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Mathematical Methods in the Physical Sciences

Mary L. Boas
John Wiley
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a very traditional text for a mathematical methods course. It focuses on problem solving, drill, and concepts, rather than detailed theory. The first edition was published in 1966 and this third edition in 2005. It is much more than a cookbook, as it does give plausibility or heuristic arguments for most of the techniques (but usually not formal proofs). One big strength of the book is that it has many real physics problems integrated into the narrative.

The book is carefully crafted for its target audience, which is sophomore physics majors, and it has the narrow purpose of teaching these students mathematical techniques that they can apply to physical problems. Although it can be used as a reference, it does not go very deep, and the intent is that the student will actually learn everything in here during a course. The exercises are nearly all drill, with some “word problems” that require a little deeper understanding of the physics involved. It is not a mathematical physics book: it teaches you math, not physics; you are supposed to already know the physics (which comes mostly from mechanics and electricity).

There is an instructor companion web site that is minimal by today’s standards: a short list of errata and an instructor’s answer book. The author died in 2010 so it’s unlikely these will be updated.

Three somewhat similar books are: Harris’s Mathematics for Physical Science and Engineering, Kreyszig’s Advanced Engineering Mathematics, and Arfken et al.’s Mathematical Methods for Physicists. Harris’s book is an unusual one, as it concentrates on computer algebra systems, but it expects that you will refer back to it many times rather than try to absorb everything. The level of difficulty and coverage is about the same as the present book. Kreyszig and Arfken, although sold as texts, really work better as references where you look up the particular topic you need and learn it on the spot. They also go much deeper and have less hand-holding than the present book.

Bottom line: a good choice for a first methods course for physics majors. Serious students will want to follow this with specialized math courses in some of these topics.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

1. Infinite Series, Power Series.

The Geometric Series.

Definitions and Notation.

Applications of Series.

Convergent and Divergent Series.

Convergence Tests.

Convergence Tests for Series of Positive Terms.

Alternating Series.

Conditionally Convergent Series.

Useful Facts about Series.

Power Series; Interval of Convergence.

Theorems about Power Series.

Expanding Functions in Power Series.

Expansion Techniques.

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 Algebra.

Complex Infinite Series.

Complex Power Series; Disk of Convergence.

Elementary Functions of Complex Numbers.

Euler’s Formula.

Powers and Roots of Complex Numbers.

The Exponential and Trigonometric Functions.

Hyperbolic Functions.


Complex Roots and Powers.

Inverse Trigonometric and Hyperbolic Functions.

Some Applications.

3. Linear Algebra.


Matrices; Row Reduction.

Determinants; Cramer’s Rule.


Lines and Planes.

Matrix Operations.

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.

Total Differentials.

Approximations using Differentials.

Chain Rule.

Implicit Differentiation.

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.

Surface Integrals.

6. Vector Analysis.


Applications of Vector Multiplication.

Triple Products.

Differentiation of Vectors.


Directional Derivative; Gradient.

Some Other Expressions Involving V.

Line Integrals.

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.

Fourier Coefficients.

Complex Form of Fourier Series.

Other Intervals.

Even and Odd Functions.

An Application to Sound.

Parseval’s Theorem.

Fourier Transforms.

8. Ordinary Differential Equations.


Separable 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.

Isoperimetric Problems.

Variational Notation.

10. Tensor Analysis.


Cartesian Tensors.

Tensor Notation and Operations.

Inertia Tensor.

Kronecker Delta and Levi-Civita Symbol.

Pseudovectors and Pseudotensors.

More about Applications.

Curvilinear Coordinates.

Vector Operators.

Non-Cartesian Tensors.

11. Special Functions.


The Factorial Function.

Gamma Function; Recursion Relation.

The Gamma Function of Negative Numbers.

Formulas Involving Gamma Functions.

Beta Functions.

Beta Functions in Terms of Gamma Functions.

The Simple Pendulum.

The Error Function.

Asymptotic Series.

Stirling’s Formula.

Elliptic Integrals and Functions.

12. Legendre, Bessel, Hermite, and Laguerre functions.


Legendre’s Equation.

Leibniz’ Rule for Differentiating Products.

Rodrigues’ Formula.

Generating Function for Legendre Polynomials.

Complete Sets of Orthogonal Functions.

Orthogonality of Legendre Polynomials.

Normalization of Legendre Polynomials.

Legendre Series.

The Associated Legendre Polynomials.

Generalized Power Series or the Method of Frobenius.

Bessel’s Equation.

The Second Solutions of Bessel’s Equation.

Graphs and Zeros of Bessel Functions.

Recursion Relations.

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.

Poisson’s Equation.

Integral Transform Solutions of Partial Differential Equations.

14. Functions of a Complex Variable.


Analytic Functions.

Contour Integrals.

Laurent Series.

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.


Sample Space.

Probability Theorems.

Methods of Counting.

Random Variables.

Continuous Distributions.

Binomial Distribution.

The Normal or Gaussian Distribution.

The Poisson Distribution.

Statistics and Experimental Measurements.

Miscellaneous Problems.


Answers to Selected Problems.