Saydanzad Group Mathematical Methods in the Physical Sciences

Course Notes

Table of Contents

Chapter 0. Math Review for Calculus-Based Physics

  1. 0.1 Real Number System and Intervals
  2. 0.2 Functions, Domain, and Range
  3. 0.3 Linear Functions and Linear Equations
  4. 0.4 Quadratic Functions and Equations
  5. 0.5 Exponential and Logarithmic Functions
  6. 0.6 Trigonometry for Physics
  7. 0.7 Calculus Essentials
  8. For the Interested Reader

Chapter 1. Infinite Series and Power Series

  1. 1.1 Definitions and Notation
  2. 1.2 Geometric Sequences and Series
  3. 1.3 Limit of the General Term as n
  4. 1.4 Testing Series for Convergence
  5. 1.5 Alternating Series
  6. Summary: Convergence Tests
  7. 1.7 Power Series
  8. 1.8 Theorems About Power Series
  9. 1.9 Taylor and Maclaurin Series
  10. 1.10 Techniques for Obtaining Power Series Expansions
  11. 1.11 Applications of Series
  12. Summary: Types of Series
  13. For the Interested Reader

Chapter 2. Complex Numbers

  1. 2.1 Introduction to Complex Numbers
  2. 2.2 Complex Algebra
  3. 2.3 Complex Infinite Series
  4. 2.4 Complex Power Series: Disk of Convergence
  5. 2.5 Euler's Formula
  6. 2.6 Elementary Functions of Complex Numbers
  7. 2.7 The Exponential Function
  8. 2.8 Powers and Roots of Complex Numbers
  9. 2.9 Trigonometric Functions of a Complex Variable
  10. 2.10 Hyperbolic Functions
  11. 2.11 Logarithms of Complex Numbers
  12. 2.12 Complex Roots and Powers
  13. 2.13 Inverse trigonometric functions
  14. 2.14 Applications of Complex Numbers
  15. Summary Table
  16. For the Interested Reader

Chapter 3. Linear Algebra

  1. 3.1 Linear Equations
  2. 3.2 System of Linear Equations
  3. 3.3 Gauss--Jordan Reduction
  4. 3.4 Determinants
  5. 3.5 Cramer's Rule
  6. 3.6 Vectors
  7. 3.7 Matrix Operations
  8. 3.8 Zero, Identity, and Transpose of Matrices
  9. 3.9 Linear Combinations and Linear Functions
  10. 3.10 Matrix Operators and Linear Transformations
  11. 3.11 Orthogonal Transformations
  12. 3.12 Rotation in 2D
  13. 3.13 Rotations and Reflections in 3D
  14. 3.14 Linear Independence
  15. 3.15 Homogeneous Equations
  16. 3.16 Eigenvalues, Eigenvectors, and Matrix Diagonalization
  17. 3.17 General Vector Spaces
  18. 3.18 Bra--Ket Notation and Inner/Outer Products
  19. 3.19 Hilbert Spaces: Inner Product, Norm, and Orthogonality
  20. 3.20 Gram--Schmidt Method of Orthonormalization
  21. 3.21 Applications of Linear Algebra
  22. Summary Table
  23. For the Interested Reader

Chapter 4. Partial Differentiation

  1. 4.1 Introduction
  2. 4.2 Applications in Thermodynamics
  3. 4.3 Power Series in Two Variables
  4. 4.4 Total Differentials
  5. 4.5 Chain Rule: Differentiating a Function of a Function
  6. 4.6 Implicit Differentiation
  7. 4.7 More on Chain Rule
  8. 4.8 Chain Rule in Polar and Rectangular Coordinates
  9. 4.9 Change of Variables: Laplace Equation in Polar Coordinates
  10. 4.10 Leibniz’s Rule
  11. 4.11 Where Partial Differentiation Is Used
  12. Summary Table
  13. For the Interested Reader

Chapter 5. Multiple Integrals; Applications of Integration

  1. 5.1 Single integral
  2. 5.2 Double and Triple Integrals
  3. 5.3 When to Integrate y-First vs. x-First
  4. 5.4 Applications of Integration
  5. 5.5 Change of Variables in Integrals
  6. 5.6 Connection with the Jacobian Matrix
  7. 5.7 Where Multiple Integrals Are Used
  8. Summary Table
  9. For the Interested Reader

Chapter 6. Vector Analysis

  1. 6.1 Applications of Vector Multiplication
  2. 6.2 Differentiation of Vectors
  3. 6.3 Fields
  4. 6.4 Directional Derivative and Gradient
  5. 6.5 Expressions Involving ∇
  6. 6.6 Maxwell’s Equations in Vacuum
  7. 6.7 Vector Integrals
  8. 6.8 The Divergence and Stokes’s Theorems
  9. 6.9 Maxwell’s Equations and the Continuity Equation
  10. Summary of Key Vector Identities
  11. 6.11 Where Vector Analysis Is Used
  12. For the Interested Reader

Chapter 7. Fourier Series and Transforms

  1. 7.1 Even and Odd Functions
  2. 7.2 Periodic Functions and Fourier Series
  3. 7.3 Fourier Transform
  4. 7.4 Where Fourier Analysis Is Used
  5. Summary Table
  6. For the Interested Reader

Chapter 8. Ordinary Differential Equations

  1. 8.1 Introduction and General Form
  2. 8.2 Classification of ODEs
  3. 8.3 First-Order ODEs
  4. 8.4 Second-Order Ordinary Differential Equations
  5. 8.5 Where Differential Equations Are Used
  6. Summary Table
  7. For the Interested Reader

Chapter 9. Series Solutions of Ordinary Differential Equations (Chapter 12)

  1. 9.1 Ordinary, Regular Singular, and Irregular Points
  2. 9.2 Legendre’s Differential Equation
  3. 9.3 Associated Legendre Functions
  4. 9.4 Bessel’s Differential Equation
  5. 9.5 Orthogonal Eigenfunction Bases in Physics
  6. 9.6 Where Series Solutions Are Used
  7. Summary Table
  8. For the Interested Reader

Chapter 10. Partial Differential Equations (Chapter 13)

  1. 10.1 Introduction
  2. 10.2 Separation of Variables and Coordinate Symmetry
  3. 10.3 Laplace’s Equation: Steady–State Temperature in a Semi–Infinite Plate
  4. 10.4 Helmholtz Equation and the Infinite Spherical Well
  5. 10.5 Where Partial Differential Equations Lead
  6. Summary Table
  7. For the Interested Reader