Mathematical Methods: For Students of Physics and Related Fields

Citation

Hassani, S. (2009). Mathematical Methods: For Students of Physics and Related Fields (2nd ed.). Springer.

Chapter Summary

The second edition runs to about 830 pages in 32 chapters. According to the publisher, it is organized into "convenient, shorter chapters" (most run between roughly 10 and 40 pages), and it adds an entirely new treatment of probability and new material on tensors and integral transforms relative to the first edition. The publisher's overview describes the order as reviews of vector algebra and of differential and integral calculus, then infinite series, vector analysis, complex algebra and analysis, and ordinary and partial differential equations, with numerical analysis, nonlinear dynamics and chaos, and the Dirac delta function as an introduction to modern topics.

Chapters 1–5: Coordinate Systems and Vectors; Differentiation; Integration: Formalism; Integration: Applications; Dirac Delta Function

The opening block (pp. 3–170) is the review of coordinates, vectors, differentiation and integration that the publisher mentions, split into a formal chapter and an applications chapter on integration. It closes with a chapter on the Dirac delta function; a reviewer of the first edition singled out the delta-function treatment, including its two- and three-dimensional forms, as a particular strength.

Chapters 6–8: Planar and Spatial Vectors; Finite-Dimensional Vector Spaces; Vectors in Relativity

An algebra-of-vectors block: vectors in the plane and in space, then finite-dimensional vector spaces, then vectors as used in relativity. The first-edition review noted that the author deliberately unifies finite-dimensional vector spaces with function-space ideas such as orthogonal polynomials, so that the notion of a function space with an orthogonal basis is familiar when Fourier series and special functions appear later.

Chapters 9–11: Infinite Series; Application of Common Series; Integrals and Series as Functions

Convergence of infinite series, the applications of common series, and the representation of functions by integrals and series.

Chapters 12–17: Vectors and Derivatives; Flux and Divergence; Line Integral and Curl; Applied Vector Analysis; Curvilinear Vector Analysis; Tensor Analysis

Vector calculus, followed by curvilinear coordinates and tensors. The first-edition review praised the use of Maxwell's equations to cap the presentation of vector calculus. Tensor analysis (pp. 439–474) is one of the topics the publisher lists as expanded in the second edition.

Chapters 18–21: Complex Arithmetic; Complex Derivative and Integral; Complex Series; Calculus of Residues

Complex analysis from arithmetic through differentiation and integration, series and residues. It begins on page 477, well into the book.

Chapters 22–28: From PDEs to ODEs; First-Order Differential Equations; Second-Order Linear Differential Equations; Laplace's Equation: Cartesian Coordinates; Laplace's Equation: Spherical Coordinates; Laplace's Equation: Cylindrical Coordinates; Other PDEs of Mathematical Physics

Differential equations. The chapter titles show the sequence: reduction of partial to ordinary equations, first- and second-order ordinary equations, and Laplace's equation in three coordinate systems, followed by other equations of mathematical physics.

Chapters 29–32: Integral Transforms; Calculus of Variations; Nonlinear Dynamics and Chaos; Probability Theory

Four closing chapters, each a separate topic: integral transforms, the calculus of variations, an introduction to nonlinear dynamics and chaos, and probability theory, the last the new section of this edition. A bibliography follows.

Key Concepts

Chapters 1–8: Coordinates, Calculus, Delta Function, Vectors

• Dirac delta function: a distribution that picks out the value of a function at a point, treated in its own chapter.
• Finite-dimensional vector space: the setting for linear algebra and for later function-space ideas.
• Vectors in relativity: the use of vectors for spacetime quantities.

Chapters 9–21: Series, Vector Analysis, Tensors, Complex Analysis

• Infinite series: sums with infinitely many terms, with their convergence and applications.
• Flux, divergence, line integral and curl: the central quantities of vector analysis.
• Tensor analysis: quantities with several indices, extended in this edition.
• Calculus of residues: the complex-analysis method for evaluating integrals.

