Average)
= 48 mo (full) + 48 mo (hollow) + 3 mo
(full)[21][22] = 8 a of 365.25 d = 2922 d
month (Greg. av.) mo
month (hollow)
mo
month (synodic) mo
octaeteris
≡ (Gℏ /c5)½
Planck time
second
s
shake
sigma
Sothic cycle
svedberg
week
S
wk
year (Gregorian)
a, y, or
yr
year (Julian)
a, y, or
yr
year (sidereal)
a, y, or
yr
year (tropical)
a, y, or
yr
99.9
≈ 2.6297 × 106 s
= 2 505 600 s
≈ 2.551 × 106 s
= 2.524 608 × 108
s
≈ 1.351 211
868 × 10−43 s
time of 9 192 631 770 periods of the
radiation corresponding to the transition
between the 2 hyperfine levels of the
(SI base unit)
ground state of the caesium 133 atom at
[8]
0 K (but other seconds are sometimes
used in astronomy)
≡ 10−8 s
= 10 ns
−6
≡ 10 s
= 1 µs
= 4.607
≡ 1 461 a of 365 d
4096 × 1010 s
≡ 10−13 s
= 100 fs
≡7d
= 604 800 s
= 365.2425 d average, calculated from
common years (365 d) plus leap years
= 31 556 952 s
(366 d) on most years divisible by 4. See
leap year for details.
= 365.25 d average, calculated from
common years (365 d) plus one leap year = 31 557 600 s
(366 d) every four years
≡ time taken for Sun to return to the
≈ 365.256 363 d ≈
same position with respect to the stars of
31 558 149.7632 s
the celestial sphere
≡ Length of time it takes for the Sun to
≈ 365.242 190 d ≈
return to the same position in the cycle of
31 556 925 s
seasons
FREQUENCY:
Name of unit Symbol
hertz (SI unit)
Hz
revolutions per
rpm
minute
Definition
≡ Number of cycles per second
≡ One unit rpm equals one rotation
completed around a fixed axis in one
minute of time.
99.10 SPEED OR VELOCITY:
Page 274 of 286
Relation to SI
units
= 1 Hz = 1/s
≈
0.104719755 rad/s
Name of unit
foot per hour
foot per
minute
foot per
second
furlong per
fortnight
inch per
minute
inch per
second
kilometre per
hour
knot
knot
(Admiralty)
Symbol
fph
≡ 1 ft/h
Relation to SI units
≈ 8.466 667 × 10−5 m/s
fpm
≡ 1 ft/min
= 5.08 × 10−3 m/s
fps
≡ 1 ft/s
= 3.048 × 10−1 m/s
≡ furlong/fortnight
≈ 1.663 095 × 10−4 m/s
ipm
≡ 1 in/min
≈ 4.23 333 × 10−4 m/s
ips
≡ 1 in/s
= 2.54 × 10−2 m/s
km/h
≡ 1 km/h
≈ 2.777 778 × 10−1 m/s
kn
≡ 1 NM/h = 1.852 km/h
≡ 1 NM (Adm)/h = 1.853 184
km/h[citation needed]
The ratio of the speed of an object
moving through a fluid to the
speed of sound in the same
medium; typically used as a
measure of aircraft speed.
≈ 0.514 444 m/s
m/s
≡ 1 m/s
= 1 m/s
mph
≡ 1 mi/h
= 0.447 04 m/s
mpm
≡ 1 mi/min
= 26.8224 m/s
mps
≡ 1 mi/s
= 1 609.344 m/s
c
≡ 299 792 458 m/s
= 299 792 458 m/s
kn
mach number M
metre per
second (SI
unit)
mile per hour
mile per
minute
mile per
second
speed of light
in vacuum
speed of
sound in air
Definition
= 0.514 773 m/s
Unitless. Actual speed of
sound varies depending on
atmospheric conditions. See
"speed of sound" below for
one specific condition.
