Guide to Essential Math

Guide to Essential Math

A Review for Physics, Chemistry and Engineering Students

1st Edition - April 24, 2008

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  • Authors: Sy Blinder, Sy Blinder
  • eBook ISBN: 9780080559674

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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
    2. 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

Product details

  • No. of pages: 312
  • Language: English
  • Copyright: © Academic Press 2008
  • Published: April 24, 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

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