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Guide to Essential Math, 2nd Edition

Book Description

This book reminds students in junior, senior and graduate level courses in physics, chemistry and engineering of the math they may have forgotten (or learned imperfectly), which is needed to succeed in science courses. The focus is on math actually used in physics, chemistry and engineering, and the approach to mathematics begins with 12 examples of increasing complexity, designed to hone the student's ability to think in mathematical terms and to apply quantitative methods to scientific problems. Detailed Illustrations and links to reference material online help further comprehension. The 2e features new problems and illustrations and features expanded chapters on matrix algebra and differential equations.



  • Use of proven pedagogical techniques developed during the author’s 40 years of teaching experience
  • New practice problems and exercises to enhance comprehension
  • Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. To the Reader
  6. Preface to Second Edition
  7. Chapter 1. Mathematical Thinking
    1. 1.1 The NCAA March Madness Problem
    2. 1.2 Gauss and the Arithmetic Series
    3. 1.3 The Pythagorean Theorem
    4. 1.4 Torus Area and Volume
    5. 1.5 Einstein’s Velocity Addition Law
    6. 1.6 The Birthday Problem
    7. 1.7 Fibonacci Numbers and the Golden Ratio
    8. 1.8 in the Gaussian Integral
    9. 1.9 Function Equal to Its Derivative
    10. 1.10 Stirling’s Approximation for!
    11. 1.11 Potential and Kinetic Energies
    12. 1.12 Riemann Zeta Function and Prime Numbers
    13. 1.13 How to Solve It
    14. 1.14 A Note on Mathematical Rigor
  8. Chapter 2. Numbers
    1. 2.1 Integers
    2. 2.2 Primes
    3. 2.3 Divisibility
    4. 2.4 Rational Numbers
    5. 2.5 Exponential Notation
    6. 2.6 Powers of 10
    7. 2.7 Binary Number System
    8. 2.8 Infinity
  9. Chapter 3. Algebra
    1. 3.1 Symbolic Variables
    2. 3.2 Legal and Illegal Algebraic Manipulations
    3. 3.3 Factor-Label Method
    4. 3.4 Powers and Roots
    5. 3.5 Logarithms
    6. 3.6 The Quadratic Formula
    7. 3.7 Imagining i
    8. 3.8 Factorials, Permutations and Combinations
    9. 3.9 The Binomial Theorem
    10. 3.10 e is for Euler
  10. Chapter 4. Trigonometry
    1. 4.1 What Use is Trigonometry?
    2. 4.2 Geometry of Triangles
    3. 4.3 The Pythagorean Theorem
    4. 4.4 in the Sky
    5. 4.5 Sine and Cosine
    6. 4.6 Tangent and Secant
    7. 4.7 Trigonometry in the Complex Plane
    8. 4.8 de Moivre’s Theorem
    9. 4.9 Euler’s Theorem
    10. 4.10 Hyperbolic Functions
  11. Chapter 5. Analytic Geometry
    1. 5.1 Functions and Graphs
    2. 5.2 Linear Functions
    3. 5.3 Conic Sections
    4. 5.4 Conic Sections in Polar Coordinates
  12. Chapter 6. Calculus
    1. 6.1 A Little Road Trip
    2. 6.2 A Speedboat Ride
    3. 6.3 Differential and Integral Calculus
    4. 6.4 Basic Formulas of Differential Calculus
    5. 6.5 More on Derivatives
    6. 6.6 Indefinite Integrals
    7. 6.7 Techniques of Integration
    8. 6.8 Curvature, Maxima and Minima
    9. 6.9 The Gamma Function
    10. 6.10 Gaussian and Error Functions
    11. 6.11 Numerical Integration