≈ 344 m/s at 20 °C, 60%
relative humidity [23]
s
99.11 FLOW (VOLUME):
Name of unit
cubic foot per minute
cubic foot per second
cubic inch per minute
cubic inch per second
cubic metre per second (SI unit)
gallon (U.S. fluid) per day
Symbol
CFM
ft3/s
in3/min
in3/s
m3/s
GPD
Definition
≡ 1 ft3/min
≡ 1 ft3/s
≡ 1 in3/min
≡ 1 in3/s
≡ 1 m3/s
≡ 1 gal/d
Page 275 of 286
Relation to SI units
= 4.719474432×10−4 m3/s
= 0.028316846592 m3/s
= 2.7311773 × 10−7 m3/s
= 1.6387064×10−5 m3/s
= 1 m3/s
= 4.381263638 × 10−8 m3/s
gallon (U.S. fluid) per hour
gallon (U.S. fluid) per minute
litre per minute
GPH
GPM
LPM
≡ 1 gal/h = 1.051503273 × 10−6 m3/s
≡ 1 gal/min = 6.30901964×10−5 m3/s
≡ 1 L/min = 1.6 × 10−5 m3/s
99.12 ACCELERATION:
Name of unit
foot per hour per second
foot per minute per second
foot per second squared
gal; galileo
inch per minute per second
inch per second squared
knot per second
metre per second squared (SI unit)
mile per hour per second
mile per minute per second
mile per second squared
standard gravity
Symbol
fph/s
fpm/s
fps2
Gal
ipm/s
ips2
kn/s
m/s2
mph/s
mpm/s
mps2
g
Definition
≡ 1 ft/(h·s)
≡ 1 ft/(min·s)
≡ 1 ft/s2
≡ 1 cm/s2
≡ 1 in/(min·s)
≡ 1 in/s2
≡ 1 kn/s
≡ 1 m/s2
≡ 1 mi/(h·s)
≡ 1 mi/(min·s)
≡ 1 mi/s2
≡ 9.806 65 m/s2
Relation to SI units
≈ 8.466 667 × 10−5 m/s2
= 5.08 × 10−3 m/s2
= 3.048 × 10−1 m/s2
= 10−2 m/s2
≈ 4.233 333 × 10−4 m/s2
= 2.54 × 10−2 m/s2
≈ 5.144 444 × 10−1 m/s2
= 1 m/s2
= 4.4704 × 10−1 m/s2
= 26.8224 m/s2
= 1.609 344 × 103 m/s2
= 9.806 65 m/s2
99.13 FORCE:
Name of unit
dyn
≡ g·cm/s2
Relation to SI
units
≈ 8.238 722
06 × 10−8 N [24]
= 10−5 N
kgf; kp;
Gf
≡ g × 1 kg
= 9.806 65 N
Symbol
≡ me· α2·c2/a0
atomic unit of force
dyne (cgs unit)
kilogram-force;
kilopond; graveforce
Definition
kip; kip-force
kip; kipf;
≡ g × 1 000 lb
klbf
milligrave-force,
gravet-force
mGf; gf
newton (SI unit)
N
ounce-force
ozf
pound
lb
pound-force
lbf
≡g×1g
= 4.448 221 615
2605 × 103 N
= 9.806 65 mN
A force capable of giving a mass of one
=1N=
kg an acceleration of one meter per
1 kg·m/s2
[25]
second, per second.
= 0.278 013 850
≡ g × 1 oz
953 7812 N
= 4.448 230 531
≡ slug·ft/s2
N
= 4.448 221 615
≡ g × 1 lb
2605 N
Page 276 of 286
poundal
pdl
≡ 1 lb·ft/s2
sthene (mts unit)
sn
≡ 1 t·m/s2
ton-force
tnf
≡ g × 1 sh tn
= 0.138 254 954
376 N
= 1 × 103 N
= 8.896 443 230
521 × 103 N
99.14 PRESSURE OR MECHANICAL STRESS:
Name of unit
Symbol
atmosphere (standard)
atm
atmosphere (technical)
at
bar
barye (cgs unit)
bar
centimetre of mercury
cmHg
Definition
≡ 1 kgf/cm2
≡ 1 dyn/cm2
≡ 13 595.1 kg/m3 × 1 cm × g
centimetre of water (4 °C) cmH2O ≈ 999.972 kg/m3 × 1 cm × g
foot of mercury
≡ 13 595.1 kg/m3 × 1 ft × g
ftHg
(conventional)
foot of water (39.2 °F)
ftH2O
≈ 999.972 kg/m3 × 1 ft × g
inch of mercury
inHg
≡ 13 595.1 kg/m3 × 1 in × g
(conventional)
inch of water (39.2 °F)
inH2O ≈ 999.972 kg/m3 × 1 in × g
kilogram-force per square
