Scientific Programming 19 (2011) 259–264 259DOI 10.3233/SPR-2011-0331IOS Press

Book Review

Adrian BrownSETI InstituteE-mail: abrown@seti.org

Abstract. Phillip M. Morse and Herman Feshbach, Professors of Physics at the MIT, published their biblical-sized textbook‘Methods of Theoretical Physics’ with McGraw-Hill in May 1953. At 1978 pages and published in two books, it is an intimidatingtwin tome that should still be atop the reading lists or the bookshelves of every mathematical physicist.

What material is covered in this book? In the most concise of terms, this book is devoted to the study of differential equationsand associated boundary conditions that describe physical fields. The thirteen chapters address what circumstances warrant theuse of which differential equations, and most often addresses the question of coordinate system transformations, for example,how do Green’s functions for Laplace’s Equation transform under different coordinate systems? Under what circumstances thesolutions can be expected to be separable? Many examples are covered to illustrate these points.

Why is this book relevant to Software Programmers? This book is part of the background that any scientific programmer islikely to need in dealing with physical fields. This book was written before personal computers became ubiquitous, however it isstill an outstanding effort to tie the methods of solving differential equations governing fields together in one book.

The book never received a second edition, however, it was reprinted to an outstanding standard by Feshbach Publishing since2004, run by the children of Herman Feshbach. Their website is feshbachpublishing.com.

The majority of this review is a mini-commentary of the book showing what is covered in a very terse fashion, which may beuseful as a summary even for those who have already read the full text. I then give a brief analysis of the approach to mathematicalphysics taken by the book. Finally, I will discuss who will benefit from reading this magnificent treatise, nearly 60 years after itwas first published.

Methods of Theoretical Physics, by Philip McCordMorse and Herman Feshbach, ISBN 0976202123.

1. Outline of the book

One approach of classifying the different sections ofthis text is to run through the table of contents. How-ever, reading through the book, I had the strong sensethat the consumable unit for Morse and Feshbach is notthe Chapter, but is the second subunit, by which I mean‘Section 1.2’ or ‘Section 3.4’. These are logical separa-tion points within the book that contain the key ideas,and are often written as standalone essays one couldextract from the book whole. For that reason, I willconcentrate on the book outline at the subunit level.

Chapter 1 is an interesting chapter. I do not considerit to be part of the core of the book, and as an introduc-tion it is extremely difficult for less experienced phys-ical mathematicians to access. Reading through laterchapters, which often do have excellent introductions,I often thought to myself that Chapter 1 was erected byMorse and Feshbach as a kind of warning about whatmight lie ahead in the book. ‘Beware, you who have

just wandering over to peek in this book – it is not forthe faint of heart’.

Section 1.1 deals with differential equations scalarfields and introduces the Laplacian. Section 1.2 intro-duces PDE’s of Vector fields, and introduces lines offlow, surface and line integrals. Singularities (a bigtopic in later chapters) are mentioned. It is in Sec-tion 1.3 that the reader is given curvilinear coordinatesto tackle, as we encounter scale factors, curvature, vol-ume element scaling, rotation and transformation ofvectors. Contravariant and Covariant vectors are in-troduced. This material is then re-used throughout thebook. Section 1.4 is a fairly standard introduction tothe ∇ operator, covering gradient, divergence, Gauss’Theorem and introduces Poisson’s Equation, Curl andVorticity, Stokes Theorem. It is at this point that thebook steps up several levels somewhat unnecessarily.Section 1.5 covers covariant and contravariant vectors,axial vectors, Christoffel symbols, Covariant Deriva-tives, Divergence and Curl Tensor notation and otherselected differential operators, all in 10 pages. This isclearly no more than a reprinting of the equations – nomeaningful introductory remarks are given here. Sec-tion 1.6 is similar in style, covering Dyadics, stress-

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260 Book Review

strain relationships, complex quaternions, abstract vec-tor spaces operators in quantum theory, Hermitian op-erators, transformation of operators, quantum mechan-ical operators, and spin operators in 38 pages. This isprobably the most excruciating section of the book, andlacks development and cohesion. Particularly disap-pointing is the introduction to Abstract Vector Spaces,which are treated through the book, but poorly intro-duced here amongst other material with a lack of co-hesion. Section 1.7 covers the Lorentz Transform, fourvectors and spinors in a more cohesive, but still rathercursory fashion, in only 14 pages. Given the grab-bagof topics Morse and Feshbach threw into Sections 1.5and 1.6 I would advise the reader not to get disheart-ened when ploughing through this section.

