Guide to Essential Math, 2nd Edition

To the Reader

Preface to Second Edition

Chapter 1. Mathematical Thinking

1.1 The NCAA March Madness Problem

1.2 Gauss and the Arithmetic Series

1.3 The Pythagorean Theorem

1.4 Torus Area and Volume

1.5 Einstein’s Velocity Addition Law

1.6 The Birthday Problem

1.7 Fibonacci Numbers and the Golden Ratio

1.8 in the Gaussian Integral

1.9 Function Equal to Its Derivative

1.10 Stirling’s Approximation for!

1.11 Potential and Kinetic Energies

1.12 Riemann Zeta Function and Prime Numbers

1.13 How to Solve It

1.14 A Note on Mathematical Rigor

Chapter 2. Numbers

2.1 Integers

2.2 Primes

2.3 Divisibility

2.4 Rational Numbers

2.5 Exponential Notation

2.6 Powers of 10

2.7 Binary Number System

2.8 Infinity

Chapter 3. Algebra

3.1 Symbolic Variables

3.2 Legal and Illegal Algebraic Manipulations

3.3 Factor-Label Method

3.4 Powers and Roots

3.5 Logarithms

3.6 The Quadratic Formula

3.7 Imagining i

3.8 Factorials, Permutations and Combinations

3.9 The Binomial Theorem

3.10 e is for Euler

Chapter 4. Trigonometry

4.1 What Use is Trigonometry?

4.2 Geometry of Triangles

4.3 The Pythagorean Theorem

4.4 in the Sky

4.5 Sine and Cosine

4.6 Tangent and Secant

4.7 Trigonometry in the Complex Plane

4.8 de Moivre’s Theorem

4.9 Euler’s Theorem

4.10 Hyperbolic Functions

Chapter 5. Analytic Geometry

5.1 Functions and Graphs

5.2 Linear Functions

5.3 Conic Sections

5.4 Conic Sections in Polar Coordinates

Chapter 6. Calculus

6.1 A Little Road Trip

6.2 A Speedboat Ride

6.3 Differential and Integral Calculus

6.4 Basic Formulas of Differential Calculus

6.5 More on Derivatives

6.6 Indefinite Integrals

6.7 Techniques of Integration

6.8 Curvature, Maxima and Minima

6.9 The Gamma Function

6.10 Gaussian and Error Functions

6.11 Numerical Integration

Chapter 7. Series and Integrals

7.1 Some Elementary Series

7.2 Power Series

7.3 Convergence of Series

7.4 Taylor Series

7.5 Bernoulli and Euler Numbers

7.6 L’Hôpital’s Rule

7.7 Fourier Series

7.8 Dirac Deltafunction

7.9 Fourier Integrals

7.10 Generalized Fourier Expansions

7.11 Asymptotic Series

Chapter 8. Differential Equations

8.1 First-Order Differential Equations

8.2 Numerical Solutions

8.3 AC Circuits

8.4 Second-Order Differential Equations

8.5 Some Examples from Physics

8.6 Boundary Conditions

8.7 Series Solutions

8.8 Bessel Functions

8.9 Second Solution

8.10 Eigenvalue Problems

Chapter 9. Matrix Algebra

9.1 Matrix Multiplication

9.2 Further Properties of Matrices

9.3 Determinants

9.4 Matrix Inverse

9.5 Wronskian Determinant

9.6 Special Matrices

9.7 Similarity Transformations

9.8 Matrix Eigenvalue Problems

9.9 Diagonalization of Matrices

9.10 Four-Vectors and Minkowski Spacetime

Chapter 10. Group Theory

10.1 Introduction

10.2 Symmetry Operations

10.3 Mathematical Theory of Groups

10.4 Representations of Groups

10.5 Group Characters

10.6 Group Theory in Quantum Mechanics

10.7 Molecular Symmetry Operations

Chapter 11. Multivariable Calculus

11.1 Partial Derivatives

11.2 Multiple Integration

11.3 Polar Coordinates

11.4 Cylindrical Coordinates

11.5 Spherical Polar Coordinates

11.6 Differential Expressions

11.7 Line Integrals

11.8 Green’s Theorem

Chapter 12. Vector Analysis

12.1 Scalars and Vectors

12.2 Scalar or Dot Product

12.3 Vector or Cross Product

12.4 Triple Products of Vectors

12.5 Vector Velocity and Acceleration

12.6 Circular Motion

12.7 Angular Momentum

12.8 Gradient of a Scalar Field

12.9 Divergence of a Vector Field

12.10 Curl of a Vector Field

12.11 Maxwell’s Equations

12.12 Covariant Electrodynamics

12.13 Curvilinear Coordinates

12.14 Vector Identities

Chapter 13. Partial Differential Equations and Special Functions

13.1 Partial Differential Equations

13.2 Separation of Variables

13.3 Special Functions

13.4 Leibniz’s Formula

13.5 Vibration of a Circular Membrane

13.6 Bessel Functions

13.7 Laplace’s Equation in Spherical Coordinates

13.8 Legendre Polynomials

13.9 Spherical Harmonics

13.10 Spherical Bessel Functions

13.11 Hermite Polynomials

13.12 Laguerre Polynomials

13.13 Hypergeometric Functions

Chapter 14. Complex Variables

14.1 Analytic Functions

14.2 Derivative of an Analytic Function

14.3 Contour Integrals

14.4 Cauchy’s Theorem

14.5 Cauchy’s Integral Formula

14.6 Taylor Series

14.7 Laurent Expansions

14.8 Calculus of Residues

14.9 Multivalued Functions

14.10 Integral Representations for Special Functions

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