PREFACE

1 The Basics
1.1 Objectives
1.2 Essential Arithmetic Principles
1.3 Notation, Notation, Notation
1.4 Basic Terms
1.4.1 Indexing and Referencing
1.4.2 Specific Mathematical Use of Terms
1.5 Functions and Equations
1.5.1 Applying Functions: The Equation of a Line
1.5.2 The Factorial Function
1.5.3 The Modulo Function
1.6 Polynomial Functions
1.7 Logarithms and Exponents
1.8 New Terminology
1.9 Chapter Appendix: It's All Greek to Me

2 Analytic Geometry
2.1 Objectives (the Width of a Circle)
2.2 Radian Measurement and Polar Coordinates
2.3 What Is Trigonometry?
2.3.1 Radian Measures for Trigonometric Functio
2.3.2 Conic Sections and Some Analytical Geome
2.4 New Terminology

3 Linear Algebra: Vectors, Matrices, and Operations 3.1 Objectives
3.2 Working with Vectors
3.2.1 Vector Norms
3.3 So What Is the Matrix?
3.3.1 Some Special Matrices
3.4 Controlling the Matrix
3.5 Matrix Transposition
3.6 Advanced Topics
3.6.1 Special Matrix Forms
3.6.2 Vectorization of Matrices
3.7 New Terminology

4 Linear Algebra Continued: Matrix Structure
4.1 Objectives
4.2 Space and Time
4.3 The Trace and Determinant of a Matrix
4.4 Matrix Rank
4.5 Matrix Norms
4.6 Matrix Inversion
4.7 Linear Systems of Equations
4.8 Eigen-Analysis of Matrices
4.9 Quadratic Forms and Descriptions
4.10 New Terminology

5 Elementary Scalar Calculus
5.1 Objectives
5.3 Understanding Rates, Changes, and Derivatives
5.2 Limits and Lines
5.4 Derivative Rules for Common Functions
5.4.1 Basic Algebraic Rules for Derivatives
5.4.2 Derivatives of Logarithms and Exponents
5.4.3 L'Hospital's Rule
5.4.4 Applications: Rolle's Theorem and the Mean Value Theorem
5.5 Understanding Areas, Slices, and Integrals
5.5.1 Riemann Integrals
5.6 The Fundamental Theorem of Calculus
5.6.1 Integrating Polynomials with Antiderivatives
5.6.2 Indefinite Integrals
5.6.3 Integrals Involving Logarithms and Exponents
5.6.4 Integration by Parts
5.7 Additional Topics: Calculus of Trigonometric Functions
5.7.1 Derivatives of Trigonometric Functions
5.7.2 Integrals of Trigonometric Functions
5.8 New Terminology

6 Additional Topics in Scalar and Vector Calculus
6.1 Objectives
6.2 Partial Derivatives
6.3 Derivatives and Partial Derivatives of Higher Order
6.4 Maxima, Minima, and Root Finding
6.4.1 Evaluating Zero-Derivative Points
6.4.2 Root Finding with Newton-Raphson
6.5 Multidimensional Integrals
6.6 Finite and Infinite Series
6.6.1 Convergence
6.7 The Calculus of Vector and Matrix Forms
6.7.1 Vector Function Notation
6.7.2 Differentiation and Integration of a Vector Function
6.8 Constrained Optimization
6.9 New Terminology

7 Probability Theory
7.1 Objectives
7.2 Counting Rules and Permutations
7.2.1 The Binomial Theorem and Pascal's Triangle
7.3 Sets and Operations on Sets
7.3.1 General Characteristics of Sets
7.3.2 A Special Set: The Empty Set
7.3.3 Operations on Sets
7.4 The Probability Function
7.5 Calculations with Probabilities
7.6 Conditional Probability and Bayes Law
7.6.1 Simpson's Paradox
7.7 Independence
7.8 Odds
7.9 New Terminology

8 Random Variables
8.1 Objectives
8.2 Levels of Measurement
8.3 Distribution Functions
8.3.1 Randomness and Variables
8.3.2 Probability Mass Functions
8.3.3 Bernoulli Trials
8.3.4 Binomial Experiments
8.3.5 Poisson Counts
8.3.6 The Cumulative Distribution Function: Discrete Version
8.3.7 Probability Density Functions
8.3.8 Exponential and Gamma PDFs
8.3.9 Normal PDF
8.3.10 The Cumulative Distribution Function: Continuous Version
8.3.11 The Uniform Distributions
8.4 Measures of Central Tendency: Mean, Median, and Mode
8.5 Measures of Dispersion: Variance, Standard Deviation, and MAD
8.6 Correlation and Covariance
8.7 Expected Value
8.8 Some Handy Properties and Rules
8.9 Inequalities Based on Expected Values
8.10 Moments of a Distribution
8.11 New Terminology

9 Markov Chains
9.1 Objectives
9.2 Defining Stochastic Processes and Markov Chains
9.2.1 The Markov Chain Kernel
9.2.2 The Stationary Distribution of a Markov Chaîn
9.3 Properties of Markov Chains
9.3.1 Homogeneity and Periodicity
9.3.2 Irreducibility
9.3.3 Recurrence
9.3.4 Stationarity and Ergodicity
9.3.5 Reversibility
9.4 New Terminology

References