Chapter 2 is entitled ‘Equations Governing Fields’.Section 2.1 is a far more reasonable introduction to thetopic of flexible string mathematics. One wonders howmuch the inventors of String Theory were inspired bythe well-explained introduction to harmonic motion,delta function on a string, wave energy and flow, waveimpedance (a big topic through the book due to theMoore’s interest in Acoustics and Sound), friction anddiffusion, and boundary conditions for forced motionand elasticity. This 26 page section is the first of theseparable essays in the book. Section 2.2 deals withtransverse and longitudinal waves in an elastic mediumin a similarly rewarding fashion. Section 2.3 is anotherfascinating essay on the Motion of Fluids, addressingthe continuity equation, stresses in incompressible flu-ids, Bernoulli’s equation and Mach-inspired sub andsupersonic flow with shock waves. Section 2.4 is anoutstanding essay on diffusive flow, introducing thediffusion equation, porosity, pressure, mean free-pathand scattering cross section, and has a large sectionon the diffusion of light, which will recur through-out the book. This section no doubt inspired MonteCarlo modelers for the rest of the 20th century. Sec-tion 2.5 deals with the Electromagnetic Field in partic-ular, Maxwell’s Equations, Lorentz Invariance, GaugeTransformations, moving charges, surfaces as bound-aries, and again the wave transmission and impedance.Section 2.6 is an essay on Quantum Mechanics, dis-cussing photons vs. waves, introducing the uncertaintyprinciple, Poisson’s Brackets, Fundamental Equationof Quantum Mechanics, transformation to momentumspace, Hamiltonian Function and Schroedinger Equa-tion, Harmonic Oscillator, and the introduction of timeinto the picture. The Klein Gordon and Dirac equationsare then derived.

Chapter 3 covers the basics of a method that featuresthroughout the book – the Variational Principle. Sec-

tion 3.1 is a little gem of an essay on the VariationalIntegral and Euler Equations. Section 3.2 beautifullyuncovers classical dynamics and Hamilton’s Principle,working up to examples of a charged particle in anEM field, then relativistic and dissipative systems. Sec-tion 3.3 covers scalar fields, deriving the Lagrangianand Hamiltonian for the flexible string, wave equa-tions, velocity potential, wave impedance, diffusion,Schroedinger and Klein Gordon equations. Section 3.4derives the Lagrangian density dyadic for isotropicelastic media, plane waves, and the EM Field. A veryuseful list of Variational equations discussed in Chap-ter 3 is provided for easy reference.

Chapter 4 is a fantastic introduction to ComplexAnalysis. Section 4.1 is an outstanding entree to con-tour integrals and Cauchy’s Theorem. Section 4.2follows this up with an amazing discussion of theCauchy–Riemann conditions, conformal representa-tions in the complex plane, Cauchy’s Principle andCauchy’s Integral Formula. The essay finishes with adiscussion of Hilbert transforms, impedance and Pois-son’s Formula. Section 4.3 covers isolated singular-ities, poles and multipoles and analytical regions ofthe Taylor and Laurent series, Liouville’s theorem andmeromorphic functions. It ends with an in-depth dis-cussion of analytic continuation, branch points, andmirror, contour and recurrence techniques for analyti-cal continuation. Section 4.4 extends the discussion tomultivalued functions and provides a worked examplealong with a rare use of a plot to show the results. Sec-tion 4.5 covers the calculus of residues to actually eval-uate integrals using complex analysis, and concludeswith a link from the complex plane to gamma and el-liptic functions. Section 4.6 discusses the method ofsteepest descent, particularly as used on asymptotic se-ries. A particularly useful worked example is coveredfor a class of series that link in with the gamma func-tion. Section 4.7 covers conformal mapping in detail,with the major result being the Schwarz–Christoffeltransformation of the inside of a polygon to the upperhalf of the w plane. Section 4.8 is an essay on Fourier,Laplace and Mellin transforms, seen through the prismof Lebesgue Integrals and including a good discussionof regions of analyticity.

