Cover
Contents
Preface
I. Revision of Fundamentals
1. Numbers and Functions
1. Real Numbers and Significant Digits
1.1. Notations and Conventions
1.2. Rounding Numbers and Significant Digits
2. Sets
3. Equations
3.1. Simple Equation
3.2. Simultaneous Equations
3.3. Inequality
4. Functions
4.1. Domain and Range
4.2. Linear Function and Modulus Function
4.3. Power Functions
4.4. Exponentials and Logarithms
4.5. Trigonometrical Functions
4.6. Composite Functions
2. Equations and Polynomials
1. Index Notation
2. Binomial Expansions
3. Floating Point Numbers
4. Quadratic Equations
5. Polynomials and Roots
II. Main Topics
3. Vectors and Matrices
1. Vectors
2. Vector Products
2.1. Dot Product
2.2. Cross Product
2.3. Triple Product of Vectors
3. Matrix Algebra
3.1. Matrix, Addition and Multiplication
3.2. Transformation and Inverse
4. System of Linear Equations
5. Eigenvalues and Eigenvectors
5.1. Eigenvalues and Eigenvectors of a Matrix
5.2. Definiteness of a Matrix
6. Tensors
6.1. Summation Notations
6.2. Tensors
6.3. Elasticity
4. Calculus I: Differentiation
1. Gradient and Derivative
2. Differentiation Rules
3. Maximum, Minimum and Radius of Curvature
4. Series Expansions and Taylor Series
5. Partial Derivatives
6. Differentiation of Vectors
6.1. Polar Coordinates
6.2. Three Basic Operators
6.3. Cylindrical Coordinates
6.4. Spherical Coordinates
7. Jacobian and Hessian Matrices
5. Calculus II: Integration
1. Integration
2. Integration by Parts
3. Integration by Substitution
4. Double Integrals and Multiple Integrals
5. Jacobian Determinant
6. Special Integrals
6.1. Line Integral
6.2. Gaussian Integrals
6.3. Error Functions
6. Complex Numbers.

1. Complex Numbers
2. Complex Algebra
3. Hyperbolic Functions
4. Analytical Functions
5. Complex Integrals
5.1. Cauchy's Integral Theorem
5.2. Residue Theorem
7. Ordinary Differential Equations
1. Differential Equations
2. First-Order Differential Equations
3. Second-Order Equations
3.1. Solution Technique
3.2. Sturm-Liouville Eigenvalue Problem
4. Higher-Order ODEs
5. System of Linear ODEs
6. Harmonic Motions
6.1. Undamped Forced Oscillations
6.2. Damped Forced Oscillations
8. Fourier Transform and Laplace Transform
1. Fourier Series
1.1. Fourier Series
1.2. Orthogonality and Fourier Coefficients
2. Fourier Transforms
3. Discrete and Fast Fourier Transforms
4. Laplace Transform
4.1. Laplace Transform Pairs
4.2. Scalings and Properties
4.3. Derivatives and Integrals
5. Solving ODE via Laplace Transform
6. Z-Transform
7. Relationships between Fourier, Laplace and Z-transforms
9. Statistics and Curve Fitting
1. Random Variables, Means and Variance
2. Binomial and Poisson Distributions
3. Gaussian Distribution
4. Other Distributions
5. The Central Limit Theorem
6. Weibull Distribution
7. Sample Mean and Variance
8. Method of Least Squares
8.1. Linear Regression and Correlation Coefficient
8.2. Linearization
9. Generalized Linear Regression
III. Advanced Topics
10. Partial Differential Equations
1. Introduction
2. First-Order PDEs
3. Classification of Second-Order PDEs
4. Classic PDEs
5. Solution Techniques
5.1. Separation of Variables
5.2. Laplace Transform
5.3. Similarity Solution
6. Integral Equations
6.1. Fredholm and Volterra Integral Equations
6.2. Solutions of Integral Equations
11. Numerical Methods and Optimization
1. Root-Finding Algorithms
2. Numerical Integration.

3. Numerical Solutions of ODEs
3.1. Euler Scheme
3.2. Runge-Kutta Method
4. Optimization
4.1. Feasible Solution
4.2. Optimality Criteria
5. Unconstrained Optimization
5.1. Univariate Functions
5.2. Multivariate Functions
6. Gradient-Based Methods
7. Nonlinear Optimization
7.1. Penalty Method
7.2. Lagrange Multipliers
7.3. Karush-Kuhn-Tucker Conditions
A. Answers to Exercises
Bibliography
Index.