Mathematical Methods in the Physical Sciences, 3rd Edition
July 2005, ©2006
Now in its third edition, Mathematical Concepts in the Physical Sciences, 3rd Edition provides a comprehensive introduction to the areas of mathematical physics. It combines all the essential math concepts into one compact, clearly written reference.This book is intended for students who have had a two-semester or three-semester introductory calculus course. Its purpose is to help students develop, in a short time, a basic competence in each of the many areas of mathematics needed in advanced courses in physics, chemistry, and engineering. Students are given sufficient depth to gain a solid foundation (this is not a recipe book). At the same time, they are not overwhelmed with detailed proofs that are more appropriate for students of mathematics. The emphasis is on mathematical methods rather than applications, but students are given some idea of how the methods will be used along with some simple applications.
The Geometric Series.
Definitions and Notation.
Applications of Series.
Convergent and Divergent Series.
Convergence Tests for Series of Positive Terms.
Conditionally Convergent Series.
Useful Facts about Series.
Power Series; Interval of Convergence.
Theorems about Power Series.
Expanding Functions in Power Series.
Accuracy of Series Approximations.
Some Uses of Series.
2. Complex Numbers.
Real and Imaginary Parts of a Complex Number.
The Complex Plane.
Terminology and Notation.
Complex Infinite Series.
Complex Power Series; Disk of Convergence.
Elementary Functions of Complex Numbers.
Powers and Roots of Complex Numbers.
The Exponential and Trigonometric Functions.
Complex Roots and Powers.
Inverse Trigonometric and Hyperbolic Functions.
3. Linear Algebra.
Matrices; Row Reduction.
Determinants; Cramer’s Rule.
Lines and Planes.
Linear Combinations, Functions, Operators.
Linear Dependence and Independence.
Special Matrices and Formulas.
Linear Vector Spaces.
Eigenvalues and Eigenvectors.
Applications of Diagonalization.
A Brief Introduction to Groups.
General Vector Spaces.
4. Partial Differentiation.
Introduction and Notation.
Power Series in Two Variables.
Approximations using Differentials.
More Chain Rule.
Maximum and Minimum Problems.
Constraints; Lagrange Multipliers.
Endpoint or Boundary Point Problems.
Change of Variables.
Differentiation of Integrals.
5. Multiple Integrals.
Double and Triple Integrals.
Applications of Integration.
Change of Variables in Integrals; Jacobians.
6. Vector Analysis.
Applications of Vector Multiplication.
Differentiation of Vectors.
Directional Derivative; Gradient.
Some Other Expressions Involving V.
Green’s Theorems in the Plane.
The Divergence and the Divergence Theorem.
The Curl and Stokes’ Theorem.
7. Fourier Series and Transforms.
Simple Harmonic Motion and Wave Motion; Periodic Functions.
Applications of Fourier Series.
Average Value of a Function.
Complex Form of Fourier Series.
Even and Odd Functions.
An Application to Sound.
8. Ordinary Differential Equations.
Linear First-Order Equations.
Other Methods for First-Order Equations.
Linear Equations (Zero Right-Hand Side).
Linear Equations (Nonzero Right-Hand Side).
Other Second-Order Equations.
The Laplace Transform.
Laplace Transform Solutions.
The Dirac Delta Function.
A Brief Introduction to Green’s Functions.
9. Calculus of Variations.
The Euler Equation.
Using the Euler Equation.
The Brachistochrone Problem; Cycloids.
Several Dependent Variables; Lagrange’s Equations.
10. Tensor Analysis.
Tensor Notation and Operations.
Kronecker Delta and Levi-Civita Symbol.
Pseudovectors and Pseudotensors.
More about Applications.
11. Special Functions.
The Factorial Function.
Gamma Function; Recursion Relation.
The Gamma Function of Negative Numbers.
Formulas Involving Gamma Functions.
Beta Functions in Terms of Gamma Functions.
The Simple Pendulum.
The Error Function.
Elliptic Integrals and Functions.
12. Legendre, Bessel, Hermite, and Laguerre functions.
Leibniz’ Rule for Differentiating Products.
Generating Function for Legendre Polynomials.
Complete Sets of Orthogonal Functions.
Orthogonality of Legendre Polynomials.
Normalization of Legendre Polynomials.
The Associated Legendre Polynomials.
Generalized Power Series or the Method of Frobenius.
The Second Solutions of Bessel’s Equation.
Graphs and Zeros of Bessel Functions.
Differential Equations with Bessel Function Solutions.
Other Kinds of Bessel Functions.
The Lengthening Pendulum.
Orthogonality of Bessel Functions.
Approximate Formulas of Bessel Functions.
Series Solutions; Fuch’s Theorem.
Hermite and Laguerre Functions; Ladder Operators.
13. Partial Differential Equations.
Laplace’s Equation; Steady-State Temperature.
The Diffusion of Heat Flow Equation; the Schrodinger Equation.
The Wave Equation; the Vibrating String.
Steady-State Temperature in a Cylinder.
Vibration of a Circular Membrane.
Steady-State Temperature in a Sphere.
Integral Transform Solutions of Partial Differential Equations.
14. Functions of a Complex Variable.
The Residue Theorem.
Methods of Finding Residues.
Evaluation of Definite Integrals.
The Point at Infinity; Residues of Infinity.
Some Applications of Conformal Mapping.
15. Probability and Statistics.
Methods of Counting.
The Normal or Gaussian Distribution.
The Poisson Distribution.
Statistics and Experimental Measurements.
Answers to Selected Problems.
• Ch. 3 also includes more detail on linear vector spaces. The discussion of basis function is continued in Ch. 7 (Fourier Series), Ch. 8 (Differential Equations), Ch. 12 (Series Solutions) , and Chapter 13 (Partial Differential Equations).
• Fourier integrals have been moved to Ch. 7 (Fourier Series). The Laplace transform and an expanded treatment of the Dirac delta function have been moved to Ch. 8 (Differential Equations).
• Throughout the book, the usefulness and also the pitfall of computer algebra systems are pointed out.
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