Chapters 22–32: Differential Equations and Further Topics

• Laplace's equation in different coordinates: separate chapters for Cartesian, spherical and cylindrical coordinates.
• Integral transforms and the calculus of variations: methods developed in Chapters 29–30.
• Nonlinear dynamics and chaos: a closing introduction to modern topics.
• Probability theory: the new section of the second edition.

Critical Analysis

Strengths

• Examples tied to physics. The publisher describes "many original, lucid, and relevant examples from the physical sciences", problems at the end of chapters and boxes that emphasize important concepts. Albano, reviewing the first edition in Physics Today (June 2001), found that each chapter opens with a preamble that places it in context, that physical examples motivate definitions, and that the book carries many notes on people and ideas.
• Broader coverage than the first edition. The first-edition review listed probability, tensors and integral transforms among the topics it missed; the publisher says the second edition adds probability and new material on tensors and integral transforms, and the contents now include chapters on tensors, integral transforms, the calculus of variations and probability theory.
• Modern topics. The delta function, nonlinear dynamics and chaos, and numerical analysis are named by the publisher as an introduction to modern mathematical physics.
• Shorter chapters. The reorganization into chapters of roughly 10 to 40 pages makes it easier to assign specific topics.

Limitations

• Review chapters up front. Albano criticized the first edition's opening chapters as review material that students should already know, and said some of the author's explanations of misconceptions led to more confusion. The second edition still begins with reviews, according to the publisher, but whether the specific passages were revised is not stated.
• Style points from the first-edition review. The reviewer also found too many footnotes and acronyms and some unwarranted generalizations; these remarks apply to the first edition and cannot be checked against the second from the sources available.
• Complex analysis comes late. Complex arithmetic does not appear until Chapter 18, at about the midpoint of the book, so a course that needs complex numbers early has to reorder the chapters. A Zentralblatt MATH review of the previous edition, quoted by the publisher, remarked on the same late arrival.
• Computing and numerical work. Albano missed any discussion of computers for symbolic or numerical work in the first edition. The publisher's overview mentions numerical analysis for the second, but none of the 32 chapter titles names it, so any treatment sits inside other chapters.

Where It Fits in the Course

Hassani is a mid-level undergraduate methods text, and it competes most directly with Boas's Mathematical Methods in the Physical Sciences. Albano's review judged the first edition most nearly comparable to Boas in target audience, pace and field coverage. The two differ in ordering. Boas, whose stated aim is to build basic competence in each area in a short time, puts complex numbers in its second chapter and linear algebra in its third, whereas Hassani delays complex numbers to Chapter 18 and places vector spaces in Chapters 6–7. Both books include tensors, the calculus of variations and probability, and Boas gives Fourier series and transforms and special functions their own chapters.

For preparation, Riley and Hobson's Foundation Mathematics for the Physical Sciences starts from school mathematics, and Schey's Div, Grad, Curl, and All That is an informal route into vector calculus that helps with Chapters 12–17. For depth beyond Hassani, Arfken, Weber and Harris's Mathematical Methods for Physicists is the graduate-level reference, and Strogatz's Nonlinear Dynamics and Chaos develops the subject of Chapter 31 at full length.

How to Use This Book

The book is written for sophomores and juniors in physics, engineering and related fields who have finished the introductory physics courses. Calculus is assumed, but Chapters 1–5 revisit it.

For a one-term methods course, a workable selection is Chapters 9–11 (series), 12–16 (vector analysis), 18–21 (complex analysis) and 22–28 (differential equations), consulting Chapters 1–5 only as needed. Chapters 6–7 (vector spaces) are worth reading early: the first-edition review notes that the author builds the idea of a function space with an orthogonal basis from them. Tensors (Chapter 17), integral transforms, the calculus of variations, chaos and probability are good second-term or self-study material. Use the end-of-chapter problems, and if complex numbers are needed earlier, read Chapter 18 first.