kgf/mm2 ≡ 1 kgf/mm2
millimetre
≡ 1 kipf/sq in
Relation to SI
units
≡ 101 325 Pa [26]
= 9.806 65 × 104
Pa [26]
≡ 105 Pa
= 0.1 Pa
≈ 1.333 22 × 103
Pa [26]
≈ 98.0638 Pa [26]
≈ 40.636
66 × 103 Pa [26]
≈ 2.988 98 × 103
Pa [26]
≈ 3.386
389 × 103 Pa [26]
≈ 249.082 Pa [26]
= 9.806 65 × 106
Pa [26]
≈ 6.894
757 × 106 Pa [26]
≈ 0.133 3224 Pa
kip per square inch
ksi
micron (micrometre) of
mercury
pound per square foot
≡ 13 595.1 kg/m3 × 1 µm × g ≈
[26]
0.001 torr
≡ 13 595.1 kg/m3 × 1 mm × g ≈ 1
mmHg
≈ 133.3224 Pa [26]
torr
≈ 999.972 kg/m3 × 1 mm × g =
mmH2O
= 9.806 38 Pa
0.999 972 kgf/m2
Pa
≡ N/m2 = kg/(m·s2)
= 1 Pa [27]
= 1 × 103 Pa = 1
pz
≡ 1 000 kg/m·s2
kPa
≈ 47.880 25 Pa
psf
≡ 1 lbf/ft2
[26]
pound per square inch
psi
poundal per square foot
pdl/sq ft ≡ 1 pdl/sq ft
millimetre of mercury
millimetre of water (3.98
°C)
pascal (SI unit)
pièze (mts unit)
µmHg
≡ 1 lbf/in2
Page 277 of 286
≈ 6.894
757 × 103 Pa [26]
≈ 1.488 164 Pa
[26]
≡ 1 sh tn × g / 1 sq ft
short ton per square foot
torr
≡ 101 325/760 Pa
torr
≈ 95.760
518 × 103 Pa
≈ 133.3224 Pa [26]
99.15 TORQUE OR MOMENT OF FORCE:
Name of unit
foot-pound force
foot-poundal
inch-pound force
metre kilogram
Newton metre (SI unit)
Symbol
ft lbf
ft pdl
in lbf
m kg
N·m
Definition
≡ g × 1 lb × 1 ft
≡ 1 lb·ft2/s2
≡ g × 1 lb × 1 in
≡N×m/g
≡ N × m = kg·m2/s2
Relation to SI units
= 1.355 817 948 331 4004 N·m
= 4.214 011 009 380 48 × 10−2 N·m
= 0.112 984 829 027 6167 N·m
≈ 0.101 971 621 N·m
= 1 N·m
99.16 ENERGY, WORK, OR AMOUNT OF HEAT:
Name of unit
barrel of oil equivalent
British thermal unit
(ISO)
British thermal unit
(International Table)
British thermal unit
(mean)
British thermal unit
(thermochemical)
British thermal unit (39
°F)
British thermal unit (59
°F)
British thermal unit (60
°F)
British thermal unit (63
°F)
calorie (International
Table)
calorie (mean)
calorie
(thermochemical)
calorie (3.98 °C)
calorie (15 °C)
calorie (20 °C)
Celsius heat unit
(International Table)
bboe
≈ 5.8 × 106 BTU59 °F
Relation to SI
units
≈ 6.12 × 109 J
BTUISO
≡ 1.0545 × 103 J
= 1.0545 × 103 J
Symbol
Definition
BTUIT
= 1.055 055 852
62 × 103 J
BTUmean
≈ 1.055 87 × 103 J
BTUth
≈ 1.054 350 × 103
J
BTU39 °F
≈ 1.059 67 × 103 J
BTU59 °F
≡ 1.054 804 × 103 J
= 1.054 804 × 103
J
BTU60 °F
≈ 1.054 68 × 103 J
BTU63 °F
≈ 1.0546 × 103 J
calIT
≡ 4.1868 J
≈ 4.190 02 J
calmean
calth
= 4.1868 J
≡ 4.184 J
cal3.98 °C
cal15 °C
cal20 °C
≡ 4.1855 J
CHUIT
≡ 1 BTUIT × 1 K/°R
Page 278 of 286
= 4.184 J
≈ 4.2045 J
= 4.1855 J
≈ 4.1819 J
= 1.899 100 534
716 × 103 J
cubic centimetre of
atmosphere; standard
cubic centimetre
cubic foot of
atmosphere; standard
cubic foot
cc atm;
scc
= 0.101 325 J
cu ft atm;
≡ 1 atm × 1 ft3
scf
= 2.869 204 480
9344 × 103 J
≡ 1 000 BTUIT
= 1.055 055 852
62 × 106 J
≡ 1 atm × 1 yd3
= 77.468 520 985
2288 × 103 J
cubic foot of natural gas
cubic yard of
atmosphere; standard
cubic yard
≡ 1 atm × 1 cm3