Chapter 5 is entitled ‘Ordinary Differential Equa-tions’ and the emphasis here is upon the separation so-lution theme that is utilized throughout the last partof the book. Section 5.1 deals with separable coordi-nates in two dimensions including boundary conditionsfor Laplace, Wave and coordinate systems (rectangu-lar, parabolic, polar, elliptic). Scale factors and sep-

Book Review 261

aration constants are discussed. The third dimensionis then introduced, alongs with the Staeckel determi-nant, quadric surfaces and the range of coordinate sys-tems is expanded. The conditions under which solu-tions are separable is discussed in depth at this point.Section 5.2 discusses separable series solutions. TheWronskian is introduced, the matter of independenceof solutions is discussed before delving into series so-lutions around ordinary and singular points and multi-ple ordinary and singular points. The Hypergeometricequation and Gegenbauer functions act as a lead in forLegendre and Tschebyscheff (Chebyshev) functions.Asymptoptic Series, Continued Fractions and Math-ieu functions are discussed and the section ends with asummation of recursion formulas for the separable so-lutions presented. Section 5.3 discusses Integral Rep-resentations of ODEs, in particular the Euler transformof the Hypergeometric Function, and this leads intodiscussions of the contours for integral representationsof Gegenbauer/Legendre (inc. associated and secondkind) functions. The Laplace transform uses a differentkernel than the Euler transform and this is again used toexamine the Hypergeometric functions. Stokes’ Phe-nomenon (where one term of the solution regularly dis-appears over part of the contour) is discussed. Anotherconfluent hypergeometric function, the Bessel func-tion, is introduced, along with Hankel and Neumannfunctions. Approximations for large order and Mathieufunctions are discussed at the end of the section. A ta-ble and diagrams of separable coordinates systems inthree dimensions is also given.

Chapter 6 introduces treatment of boundary con-ditions and Eigenfunctions. Section 6.1 is an excel-lent discussion of the genera of differential equa-tions, Characteristic Curves and Dirichlet, Neumannand Cauchy conditions, and then Elliptic and Parabolicequations complete the section. Section 6.2 discussesdiscretization into difference equations on nets and theeffect on boundary conditions. Green’s Functions arethen introduced for the first time. Satisfactory bound-ary conditions for hyperbolic, elliptic and parabolicequations are summarized at the end of the section.Section 6.3 introduces Eigenfunctions and how theymay be utilized in order to satisfy competing bound-ary conditions along coordinate surfaces. The Sturm–Liouville Equation and its factorization into linearPDE’s is discussed, the variation principle and com-pleteness are mentioned. The Gibbs Phenomenon fordiscontinuous curves is discussed. Generating Func-tions are introduced and applied to Legendre Polyno-mials. Eigenfunctions for PDEs are mentioned, along

with a discussion of the density of eigenfunctions.This is a very well written chapter, although the in-clusion of the section on Abstract Vector Spaces atthe end was perhaps questionable. A table discussesusing the Schmidt method to generate Eigenfunctionsfor the Sturm–Liouville Equation, generating Gegen-bauer (Legendre, Tschebyscheff), Laguerre and Her-mite Polynomials, their Generating Functions and re-currence formulas.

Chapter 7 is devoted to Green’s functions. Sec-tion 7.1 very nicely discusses generating Green’s func-tions for source points and boundary points, and the re-lation between volume and surface Green’s functions.Section 7.2 discusses Green’s Functions for steadywaves, covering the Helmholtz Equation, Inhomoge-nous Equations, and the effect of boundary conditions.The Method of Images is introduced and expansions ofthe Green’s function in Eigenfunctions and on the In-finite Domain and in Polar Coordinates are discussed.A very useful general technique for the production ofsteady-wave Green’s functions for the Sturm Liouvilleoperator (and hence steady state D.E.s in general) ispresented. Section 7.3 discusses the Green’s functionfor the Scalar Wave Equation. The very useful Reci-procity Relation G(r|r0) = G(r0 |r) is introduced anddiscussed. Two and One dimensional Green’s func-tions for the wave equation are given. The mathe-matical expression of Huygens’ Principle is demon-strated. Boundaries and Eigenfunction expansions arementioned and then the example of transient motionof a circular membrane is given. Finally, the Green’sfunction for the Klein–Gordon equation is addressed.Section 7.4 discusses Green’s functions for Diffusion,discussing causality and reciprocity, inhomogeneousboundary conditions, and infinite and finite boundaries.Section 7.5 addresses the Green’s function in AbstractOperator form, extending the applicability of the afore-mentioned equations to any linear equation in physics.To achieve this, adjoint operators in differential and in-tegral form are introduced. The chapter closes with de-velopment of the Green’s operator for adjoint, conju-gate and Hermitian and non-Hermitian (biorthogonal)cases. A table of Green’s functions is provided.

