# Guide to Essential Math

## 1st Edition

### A Review for Physics, Chemistry and Engineering Students

**Authors:**Sy Blinder Sy Blinder

**eBook ISBN:**9780080559674

**Imprint:**Academic Press

**Published Date:**24th April 2008

**Page Count:**312

## 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. By the author's design, no problems are included in the text, to allow the students to focus on their science course assignments.

## Key Features

- Highly accessible presentation of fundamental mathematical techniques needed in science and engineering courses
- Use of proven pedagogical techniques develolped during the author’s 40 years of teaching experience
- illustrations and links to reference material on World-Wide-Web
- Coverage of fairly advanced topics, including vector and matrix algebra, partial differential equations, special functions and complex variables

## Readership

Upper-level undergraduates and graduate students in physics, chemistry and engineering

## Table of Contents

To the Student
1 Mathematical Thinking
1.1 The NCAA 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 p¼ in the Gaussian Integral
1.8 Function Equal to its Derivative
1.9 Log of N Factorial for Large N
1.10 Potential and Kinetic Energies
1.11 Lagrangian Mechanics
1.12 Riemann Zeta Function and Prime Numbers 1.13 How to Solve It
1.14 A Note on Mathematical Rigor

- Numbers 2.1 Integers 2.2 Primes 2.3 Divisibility 2.4 Fibonacci Numbers 2.5 Rational Numbers 2.6 Exponential Notation 2.7 Powers of 10 2.8 Binary Number System 2.9 Infinity 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 4 Trigonometry 4.1 What Use is Trigonometry? 4.2 The Pythagorean Theorem 4.3 ¼ in the Sky 4.4 Sine and Cosine 4.5 Tangent and Secant 4.6 Trigonometry in the Complex Plane 4.7 De Moivre's Theorem 4.8 Euler's Theorem 4.9 Hyperbolic Functions 5 Analytic Geometry 5.1 Functions and Graphs 5.2 Linear Functions 5.3 Conic Sections 5.4 Conic Sections in Polar Coordinates 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 7 Series and Integrals 7.1 Some Elementary Series 7.2 Power Series 7.3 Convergence of Series 7.4 Taylor Series 7.5 L'H'opital's Rule 7.6 Fourier Series 7.7 Dirac Deltafunction 7.8 Fourier Integrals 7.9 Generalized Fourier Expansions 7.10 Asymptotic Series 8 Differential Equations 8.1 First-Order Differential Equations 8.2 AC Circuits 8.3 Second-Order Differential Equations 8.4 Some Examples from Physics 8.5 Boundary Conditions 8.6 Series Solutions 8.7 Bessel Functions 8.8 Second Solution 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 Eigenvalue Problems 9.9 Group Theory 9.10 Minkowski Spacetime 10 Multivariable Calculus 10.1 Partial Derivatives 10.2 Multiple Integration 10.3 Polar Coordinates 10.4 Cylindrical Coordinates 10.5 Spherical Polar Coordinates 10.6 Differential Expressions 10.7 Line Integrals 10.8 Green's Theorem 11 Vector Analysis 11.1 Scalars and Vectors 11.2 Scalar or Dot Product 11.3 Vector or Cross Product 11.4 Triple Products of Vectors 11.5 Vector Velocity and Acceleration 11.6 Circular Motion 11.7 Angular Momentum 11.8 Gradient of a Scalar Field 11.9 Divergence of a Vector Field 11.10 Curl of a Vector Field 11.11 Maxwell's Equations 11.12 Covariant Electrodynamics 11.13 Curvilinear Coordinates 11.14 Vector Identities 12 Special Functions 12.1 Partial Differential Equations 12.2 Separation of Variables 12.3 Special Functions 12.4 Leibniz's Formula 12.5 Vibration of a Circular Membrane 12.6 Bessel Functions 12.7 Laplace Equation in Spherical Coordinates 12.8 Legendre Polynomials 12.9 Spherical Harmonics 12.10 Spherical Bessel Functions 12.11 Hermite Polynomials 12.12 Laguerre Polynomials 13 Complex Variables 13.1 Analytic Functions 13.2 Derivative of an Analytic Function 13.3 Contour Integrals 13.4 Cauchy's Theorem 13.5 Cauchy's Integral Formula 13.6 Taylor Series 13.7 Laurent Expansions 13.8 Calculus of Residues 13.9 Multivalued Functions 13.10 Integral Representations for Special Functions

## Details

- No. of pages:
- 312

- Language:
- English

- Copyright:
- © Academic Press 2008

- Published:
- 24th April 2008

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080559674

## About the Authors

### Sy Blinder

Professor Blinder is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor and a senior scientist with Wolfram Research Inc., Champaign, IL.. After receiving his A.B. in Physics and Chemistry from Cornell University, he went on to receive an A. M in Physics, and a Ph. D. in Chemical Physics from Harvard University under Professors W. E. Moffitt and J. H. Van Vleck. He has held positions at Johns Hopkins University, Carnegie-Mellon University, Harvard University, University College London, Centre de Méchanique Ondulatoire Appliquée in Paris, the Mathematical Institute in Oxford, and the University of Michigan. Prof Blinder has won multiple awards for his work, published 4 books, and over 100 journal articles. His research interests include Theoretical Chemistry, Mathematical Physics, applications of quantum mechanics to atomic and molecular structure, theory and applications of Coulomb Propagators, structure and self-energy of the electron, supersymmetric quantum field theory, connections between general relativity and quantum mechanics.

### Affiliations and Expertise

Professor Emeritus of Chemistry and Physics at the University of Michigan, USA, and Senior Scientist with Wolfram Research, Illinois, USA

### Sy Blinder

Professor Blinder is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor and a senior scientist with Wolfram Research Inc., Champaign, IL.. After receiving his A.B. in Physics and Chemistry from Cornell University, he went on to receive an A. M in Physics, and a Ph. D. in Chemical Physics from Harvard University under Professors W. E. Moffitt and J. H. Van Vleck. He has held positions at Johns Hopkins University, Carnegie-Mellon University, Harvard University, University College London, Centre de Méchanique Ondulatoire Appliquée in Paris, the Mathematical Institute in Oxford, and the University of Michigan. Prof Blinder has won multiple awards for his work, published 4 books, and over 100 journal articles. His research interests include Theoretical Chemistry, Mathematical Physics, applications of quantum mechanics to atomic and molecular structure, theory and applications of Coulomb Propagators, structure and self-energy of the electron, supersymmetric quantum field theory, connections between general relativity and quantum mechanics.

### Affiliations and Expertise

Professor Emeritus of Chemistry and Physics at the University of Michigan, USA, and Senior Scientist with Wolfram Research, Illinois, USA