Foundation Mathematics for the Physical Sciences

Citation

Riley, K. F., & Hobson, M. P. (2011). Foundation Mathematics for the Physical Sciences. Cambridge University Press.

Chapter Summary

The book has fifteen chapters and six short appendices. Each chapter ends with a summary of its main results and a set of problems with hints and answers, and each section ends with a few short exercises. The preface describes the book as an alternative edition of the authors' longer Mathematical Methods for Physics and Engineering: it keeps the earlier, tool-building part of that book and leaves out the topics that are truly "methods".

Chapters 1–2: Arithmetic and Geometry; Preliminary Algebra

The opening chapters cover material many readers will have met at school: powers, exponential and logarithmic functions, physical dimensions, the binomial expansion, trigonometric identities and inequalities, followed by polynomial equations, coordinate geometry, partial fractions and some methods of proof. The preface explains that this introductory material was added to match the starting level of the book's readers.

Chapters 3–4: Differential Calculus; Integral Calculus

Chapter 3 treats differentiation, Leibnitz's theorem, special points and curvature of a function, theorems of differentiation and graph sketching. Chapter 4 covers integration methods, integration by parts, reduction formulae, improper integrals, plane polar coordinates, integral inequalities and applications of integration.

Chapters 5–6: Complex Numbers and Hyperbolic Functions; Series and Limits

Chapter 5 motivates complex numbers, develops their algebra and polar form, De Moivre's theorem and complex logarithms and powers, and ends with hyperbolic functions. Chapter 6 deals with series: summation, convergence tests, operations with series, power series, Taylor series and the evaluation of limits.

Chapters 7–8: Partial Differentiation; Multiple Integrals

Chapter 7 moves to functions of several variables: the total differential, exact and inexact differentials, the chain rule, change of variables, stationary values with and without constraints, envelopes, thermodynamic relations and differentiation of integrals. Chapter 8 introduces double integrals, their applications and changes of variables.

Chapters 9–10: Vector Algebra; Matrices and Vector Spaces

Chapter 9 covers scalar and vector products, triple products, lines, planes and spheres, distances and reciprocal vectors. Chapter 10 is the longest in the book and runs from vector spaces and linear operators through determinants, inverses, rank and simultaneous equations to eigenvalues, diagonalisation, quadratic and Hermitian forms and the summation convention.

Chapters 11–12: Vector Calculus; Line, Surface and Volume Integrals

Chapter 11 develops differentiation of vectors, scalar and vector fields, vector operators, cylindrical and spherical polar coordinates and general curvilinear coordinates. Chapter 12 turns to line, surface and volume integrals, Green's theorem in a plane, conservative fields and potentials, and the divergence and Stokes' theorems.

Chapters 13–15: Laplace Transforms; Ordinary Differential Equations; Elementary Probability

Chapter 13 introduces Laplace transforms, the Dirac delta function and the Heaviside step function. Chapter 14 handles first-order and higher-order linear ordinary differential equations and linear recurrence relations. Chapter 15 gives an elementary treatment of probability, from Venn diagrams and permutations to common discrete and continuous distributions and joint distributions.

Appendices A–F

The appendices hold the more formal developments (the base of natural logarithms, sinusoidal definitions, Leibnitz's theorem, the summation convention), a table of physical constants, and the answers to the footnote questions described below.

Key Concepts

Chapters 1–6: Algebra, calculus, complex numbers and series

• Physical dimensions: a section of Chapter 1 devoted to the dimensions of physical quantities.
• Integration methods: substitution, parts and reduction formulae, treated in Chapter 4.
• Complex numbers: the polar representation and De Moivre's theorem of Chapter 5.
• Taylor series and limits: the power-series material of Chapter 6.

Chapters 7–12: Several variables, matrices and vector calculus

• Exact and inexact differentials: the language of Chapter 7, which includes a section on thermodynamic relations.
• Eigenvectors and eigenvalues: Chapter 10 and the diagonalisation that follows from them.
• Vector operators: gradient, divergence and curl as defined in Chapter 11.
• Divergence and Stokes' theorems: the integral theorems that close Chapter 12.

Chapters 13–15: Transforms, differential equations and probability

• Laplace transforms: a method for turning differential equations into algebra.
• Linear ODEs with constant coefficients: the core of Chapter 14.
• Random variables and distributions: the framework of Chapter 15.

Critical Analysis

Strengths

• Explanation-led style. The preface says the authors moved the text towards more explanation in words, using a qualitative, physically based introduction to each topic, then a formal treatment, then one or two fully worked examples.
• Footnotes that ask questions. Roughly half of the more than 300 footnotes pose a question or an instruction to the reader, with specimen answers in Appendix F. The authors present this as a substitute for the quick checking questions of a tutorial.
• Layered practice. Short exercises follow each section (more than 300 in total), a summary closes each chapter, and more demanding problems (more than 450) come at the end of each chapter. Abbreviated solutions to odd-numbered problems are printed in the book, and a separate Student Solution Manual gives full solutions to the same problems.
• Gentle on-ramp. Because the first two chapters revisit powers, logarithms, dimensions and trigonometry, a reader with a weak school background has a place to start.

Limitations

• Deliberately incomplete. The preface says the book omits the topics that truly are methods. The table of contents has no chapters on Fourier series, partial differential equations or special functions, so a physics course will need a second mathematics text.
• Long for an "introduction". The index begins on page 706, and the book is organised so that a single course will not cover all fifteen chapters.
• Summaries are reminders, not teaching. The preface says the chapter summaries cannot state every condition attached to a result, so they should be used as a check-list and formula source after reading the main text.

Where It Fits in the Course

This is the starting point of the mathematical methods list. Boas's Mathematical Methods in the Physical Sciences is described on this site as a text for students who want to use a method before meeting its full theory, covering series, complex numbers, linear algebra, multivariable and vector calculus, Fourier methods, differential equations, special functions and tensors. Riley and Hobson stop earlier and spend more space on school-level foundations and on explanation, so the usual order is this book first, then Boas or Hassani's Mathematical Methods, which motivates each technique with a physical problem.

Schey's Div, Grad, Curl, and All That covers the same territory as Chapters 11–12 on its own, in a short and informal way built around electrostatics, and works well beside those chapters. Arfken, Weber and Harris is a reference and graduate-level text for after the undergraduate sequence, and Strogatz's Nonlinear Dynamics and Chaos is a separate subject text rather than a general toolkit.

How to Use This Book

The book suits early undergraduate students of physics and related sciences who want mathematics explained at length, and self-studying readers who need to refresh school mathematics before a methods course. The preface presents it as a gentle introduction to university-level mathematics whose starting level is meant to match its readers.

A practical plan:

• Core: Chapters 1–4 and 6–7 for calculus fluency, Chapters 9–12 for vectors, matrices and vector calculus, and Chapter 14 for ordinary differential equations. Readers confident at school level can skim Chapters 1–2.
• As needed: Chapter 5 before any work with waves or oscillations; Chapter 8 before multi-variable physics; Chapter 13 when Laplace transforms appear in a course; Chapter 15 before a statistics or statistical physics course.
• Working method: answer the footnote questions as they arise, try the end-of-section exercises straight after each section, and keep the end-of-chapter problems for revision. Check odd-numbered answers against the book and use the Student Solution Manual for the worked versions.
• Appendices: read them when the main text refers to them, or defer them until the main chapters feel comfortable, as the preface allows.