Chapter 8 is on Integral Equations (as opposed todifferential equations) to describe fields. Section 8.1opens by classifying the Integral Equations of transporttheory, acoustics and wave mechanics physics into ho-mogenous and inhomogenous Fredholm and Volterraequations of the first and second kind, with posi-tive/negative definite kernels. Section 8.2 discusses thegeneral properties of Integral Equations based on their

262 Book Review

kernel. Properties of positive definite, semi-definite, in-definite, non-real-non-definite and singular kernels arediscussed. Section 8.3 discusses the exact solutions ofthe Fredholm Equation of the first kind using seriessolutions and Schmidt orthogonalization. Biorthogo-nal, Gegenbauer Polynomials, and the Moment Prob-lem round out the section. Section 8.4 discusses exactsolutions for Integral Equations of the second kind –the same series are used as in Section 8.3 though themanipulation of the equation is different. Section 8.5explores Fourier Transforms, Hankel Transforms, anddisplaced kernels such as v(x − x0), Laplace Trans-forms and discusses the Method of Weiner and Hopfwhere Integral Equation boundaries are semi-infiniterather than infinite. It illustrates the method, and thenends with the Weiner–Hopf solution for the radiativetransfer in a homogenous medium with isotropic scat-tering (Milne) problem. A table is given at the endof the chapter summarizing all types of integral equa-tions.

Chapter 9 delves into approximate solutions (pri-marily perturbational and variational) for differentialequations. Section 9.1 is an excellent essay describ-ing the perturbational method, which is best used whena solution is sought that is close to a known exactsolution. The improvement from iterative–perturbativeto Feenberg to Fredholm to the Variation–Iterationto modified iterative–perturbative type schemes is il-lustrated through using a Mathieu function example.Finally the use of non-orthogonal basis functions isdiscussed. Section 9.2 discusses perturbations of theboundary conditions and the perturbation of the bound-ary shape, using successive approximations (includ-ing improved iteration–perturbation and Feenberg) andsecular determinant types (Fredholm and variation–perturbation). The scalar Helmholtz equation is used asan exemplar. Section 9.3 discusses perturbation meth-ods for scattering and diffraction. This excellent es-say discusses a first improvement to the Born approx-imation (Kirchoff approximation) which is the equiv-alent to the first iteration–perturbation solution, andthen higher order Born approximations. It then dis-cusses Fredholm series improvements, and long wave-length approximations. The short wavelength WKBJapproximation is then discussed in one and three di-mensions for potential barrier and bound systems. Sec-tion 9.4 is a long (52 page) essay on Variational meth-ods (suitable for larger variations than perturbative the-ory). This method inserts trial functions into a La-grangian that satisfy the boundary conditions and thenattempts to constrain their coefficients. An example of

a circular membrane is used and phase shifts and non-linear coefficients, used on potential barrier and surfaceperturbation and radiation problems are featured. TheVariation–Iteration method is then re-introduced to en-able an estimation of the accuracy of the results, thecircular membrane is used as an example. The sectionends with a discussion of the Method of Minimized It-erations. A useful summary table of approximate meth-ods is given at the end of the chapter.

Chapter 10 moves into a different mode that willbe followed in the remainder of the book. The gen-eral study of fields and their behavior is over, from thispoint forward, the techniques introduced are applied tospecific physical problems.

Chapter 10 is intended to cover solutions of La-place’s and Poisson’s Equations, ∇2ψ = 0 and ∇2ψ =−4πρ. Section 10.1 discusses solutions in 2 dimen-sions, first in Cartesian coordinates (prism heated ona side), and then polar coordinates (flow of viscousliquids, internal heating of cylinders, potential near aslotted cylinder) then elliptic coordinates (viscous flowthrough a slit, elliptic cylinders in uniform fields, po-tential inside cylinder with a narrow slot), paraboliccoordinates, bipolar coordinates (two cylinders in auniform field). In each case the Green’s Function is pre-sented. Section 10.2 discusses complex variables andthe 2D Laplace Equation, which is a special case thatallows use of a powerful technique utilizing the com-plex variable and analytic functions. Examples of liftand circulation during non-viscous flow, distributionsof line sources, grids, and linear arrays, periodic dis-tributions of images, potentials around prisms, paral-lel plate and variable condensers and other rectangu-lar shapes are tackled. Section 10.3 discusses Poissonand Laplace equation solutions for three dimensions,covering rectangular coordinates, circular cylindricalcoordinates, spherical coordinates (field of chargeddisks, currents in wire loops, charged spherical caps,dipoles, quadrupoles and multipoles, spherical shellwith a hole), prolate cylindrical coordinates (oblatespheroids, flow through an orifice). In each of thesecases integral representation and Green’s functions arepresented. Parabolic, bispherical, toroidal, ellipsoidalcoordinates are also discussed at the end of the chap-ter. A table of Trigonometric, Hyperbolic, Bessel andLegendre Functions is also given.

