Guide to Essential Math, 2nd Edition
A Review for Physics, Chemistry and Engineering Students

Author : S  Blinder



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) that 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 second edition features new problems and illustrations and features expanded chapters on matrix algebra and differential equations.

Key Features

• 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

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

About the Author

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