Guide to Essential Math by Blinder, Sy M. -- Read -- Imperial Library of Trantor

Index

Cover image Title page Table of Contents Copyright 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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