Chapter 11 introduces time by examining the WaveEquation. Section 11.1 is on the Wave Equation inone dimension – first looking at Fourier and LaplaceTransform techniques and then covering several ex-amples with Green’s functions (string with friction,

Book Review 263

string with elastic support and non-rigid supports, re-flection from a frictional support, sound waves in atube, tube with varying cross-section, acoustic circuitelements, free waves and moveable supports). Sec-tion 11.2 moves to waves in two dimensions, startingwith a great discussion on the link between Green’sfunctions and Fourier transforms, then rectangular co-ordinates (variable boundary admittance), polar coor-dinates (waves in a circular boundary, radiation fromcircular boundary, scattering of a plane wave from acylinder and a knife edge, Fresnel diffraction from aknife edge and from a cylinder with a slit), paraboliccoordinates (waves outside parabolic boundaries), el-liptical coordinates (waves inside an elliptic bound-ary, radiation from a vibrating strip, radiation from acurrent-carrying strip, scattering of waves from strips,diffraction through a slit – Babinet’s principle) are dis-cussed. Section 11.3 moves to into a discussion of threedimensional scenarios. Starting with rectangular coor-dinates (distortion of standing wave by strip, trans-mission through ducts, acoustic transients in a rect-angular duct, constriction of a duct, wave transmis-sion around corner, membrane in circular tube, ra-diation from tube termination, transmission in elastictubes) spherical coordinates and the radial functions(waves inside a sphere, vibrations in a hollow, flexi-ble sphere, vibrating string in a sphere, radiation froma sphere, radiation from a piston in a sphere, scat-tering of a plane wave from sphere, scattering froma Helmholtz resonator, scattering from an ensembleof scatterers, scattering of sound from air bubbles)spheroidal and oblate spheroidal coordinates are allmentioned. Section 11.4 addresses Integral and Varia-tional Techniques. Examples include iris diaphragm inpipe, hole in an infinite plane, reflection in a lined duct,radiation from end of circular pipe and the Fouriertransform is used to examine radiation from vibratingsource. The Variational principle is then used to exam-ine the scattering of waves and scattering from a strip).Tables of Cylindrical Bessel Functions, Weber Func-tions, Mathieu Functions, Spherical Bessel Functionsand Spheroidal Functions and a short table of LaplaceFunctions of common functions are given.

Chapter 12 is a slightly different chapter – it tacklesdiffusion and quantum wave mechanics. Section 12.1addresses solutions of the diffusion equation (transientsurface heating of a slab, radiative heating, transientinternal heating of a slab). A long discussion of dif-fusion and absorption of particles in a nuclear reac-tor is given. Heating of a sphere completes the sec-tion. Section 12.2 examines the distribution functions

for Milne and Chandresekhar’s radiation problems, in-cluding uniform, forward scattering approximations,diffuse scattering and emission. The Variational Cal-culation of Density, loss of energy on collision, uni-form space density and Age Theory – all methods suit-able for Monte Carlo transport problems are discussed.Departing from classical physics, Section 12.3 delvesinto solutions of the Schroedinger equation (The Har-monic Oscillator, perturbation theory, bound and freestates, reflection and transmission, potential barriers,central force fields and angular momentum, inversecube force and Coulomb field, Rutherford Scattering,perturbations of degenerate systems, the Stark Effect,scattering form central fields, Ramsauer and other res-onance effects, slow incident particles, the Born ap-proximation and structure factors, Variation–Iterationmethod, Variation and Variation–Iteration Methods forScattering, two particles-one dimension, coupled har-monic oscillators, several particles, Inversion and Par-ity, Symmetrizing two-particle systems). Finally, thescattering of an Electron from Hydrogen Atom is dis-cussed (elastic or inelastic and the possible exchangeof the particles). A table of Jacobi functions and Semi-cylindrical Functions is given.

