Handbook of Mathematical Concepts and Formulas for Students in Science and Engineering

Contents
Preface
About the Authors
Acknowledgments
I Elementary Set Theory, Algebra, and Geometry
	1. Set Theory
		1.1 Basic Concepts
			1.1.1 Product set
		1.2 Sets of Numbers
		1.3 Cardinality
	2. Elementary Algebra
		2.1 Basic Concepts
		2.2 Rules of Arithmetics
			2.2.1 Fundamental algebraic rules
			2.2.2 The binomial theorem
		2.3 Polynomials in One Variable
		2.4 Rational Expression
			2.4.1 Expansion of rational expression
		2.5 Inequalities
			2.5.1 Absolute value
		2.6 Complex Numbers
			2.6.1 To solve equations of second degree with complex coefficients
			2.6.2 Complex numbers in polar form
		2.7 Powers and Logarithms
			2.7.1 Powers
			2.7.2 Logarithms
	3. Geometry and Trigonometry
		3.1 Plane Geometry
			3.1.1 Angle
			3.1.2 Units of different angular measurements
			3.1.3 Reflection in point and line
			3.1.4 Polygon
			3.1.5 Types of triangles
			3.1.6 Regular polygons
			3.1.7 Circle and ellipse
		3.2 Space Geometry
			3.2.1 Names and volumes of some common bodies
			3.2.2 Parallelepiped and the five regular polyhedra
		3.3 Coordinate System (R2)
		3.4 Trigonometry
			3.4.1 Basic theorems
		3.5 Addition Formulas
			3.5.1 Addition formulas for sine and cosine functions
			3.5.2 Addition formulas for tangent
			3.5.3 Phase–amplitude form
			3.5.4 Identities for double and half angles
			3.5.5 Some exact values
		3.6 Inverse Trigonometric Functions
		3.7 Trigonometric Equations
		3.8 Solving Triangles
		3.9 Coordinate System (R3)
			3.9.1 Identities in spherical trigonometry
			3.9.2 Triangle solution of spheric triangle
	4. Vector Algebra
		4.1 Basic Concepts
			4.1.1 Line in R2
		4.2 Vectors in R3
			4.2.1 Cross product and scalar triple product
			4.2.2 Plane in R3
			4.2.3 Distance between some objects in R3
			4.2.4 Intersection, projection, lines, and planes
	5. Linear Algebra
		5.1 Linear Equation Systems
			5.1.1 Solution of linear system of equations with matrices
			5.1.2 Column, row, and null-spaces
		5.2 Matrix Algebra
			5.2.1 Inverse matrix
			5.2.2 Elementary matrices
			5.2.3 LU-factorization
			5.2.4 Quadratic form
		5.3 Determinant
			5.3.1 Number of solutions for ES, determinant, and rank
			5.3.2 Computing the determinant using sub-determinants
			5.3.3 Cramer’s rule
			5.3.4 Determinant and row operations
			5.3.5 Pseudoinverse
			5.3.6 Best LS solutions for some common functions
			5.3.7 Eigenvalues and eigenvectors
			5.3.8 Diagonalization of matrix
			5.3.9 Matrices with complex elements
			5.3.10 Base
			5.3.11 Basis and coordinate change
		5.4 The Quaternion Ring
			5.4.1 Splitting a quaternion q in its scalar and vector parts
			5.4.2 Matrix representation
		5.5 Optimization
			5.5.1 Linear optimization
			5.5.2 Convex optimization
	6. Algebraic Structures
		6.1 Overview
		6.2 Homomorphism and Isomorphism
		6.3 Groups
			6.3.1 Examples of groups
		6.4 Rings
			6.4.1 Examples of rings
	7. Logic and Number Theory
		7.1 Combinatorics
			7.1.1 Sum and product
			7.1.2 Factorials
			7.1.3 Permutations and combinations
		7.2 Proof by Induction
			7.2.1 Strong induction
		7.3 Relations
		7.4 Expressional Logic
			7.4.1 Tautology and contradiction
			7.4.2 Methods of proofs
		7.5 Predicate Logic
		7.6 Boolean Algebra
			7.6.1 Graph theory
			7.6.2 Trees
		7.7 Difference Equations
		7.8 Number Theory
			7.8.1 Introductory concepts
			7.8.2 Some results
			7.8.3 RSA encryption
	8. Calculus of One Variable
		8.1 Elementary Topology on R
		8.2 Real Functions
			8.2.1 Symmetry; even and odd functions (I)
		8.3 The Elementary Functions
			8.3.1 Algebraic functions
			8.3.2 Transcendental functions
			8.3.3 Polynomial
			8.3.4 Power functions
			8.3.5 Exponential functions
			8.3.6 Logarithmic functions
			8.3.7 The trigonometric functions
			8.3.8 The arcus functions
			8.3.9 Composition of functions and (local) inverses
			8.3.10 Tables of elementary functions