cu yd
atm; scy
electronvolt
eV
≡e×1V
erg (cgs unit)
erg
≡ 1 g·cm2/s2
foot-pound force
ft lbf
≡ g × 1 lb × 1 ft
foot-poundal
ft pdl
≡ 1 lb·ft2/s2
gallon-atmosphere
(imperial)
imp gal
atm
US gal
gallon-atmosphere (US)
atm
hartree, atomic unit of
Eh
energy
≡ 1 atm × 1 gal (imp)
horsepower-hour
hp·h
≡ 1 hp × 1 h
inch-pound force
in lbf
≡ g × 1 lb × 1 in
≡ 1 atm × 1 gal (US)
≡ me· α2·c2 (= 2 Ry)
≈ 1.602 177
33 × 10−19 ±
4.9 × 10−26 J
= 10−7 J
= 1.355 817 948
331 4004 J
= 4.214 011 009
380 48 × 10−2 J
= 460.632 569 25
J
= 383.556 849
0138 J
≈ 4.359
744 × 10−18 J
= 2.684 519 537
696 172 792 × 106
J
= 0.112 984 829
027 6167 J
The work done when a force of
one newton moves the point of
= 1 J = 1 m·N =
joule (SI unit)
J
its application a distance of one
1 kg·m2/s2
meter in the direction of the
force.[25]
kilocalorie; large calorie kcal; Cal ≡ 1 000 calIT
= 4.1868 × 103 J
kilowatt-hour; Board of kW·h;
≡ 1 kW × 1 h
= 3.6 × 106 J
Trade Unit
B.O.T.U.
litre-atmosphere
l atm; sl ≡ 1 atm × 1 L
= 101.325 J
= 1.055 055 852
quad
≡ 1015 BTUIT
62 × 1018 J
≈ 2.179
rydberg
Ry
≡ R∞· ℎ ·c
872 × 10−18 J
= 105.505 585
therm (E.C.)
≡ 100 000 BTUIT
262 × 106 J
Page 279 of 286
therm (U.S.)
thermie
ton of coal equivalent
ton of oil equivalent
ton of TNT
th
TCE
TOE
tTNT
≡ 100 000 BTU59 °F
≡ 1 McalIT
≡ 7 Gcalth
≡ 10 Gcalth
≡ 1 Gcalth
= 105.4804 × 106 J
= 4.1868 × 106 J
= 29.3076 × 109 J
= 41.868 × 109 J
= 4.184 × 109 J
99.17 POWER OR HEAT FLOW RATE:
Name of unit
atmosphere-cubic
centimetre per minute
atmosphere-cubic
centimetre per second
atmosphere-cubic foot
per hour
atmosphere-cubic foot
per minute
atmosphere-cubic foot
per second
BTU (International
Table) per hour
BTU (International
Table) per minute
BTU (International
Table) per second
calorie (International
Table) per second
foot-pound-force per
hour
foot-pound-force per
minute
foot-pound-force per
second
horsepower (boiler)
horsepower
(European electrical)
horsepower (Imperial
electrical)
horsepower (Imperial
mechanical)
horsepower (metric)
litre-atmosphere per
minute
Symbol
Definition
Relation to SI units
atm ccm
≡ 1 atm × 1 cm3/min
= 1.688 75 × 10−3 W
atm ccs
≡ 1 atm × 1 cm3/s
= 0.101 325 W
atm cfh
≡ 1 atm × 1 cu ft/h
atm·cfm
≡ 1 atm × 1 cu ft/min
atm cfs
≡ 1 atm × 1 cu ft/s
= 0.797 001 244 704
W
= 47.820 074 682 24
W
= 2.869 204 480
9344 × 103 W
BTUIT/h
≡ 1 BTUIT/h
BTUIT/min ≡ 1 BTUIT/min
≈ 0.293 071 W
≈ 17.584 264 W
BTUIT/s
≡ 1 BTUIT/s
= 1.055 055 852
62 × 103 W
calIT/s
≡ 1 calIT/s
= 4.1868 W
ft lbf/h
≡ 1 ft lbf/h
ft lbf/min ≡ 1 ft lbf/min
≈ 3.766 161 × 10−4
W
= 2.259 696 580 552
334 × 10−2 W
= 1.355 817 948 331
4004 W
≈ 9.810 657 × 103
W
ft lbf/s
≡ 1 ft lbf/s
bhp
≈ 34.5 lb/h × 970.3 BTUIT/lb
hp
≡ 75 kp·m/s
= 736 W
hp
≡ 746 W
= 746 W
hp
≡ 550 ft lbf/s
hp
≡ 75 m kgf/s
= 745.699 871 582
270 22 W
= 735.498 75 W
L·atm/min ≡ 1 atm × 1 L/min
Page 280 of 286
= 1.688 75 W
litre-atmosphere per