  13. Chapter 7. Series and Integrals
    1. 7.1 Some Elementary Series
    2. 7.2 Power Series
    3. 7.3 Convergence of Series
    4. 7.4 Taylor Series
    5. 7.5 Bernoulli and Euler Numbers
    6. 7.6 L’Hôpital’s Rule
    7. 7.7 Fourier Series
    8. 7.8 Dirac Deltafunction
    9. 7.9 Fourier Integrals
    10. 7.10 Generalized Fourier Expansions
    11. 7.11 Asymptotic Series
  14. Chapter 8. Differential Equations
    1. 8.1 First-Order Differential Equations
    2. 8.2 Numerical Solutions
    3. 8.3 AC Circuits
    4. 8.4 Second-Order Differential Equations
    5. 8.5 Some Examples from Physics
    6. 8.6 Boundary Conditions
    7. 8.7 Series Solutions
    8. 8.8 Bessel Functions
    9. 8.9 Second Solution
    10. 8.10 Eigenvalue Problems
  15. Chapter 9. Matrix Algebra
    1. 9.1 Matrix Multiplication
    2. 9.2 Further Properties of Matrices
    3. 9.3 Determinants
    4. 9.4 Matrix Inverse
    5. 9.5 Wronskian Determinant
    6. 9.6 Special Matrices
    7. 9.7 Similarity Transformations
    8. 9.8 Matrix Eigenvalue Problems
    9. 9.9 Diagonalization of Matrices
    10. 9.10 Four-Vectors and Minkowski Spacetime
  16. Chapter 10. Group Theory
    1. 10.1 Introduction
    2. 10.2 Symmetry Operations
    3. 10.3 Mathematical Theory of Groups
    4. 10.4 Representations of Groups
    5. 10.5 Group Characters
    6. 10.6 Group Theory in Quantum Mechanics
    7. 10.7 Molecular Symmetry Operations
  17. Chapter 11. Multivariable Calculus
    1. 11.1 Partial Derivatives
    2. 11.2 Multiple Integration
    3. 11.3 Polar Coordinates
    4. 11.4 Cylindrical Coordinates
    5. 11.5 Spherical Polar Coordinates
    6. 11.6 Differential Expressions
    7. 11.7 Line Integrals
    8. 11.8 Green’s Theorem
  18. Chapter 12. Vector Analysis
    1. 12.1 Scalars and Vectors
    2. 12.2 Scalar or Dot Product
    3. 12.3 Vector or Cross Product
    4. 12.4 Triple Products of Vectors
    5. 12.5 Vector Velocity and Acceleration
    6. 12.6 Circular Motion
    7. 12.7 Angular Momentum
    8. 12.8 Gradient of a Scalar Field
    9. 12.9 Divergence of a Vector Field
    10. 12.10 Curl of a Vector Field
    11. 12.11 Maxwell’s Equations
    12. 12.12 Covariant Electrodynamics
    13. 12.13 Curvilinear Coordinates
    14. 12.14 Vector Identities
  19. Chapter 13. Partial Differential Equations and Special Functions
    1. 13.1 Partial Differential Equations
    2. 13.2 Separation of Variables
    3. 13.3 Special Functions
    4. 13.4 Leibniz’s Formula
    5. 13.5 Vibration of a Circular Membrane
    6. 13.6 Bessel Functions
    7. 13.7 Laplace’s Equation in Spherical Coordinates
    8. 13.8 Legendre Polynomials
    9. 13.9 Spherical Harmonics
    10. 13.10 Spherical Bessel Functions
    11. 13.11 Hermite Polynomials
    12. 13.12 Laguerre Polynomials
    13. 13.13 Hypergeometric Functions
  20. Chapter 14. Complex Variables
    1. 14.1 Analytic Functions
    2. 14.2 Derivative of an Analytic Function
    3. 14.3 Contour Integrals
    4. 14.4 Cauchy’s Theorem
    5. 14.5 Cauchy’s Integral Formula
    6. 14.6 Taylor Series
    7. 14.7 Laurent Expansions
    8. 14.8 Calculus of Residues
    9. 14.9 Multivalued Functions
    10. 14.10 Integral Representations for Special Functions
  21. About the Author