		8.4 Some Specific Functions
			8.4.1 Some common function classes
		8.5 Limit and Continuity
			8.5.1 Calculation rules for limits
			8.5.2 Corollary from the limit laws
			8.5.3 The size order between exp-, power-, and logarithm functions
			8.5.4 Limits for the trigonometric functions
			8.5.5 Some special limits
			8.5.6 Some derived limits
		8.6 Continuity
			8.6.1 Definition
			8.6.2 Calculus rules for continuity
			8.6.3 Some theorems about continuity
			8.6.4 Riemann’s z-function
	9. Derivatives
		9.1 Directional Coefficient
			9.1.1 The one- and two-point formulas
			9.1.2 Continuity and differentability
			9.1.3 Tangent, normal, and asymptote
		9.2 The Differentiation Rules
		9.3 Applications of Derivatives
			9.3.1 Newton–Raphson iteration method
			9.3.2 L’Hôspital’s rule
			9.3.3 Lagrange’s mean value theorem
			9.3.4 Derivative of inverse function and implicit derivation
			9.3.5 Second derivative of inverse function
			9.3.6 Implicit differentiation
			9.3.7 Convex and concave functions
		9.4 Tables
	10. Integral
		10.1 Definitions and Theorems
			10.1.1 Lower and upper sums
		10.2 Primitive Function
		10.3 Rules of Integral Calculus
			10.3.1 Linearity of integral
			10.3.2 Area between function curves
			10.3.3 The integral mean value theorem (I)
			10.3.4 The integral mean theorem (II)
			10.3.5 Some common inequalities for integrals
		10.4 Methods of Integration
			10.4.1 Symmetry; even and odd functions (II)
			10.4.2 Integration by parts
			10.4.3 Variable substitution
			10.4.4 The tan x/2−substitution
		10.5 Improper Integral
		10.6 Tables
			10.6.1 Common indefinite integrals with algebraic integrand
			10.6.2 Common indefinite integrals with non-algebraic integrands
			10.6.3 Some integrals with trigonometric integrands
			10.6.4 Recursion formulas
			10.6.5 Tables of some definite integrals
			10.6.6 Tables of improper integrals
			10.6.7 Tables of some non-elementary integrals
			10.6.8 The Dirac function
		10.7 Numerical Integration
	11. Differential Equations
		11.1 ODEs of Order 1 and 2
		11.2 Linear ODE
			11.2.1 Linear ODE of first order
		11.3 Linear DE with Constant Coefficients
			11.3.1 Solution of linear DE
			11.3.2 Ansatz to determine yp
		11.4 Linear DE with Continuous Coefficients
			11.4.1 Linear ODE of second order
			11.4.2 Some special ODEs of second order
			11.4.3 Linear system of differential equations
		11.5 Existence and Uniqueness of the Solution
		11.6 Partial Differential Equations (PDEs)
			11.6.1 The most common initial and boundary value problems
			11.6.2 Representation with orthogonal series
			11.6.3 Green’s functions
	12. Numerical Analysis
		12.1 Computer Language Approach
		12.2 Numerical Differentiation and Integration
			12.2.1 Numerical differentiation
			12.2.2 Numerical integration
		12.3 Solving f(x) = 0
			12.3.1 Linear case
			12.3.2 Numerical solution of nonlinear equations
			12.3.3 Common methods of iterations
		12.4 Ordinary Differential Equations (ODEs)
			12.4.1 The initial value problems
			12.4.2 Some common methods
			12.4.3 More accurate methods
		12.5 Finite Element Method (FEM)
		12.6 Monte Carlo Methods
			12.6.1 Monte Carlo method for DEs (indirect method)
			12.6.2 Examples of finite difference approximations for parabolic equations
			12.6.3 Monte Carlo for elliptic equations
	13. Differential Geometry
		13.1 Curve
			13.1.1 Examples of curves and surfaces in R2
		13.2 R3
			13.2.1 Notations in R3
			13.2.2 Curve and surface in R3
			13.2.3 Slice method
			13.2.4 Volume of rotation bodies
			13.2.5 Guldin’s rules
	14. Sequence and Series
		14.1 General Theory
		14.2 Positive Series
			14.2.1 Examples of sequences and series
		14.3 Function Sequences and Function Series
			14.3.1 General theory
			14.3.2 Power series
			14.3.3 Taylor expansions
			14.3.4 Fourier series
			14.3.5 Some sums, series, and inequalities
		14.4 Some Important Orthogonal Functions