second
lusec
poncelet
square foot equivalent
direct radiation
ton of air conditioning
ton of refrigeration
(Imperial)
ton of refrigeration
(IT)
watt (SI unit)
L·atm/s
≡ 1 atm × 1 L/s
= 101.325 W
lusec
p
≡ 1 L·µmHg/s [14]
≡ 100 m kgf/s
≈ 1.333 × 10−4 W
= 980.665 W
sq ft EDR ≡ 240 BTUIT/h
≈ 70.337 057 W
≡ 1 t ice melted / 24 h
≡ 1 BTUIT × 1 lng tn/lb ÷ 10
min/s
≡ 1 BTUIT × 1 sh tn/lb ÷ 10
min/s
The power which in one second
of time gives rise to one joule of
energy.[25]
W
≈ 3 504 W
≈ 3.938 875 × 103
W
≈ 3.516 853 × 103
W
= 1 W = 1 J/s =
1 N·m/s =
1 kg·m2/s3
99.18 ACTION:
Name of unit
Symbol Definition
atomic unit of action au
Relation to SI units
≡ ℏ = ℎ /2π ≈ 1.054 571 68 × 10−34 J·s[28]
99.19 DYNAMIC VISCOSITY:
Name of unit
pascal second (SI unit)
poise (cgs unit)
pound per foot hour
pound per foot second
pound-force second per square foot
pound-force second per square inch
Symbol
Pa·s
P
lb/(ft·h)
lb/(ft·s)
lbf·s/ft2
lbf·s/in2
Definition
≡ N·s/m2 , kg/(m·s)
≡ 10−1 Pa·s
≡ 1 lb/(ft·h)
≡ 1 lb/(ft·s)
≡ 1 lbf·s/ft2
≡ 1 lbf·s/in2
Relation to SI units
= 1 Pa·s
= 0.1 Pa·s
≈ 4.133 789 × 10−4 Pa·s
≈ 1.488164 Pa·s
≈ 47.88026 Pa·s
≈ 6,894.757 Pa·s
99.20 KINEMATIC VISCOSITY:
Name of unit
square foot per second
square metre per second (SI unit)
stokes (cgs unit)
Symbol
ft2/s
m2/s
St
Definition
≡ 1 ft2/s
≡ 1 m2/s
≡ 10−4 m2/s
Relation to SI units
= 0.09290304 m2/s
= 1 m2/s
= 10−4 m2/s
99.21 ELECTRIC CURRENT:
Name of unit
ampere (SI base
unit)
Symbol
A
Definition
Relation to SI
units
≡ The constant current needed to
produce a force of 2 × 10−7 newton per = 1 A
metre between two straight parallel
Page 281 of 286
conductors of infinite length and
negligible circular cross-section placed
one metre apart in a vacuum.[8]
electromagnetic
unit; abampere (cgs abamp ≡ 10 A
unit)
esu per second;
statampere (cgs
esu/s
≡ (0.1 A·m/s) / c
unit)
= 10 A
≈
3.335641×10−10 A
99.22 ELECTRIC CHARGE:
Name of unit
Symbol
Definition
abcoulomb;
electromagnetic unit (cgs abC; emu ≡ 10 C
unit)
Relation to SI
units
= 10 C
atomic unit of charge
au
≡e
≈ 1.602 176
462 × 10−19 C
coulomb (SI unit)
C
≡ The amount of electricity
carried in one second of time by
one ampere of current.[25]
= 1 C = 1 A·s
faraday
F
≡ 1 mol × NA·e
≈ 96 485.3383
C
statcoulomb; franklin;
electrostatic unit (cgs
unit)
statC; Fr;
≡ (0.1 A·m) / c
esu
≈ 3.335
641 × 10−10 C
99.23 ELECTRIC DIPOLE:
Name of unit
Symbol Definition
Relation to SI units
atomic unit of electric dipole moment ea0
≈ 8.478 352 81 × 10−30 C·m
99.24 ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE:
Name of
Symbol
Definition
unit
abvolt
abV
≡ 1 × 10−8 V
(cgs unit)
statvolt
statV ≡ c· (1 µJ/A·m)
(cgs unit)
The difference in electric potential across two
volt (SI
points along a conducting wire carrying one ampere
V
unit)
of constant current when the power dissipated
between the points equals one watt.