Chapter 13 discusses Vector Fields. Many of the ex-amples are extensions of those in Chapter 11. Sec-tion 13.1 discusses Vector Boundary Conditions, Ei-genfunctions and Green’s functions. The vector Helm-holtz equation is used to generate examples of thesefunctions (including the Green’s function which is ac-tually a Green’s dyadic). The same is done for the vec-tor Laplace equation. Section 13.2 looks at static andsteady-state solutions, first in two dimensions with po-lar coordinates, then circular cylindrical coordinatesand finally spherical coordinates. Viscous flow arounda sphere is examined, as well as elastic deformationof a sphere. Section 13.3 discusses vector wave solu-tions (reflection of plane waves from a plane, waves ina duct – 1. including generation of waves by a wire2. reflection of waves from the end of the tube 3. changeof duct size and reflection from a wire across the tube,then elastic waves along a bar, torsional forced mo-tions of a rod, nonstationary viscous flow in tube, elec-tromagnetic resonators including 1. driving currents,2. wave guides and 3. klystron cavities, then scatter-ing from cylinders, spherical waves, radiation fromdipoles and multipoles, standing waves in a spheri-cal enclosure, vibrations of an elastic sphere, radiationfrom a vibrating current distribution, radiation from ahalf-wave antenna and current loop, scattering from asphere, distortion of a field by small object).

264 Book Review

Many numerical tables for functions in different co-ordinate systems are given in the final appendices ofthe book. These will likely not be of interest to themodern reader.

2. Pedagogical approach

Morse and Feshbach wrote the longest book writ-ten to date on theoretical physics – it is a complex andall-encompassing tome. These days one is far morelikely to see shorter works that take on just part of thetask – books on Electromagnetic Theory, Vector Anal-ysis, Partial Differential Equations, Complex Analysisor Linear Analysis.

This book was written before personal computersbecame readily available to scientists, so it is amazingthat is has managed to maintain its relevance.

Morse and Feshbach write several times that theyare not interested in the physics of the problems theytackle – they are interested in the mathematical ap-proach to finding a solution. They stick to this mantrathroughout the book, and it is this discipline that allowsthe book to stay relevant. When one reaches for Morseand Feshbach, one knows what to expect.

The authors clearly agonized over what to leave inand out of the book. They said as much in the Preface:

A discussion of the physical concepts and exper-imental procedures in all the branches of physicswhich use fields for their description would itselfresult in an overlong shelf, duplicating the subjectmatter of many excellent texts, and by its prolixity,disguising the fundamental unity of the subject.

I think that is the essence of way Morse and Fesh-bach remains a crucial read – because of its breadth andscope, it is able to bring together almost all of the tech-niques for mathematical analysis of Fields in Physics,and demonstrate to the reader their interrelations. Forexample – how a two dimensional solution of the Pois-son equation can be extended to three dimensions andremain separable in certain coordinate systems.

The standard of the problems at the end of the chap-ters is extremely demanding – the problems will pro-vide plenty of work for the motivated student. No an-swers are provided to the problems.

Morse and Feshbach provide plenty of tables for ref-erence at the end of relevant chapters and at the endof the book. Most of these are unnecessary in today’scomputer-driven environment.

3. Who will this book appeal to and why?

This book is definitely not an introductory book, andwastes little time in exacting explanations of how tostep through the derivations presented. Presumably thiswas done because the perceived audience of the bookwere science or engineering professionals with strongmathematical backgrounds, and not college students.

It would be easy to list the number of physical fieldsthat have developed, matured, or simply appeared sinceMorse and Feshbach was published. The book containsthe fundamentals for Monte Carlo approaches and fi-nite difference and finite element theory, which havereceived significant attention with the advent of com-puting. Quantum Field Theory is of course absent, andgroup theory and chaos theory do not make an appear-ance.

Even so, in 2011, almost 60 years after first publi-cation, it is still hard to imagine a practicing appliedmathematician or applied physics professor who doesnot have a copy of this book nearby. The incrediblebreadth and sheer number of problems covered make itan invaluable resource that is still unmatched in scope.

Texts that are more suitable for the beginning stu-dent are now available (those by Haberman or Farlowfor example) however Morse and Feshbach providesolutions where others fear to tread. As I have notedabove, Morse and Feshbach sought to include enoughmathematical material to enable the reader to grasp theconnections between the disparate theoretical methodsof physics. No other book can claim that to this day.

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