			14.4.1 Generation of the most common polynomial classes
			14.4.2 Hypergeometric functions
		14.5 Products
			14.5.1 Basic examples
			14.5.2 Infinite products
	15. Transform Theory
		15.1 Fourier Transform
			15.1.1 Cosine and sine transforms
			15.1.2 Relations between Fourier transforms
			15.1.3 Special symbols
			15.1.4 Fourier transform in signal and system
			15.1.5 Table of discrete Fourier transform
		15.2 The jω-Method
		15.3 The z -Transform
		15.4 The Laplace Transform
		15.5 Distributions
	16. Complex Analysis
		16.1 Curves and Domains in the Complex Plane C
		16.2 Functions on the Complex Plane C
			16.2.1 Elementary functions
		16.3 Lines, Circles, and Möbius Transforms
			16.3.1 Preliminaries: The Riemann sphere
		16.4 Some Simple Mappings
			16.4.1 Möbius mappings
			16.4.2 Angle preserving functions
		16.5 Some Special Mappings
			16.5.1 Applications in potential theory
		16.6 Harmonic Functions
		16.7 Laurent Series, Residue Calculus
	17. Multidimensional Analysis
		17.1 Topology in Rn
			17.1.1 Subsets of Rn
			17.1.2 Connected sets, etc.
		17.2 Functions Rm −→ Rn
			17.2.1 Functions Rn −→ R
			17.2.2 Some common surfaces
			17.2.3 Level curve and level surface
			17.2.4 Composite function and its derivatives
			17.2.5 Some special cases of chain rule
		17.3 Taylor’s Formula
		17.4 Maximum and Minimum Values of a Function
			17.4.1 Max and min with constraints
		17.5 Optimization Under Constraints for Linear or Convex Function
			17.5.1 Convex optimization
		17.6 Integral Calculus
			17.6.1 Variable substitution in multiple integral
	18. Vector Analysis
		18.1 Differential Calculus in Rn
		18.2 Types of Differential Equations
	19. Topology
		19.1 Definitions and Theorems
			19.1.1 Variants of compactness
		19.2 The Usual Topology on Rn
			19.2.1 A comparison between two topologies
		19.3 Axioms
			19.3.1 The parallel axiom
			19.3.2 The induction axiom
			19.3.3 Axiom of choice
		19.4 The Supremum Axiom with Some Applications
			19.4.1 The supremum axiom
			19.4.2 Compact set in Rn
			19.4.3 Three theorems about continuity on compact, connected set K ⊆ Rn
		19.5 Map of Topological Spaces
	20. Integration Theory
		20.1 The Riemann Integral
			20.1.1 Definition of the Riemann integral
			20.1.2 Integrability of continuous functions
			20.1.3 Comments about the Riemann integral
		20.2 The Lebesgue Integral
			20.2.1 General theory
			20.2.2 The Lebesgue integral on Rn
	21. Functional Analysis
		21.1 Topological Vector Space
		21.2 Some Common Function Spaces
			21.2.1 Hilbert space
			21.2.2 Hilbert space and Fourier series
			21.2.3 A criterion for Banach space
			21.2.4 Fourier transform
		21.3 Distribution Theory
			21.3.1 Generalized function
		21.4 Distributions
			21.4.1 Tempered distribution
	22. Mathematical Statistics
		22.1 Elementary Probability Theory
		22.2 Descriptive Statistics
			22.2.1 Class sample
		22.3 Distributions
			22.3.1 Discrete distribution
			22.3.2 Some common discrete distributions
			22.3.3 Continuous distributions
			22.3.4 Some common continuous distributions
			22.3.5 Connection between arbitrary normal distribution and the standard normal distribution
			22.3.6 Approximations
		22.4 Location and Spread Measures
		22.5 Multivariate Distributions
			22.5.1 Discrete distributions
			22.5.2 Bivariate continuous distribution
		22.6 Conditional Distribution
		22.7 Linear Combination of Random Variables
		22.8 Generating Functions
		22.9 Some Inequalities
		22.10 Convergence of Random Variables
			22.10.1 Table of some probability and moment generating functions
		22.11 Point Estimation of Parameters
			22.11.1 Expectancy accuracy and efficiency
		22.12 Interval Estimation
			22.12.1 Confidence interval for μ in normal distribution: X ∈ N(μ, σ)
			22.12.2 Confidence interval for σ2 in normal distribution
			22.12.3 Sample in pair and two samples
		22.13 Hypothesis Testing of μ in Normal Distribution σ Known
			22.13.1 σ unknown
		22.14 F-Distribution and F-Test