Page 282 of 286
Relation to SI
units
= 1 × 10−8 V
= 299.792 458 V
= 1 V = 1 W/A =
1 kg·m2/(A·s3)
99.25 ELECTRICAL RESISTANCE:
Name
Symbol
of unit
Definition
The resistance between two points in a conductor
when one volt of electric potential difference,
applied to these points, produces one ampere of
current in the conductor.
ohm (SI
Ω
unit)
Relation to SI
units
= 1 Ω = 1 V/A = 1
kg·m2/(A2·s3)
99.26 CAPACITANCE:
Name
Symbol
of unit
Definition
The capacitance between two parallel plates that
results in one volt of potential difference when
charged by one coulomb of electricity.
farad
F
(SI unit)
Relation to SI
units
= 1 F = 1 C/V = 1
A2·s4/(kg·m2)
99.27 MAGNETIC FLUX:
Name of
Symbol
Definition
unit
maxwell
Mx
≡ 10−8 Wb
(CGS unit)
Magnetic flux which, linking a circuit of one
weber (SI
turn, would produce in it an electromotive force
Wb
unit)
of 1 volt if it were reduced to zero at a uniform
rate in 1 second.
Relation to SI
units
= 1 × 10−8 Wb
= 1 Wb = 1 V·s =
1 kg·m2/(A·s2)
99.28 MAGNETIC FLUX DENSITY:
Name of unit Symbol
Definition
Relation to SI units
2
−4
gauss (CGS unit) G
≡ Mx/cm = 10 T = 1 × 10−4 T
tesla (SI unit)
T
≡ Wb/m2
= 1 T = 1 Wb/m2 = 1 kg/(A·s2)
99.29 INDUCTANCE:
Name
Symbol
of unit
henry
H
(SI unit)
Definition
Relation to SI
units
The inductance of a closed circuit that produces one
volt of electromotive force when the current in the = 1 H = 1 Wb/A =
circuit varies at a uniform rate of one ampere per
1 kg·m2/(A·s)2
second.
99.30 TEMPERATURE:
Name of
Symbol
Definition
Page 283 of 286
Conversion to
unit
degree
Celsius
degree
Delisle
degree
Fahrenheit
degree
Newton
degree
Rankine
degree
Réaumur
kelvin
°C = K − 273.15. A unit of °C is the same size
as a unit of K; however, their numerical values
differ as the zero point of Celsius is set at
273.15 K (the ice point).
°C
[K] = 373.15 −
[°De] × 2/3
0 °F ≡ freezing pt. of H2O+NaCl, 180°F
[K] = ([°F] +
between freezing and boiling pt of H2O @ 1atm 459.67) × 5/9
[K] = [°N] ×
100/33 + 273.15
°De
°F
°N
°R; °Ra 0 °R ≡ absolute zero
[K] = [°R] × 5/9
[K] = [°Ré] × 5/4
+ 273.15
[K] = ([°Rø] −
7.5) × 40/21 +
273.15
°Ré
degree
Rømer
°Rø
kelvin (SI
base unit)
K
[K] = [°C] +
273.15
≡ 1/273.16 of the thermodynamic temperature
of the triple point of water.