		22.15 Markov Chains
II Appendices
	A. Mechanics
		A.1 Definitions, Formulas, etc
			A.1.1 Newton’s motion laws
			A.1.2 Linear momentum
			A.1.3 Impulse momentum and moment of inertia
			A.1.4 Table of center of mass, and moment of inertia of some homogenous bodies
			A.1.5 Physical constants
	B. Varia
		B.1 Greek Alphabet
			B.1.1 Uppercase
			B.1.2 Lowercase
			B.1.3 The numbers π and e
			B.1.4 Euler constant
	C. Programming Mathematica (Mma)
		C.1 Elementary Syntax
			C.1.1 Parentheses
			C.1.2 Operations
			C.1.3 Equalities and defining concepts
			C.1.4 Elementary algebra
		C.2 Linear Algebra
		C.3 Calculus
			C.3.1 Calculus in several variables
		C.4 Ordinary Differential Equations
		C.5 Mathematical Statistics
		C.6 Difference or Recurrence Equations (RE)
			C.6.1 List of common commands
	D. The Program Matlab
		D.1 Introduction
			D.1.1 Accessing MATLAB
			D.1.2 Arithmetic operations
			D.1.3 Elementary functions
			D.1.4 Variables
			D.1.5 Editing and formatting
		D.2 Help in MATLAB
			D.2.1 Description of help command
			D.2.2 Example for how to use help
			D.2.3 The error message
			D.2.4 The command look for
			D.2.5 Demos and documentation
		D.3 Row Vectors and Curve Plotting
			D.3.1 Operations with row vectors
			D.3.2 Generating arithmetic sequences
			D.3.3 Plotting curves
			D.3.4 Plotting graphs of functions
			D.3.5 Several graphs/curves in the same figure
			D.3.6 Dimensioning of the coordinate axes
		D.4 Good to Know
			D.4.1 Strings and the command eval
			D.4.2 The command fplot
			D.4.3 Complex numbers
			D.4.4 Polynomials
			D.4.5 To save, delete, and recover data
			D.4.6 Text in figures
			D.4.7 Three-dimensional graphics
		D.5 To Create Own Commands
			D.5.1 Textfiles
			D.5.2 Function files
			D.5.3 How to write own help command
			D.5.4 Some simple but important recommendations
		D.6 Matrix Algebras
			D.6.1 Basic matrix operations
			D.6.2 System of equations (matrix division)
			D.6.3 Rows, columns, and individual matrixelements
			D.6.4 A guide for a better way to work with matrices
			D.6.5 Inverse and identity matrix
			D.6.6 Determinants, eigenvalues, and eigenvectors
			D.6.7 Functions of matrices
		D.7 Programming in MATLAB
			D.7.1 General function files
			D.7.2 Choice and condition
			D.7.3 Loops
			D.7.4 Input and output
			D.7.5 Functions as in-variables
			D.7.6 Efficient programming
			D.7.7 Search command and related topics
			D.7.8 Examples of some programs
			D.7.9 List of most important command categories
		D.8 Algorithms and MATLAB Codes
			D.8.1 The bisection method
			D.8.2 An algorithm for the bisection method
			D.8.3 An algorithm for the secant method
			D.8.4 An algorithm for the Newton’s method
			D.8.5 An algorithm for L2-projection
			D.8.6 A Matlab code to compute the mass matrix M for a non-uniform mesh
			D.8.7 A Matlab routine to compute the load vector b
			D.8.8 Matlab routine to compute the L2-projection
			D.8.9 A Matlab routine for the composite midpoint rule
			D.8.10 A Matlab routine for the composite trapezoidal rule
			D.8.11 A Matlab routine for the composite Simpson’s rule
			D.8.12 A Matlab routine assembling the stiffness matrix
			D.8.13 A Matlab routine to assemble the convection matrix
			D.8.14 Matlab routines for Forward-, Backward-Euler and Crank–Nicolson
			D.8.15 A Matlab routine for mass-matrix in 2D
			D.8.16 A Matlab routine for a Poisson assembler in 2D
III Tables
	E. Tables
		E.1 Some Mathematical Constants
		E.2 Table of the CDF of N(0, 1)
		E.3 Table of Some Quantiles of t-Distribution
		E.4 Table of the χ2-Distribution
		E.5 F-Table
	F. Key Concepts
		F.1 Symbols
		F.2 General Notation
		F.3 Derivatives and Differential Calculus
		F.4 Differential Geometry
			F.4.1 General notations for a vector space
			F.4.2 Notation of special vector spaces
		F.5 Generalized Functions
			F.5.1 Finite element related concepts
		F.6 Filter
Bibliography
Supplementary Material
Index