1K
99.31 INFORMATION ENTROPY:
Name of unit
Symbol
Definition
SI unit
J/K
≡ J/K
nat; nip; nepit
nat
≡ kB
bit; shannon
bit; b; Sh ≡ ln(2) × kB
ban;
≡ ln(10) ×
Hart
kB
ban; hartley
≡ 4 bits
nibble
byte
B
≡ 8 bits
kilobyte (decimal) kB
≡ 1 000 B
kilobyte
KB; KiB ≡ 1 024 B
(kibibyte)
Relation to SI units
Relation to
bits
= 1 J/K
= 1.380 650 5(23) × 10−23
J/K
= 9.569 940 (16) × 10−24 J/K = 1 bit
= 3.179 065 3(53) × 10−23
J/K
= 3.827 976 0(64) × 10−23
= 22 bit
J/K
= 7.655 952 (13) × 10−23 J/K = 23 bit
= 7.655 952 (13) × 10−20 J/K
= 7.839 695 (13) × 10−20 J/K = 210 bit
99.32 LUMINOUS INTENSITY:
Name of unit Symbol
candela (SI
base unit);
candle
cd
Definition
The luminous intensity, in a given direction, of a
source that emits monochromatic radiation of
frequency 540 × 1012 hertz and that has a radiant
Page 284 of 286
Relation
to SI units
= 1 cd
intensity in that direction of 1/683 watt per
steradian.
≡ cd The use of candlepower as a unit is
discouraged due to its ambiguity.
Varies and is poorly reproducible. Approximately
0.981 cd.
candlepower
cp
(new)
candlepower
cp
(old, pre-1948)
= 1 cd
≈ 0.981 cd
99.33 LUMINANCE:
Name of unit
candela per square foot
candela per square inch
candela per square metre (SI unit);
footlambert
lambert
stilb (CGS unit)
Symbol
cd/ft2
cd/in2
cd/m2
fL
L
sb
Definition
≡ cd/ft2
≡ cd/in2
≡ cd/m2
≡ (1/π) cd/ft2
≡ (104/π) cd/m2
≡ 104 cd/m2
Relation to SI units
≈ 10.763910417 cd/m2
≈ 1,550.0031 cd/m2
= 1 cd/m2
≈ 3.4262590996 cd/m2
≈ 3,183.0988618 cd/m2
≈ 1 × 104 cd/m2
99.34 LUMINOUS FLUX:
Name of unit Symbol Definition Relation to SI units
lumen (SI unit) lm
≡ cd·sr
= 1 lm = 1 cd·sr
99.35 ILLUMINANCE:
Name of unit
footcandle; lumen per square foot
lumen per square inch
lux (SI unit)
phot (CGS unit)
Symbol
fc
lm/in2
lx
ph
Definition
≡ lm/ft2
≡ lm/in2
≡ lm/m2
≡ lm/cm2
Relation to SI units
= 10.763910417 lx
≈ 1,550.0031 lx
= 1 lx = 1 lm/m2
= 1 × 104 lx
99.36 RADIATION - SOURCE ACTIVITY:
Name of unit
becquerel (SI unit)
curie
rutherford (H)
Symbol
Definition
Bq
≡ Number of disintegrations per second
Ci
≡ 3.7 × 1010 Bq
rd
≡ 1 MBq
99.37 RADIATION – EXPOSURE:
Name of unit Symbol
Definition
Relation to SI units
−4
roentgen
R
1 R ≡ 2.58 × 10 C/kg = 2.58 × 10−4 C/kg
99.38 RADIATION - ABSORBED DOSE:
Page 285 of 286
Relation to SI units
= 1 Bq = 1/s
= 3.7 × 1010 Bq
= 1 × 106 Bq
Name of unit Symbol
Definition
Relation to SI units
2 2
gray (SI unit) Gy
≡ 1 J/kg = 1 m /s = 1 Gy
rad
rad
≡ 0.01 Gy
= 0.01 Gy
99.39 RADIATION - EQUIVALENT DOSE:
Name of unit
Symbol Definition Relation to SI units
Röntgen equivalent man rem
≡ 0.01 Sv = 0.01 Sv
sievert (SI unit)
Sv
≡ 1 J/kg = 1 Sv
Page 286 of 286