Harvey Greenspan's books

Harvey Greenspan wrote two books, one a research monograph and the other a calculus teaching text. The teaching text takes an "applied" approach and receives mixed reviews, but it ran to a second edition which we list as a separate book. We give below extracts from prefaces and reviews of these books.

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The Theory of Rotating Fluids (1968)

Calculus - An Introduction to Applied Mathematics (1973), with David J Benney

Calculus - An Introduction to Applied Mathematics (Second Edition) (1987), with David J Benney

1. The Theory of Rotating Fluids (1968), by H P Greenspan.
1.1. From the Preface.

The selection of subject matter and the manner of its presentation are not issues on which there can be exact agreement, much less so when the field in question is in the process of rapid and diverse growth. Perhaps, then, it would be of interest for me to describe the scope of this work, its rationale and some of the decisions and compromises that I have made or accepted.

It is my intention to provide a basic foundation for the support and promotion of research in rotating fluids. Because the subject has so many separate branches, I have tried to concentrate on those topics which I consider fundamental, of central importance to most, if not all, the areas of application. Practically speaking, this has been translated to mean the study of rotating fluids in quite ordinary circumstances, unembellished by very special and exotic effects - the motion of a contained, incompressible, viscous fluid such as water in a simple controlled environment. Furthermore, attention is focused almost exclusively upon primary phenomena, those that occur only in a rotating medium. To restrict the length of this monograph, I have also severely curtailed, and at times omitted, the discussion of material which receives extensive coverage in other books.

The amount of detail to be included in a theoretical development is a problem having no generally satisfactory solution. What is sufficient for one reader is inadequate or superfluous for another. My policy is as follows. If the material is new, the topic pregnant with possibility, or the methods basic, then the exposition is rather complete. On the other hand, I have not hesitated merely to highlight or to summarise, difficult, extended or inconclusive analyses, taking care to cite all the pertinent references for those interested. The intricate details of elaborate experimental and numerical programmes are also omitted from the text. However, it is strongly recommended that the serious student attempt. on his own, some of the experiments which can be performed with a modest outlay, and sufficient detail is provided for this purpose. These demonstrations really give the subject life and their role in developing intuition cannot be overestimated.

I have attempted to make as many sections as possible reasonably self-contained. The basic equations of motion are repeated often for this purpose as are certain important definitions and formulas. I regard this duplication of effort a minor penalty for the afforded convenience.

In all of this, there is the large undeniable element of personal preference. This is an author's prerogative even if it is, perhaps. a challenge to some to exercise their own literary judgments.

H. P. G.
Cambridge, Mass.

1.2. Review by: M James Lighthill.
Science, New Series 164 (3882) (1969), 938.

The body of science so built up, laying emphasis on comparing theory and experiment for such movements of rotating masses of homogeneous liquid as can he realised in the laboratory, but referring often to specific geophysical phenomena that may be in part illuminated by individual studies of this kind, is excellently described in this well-written book. On the even more complicated subject of rotating stratified fluids, the book includes only a small amount of material, but enough to make the reader quite clearly aware of the possibility that particular phenomena may be grossly altered by stratification and of the need for careful scrutiny of this aspect in geophysical applications.

The central phenomena described, each from several points of view, are Ekman layers, Taylor columns, spin-up, inertial waves, Rossby waves, forced motions (and decaying disturbances) within a uniformly rotating container, and motion within a precessing container. Most of these important but complex and to some extent "unexpected" phenomena are first introduced in chapter 1 through simple experiments and elementary theoretical ideas. Their nature in certain limiting cases that possess considerable practical significance, while allowing simplification of theory through linearisation, is described in the very long second chapter. The understanding so achieved is then used as a framework for explaining the meaning of various harder pieces of theory, valid in other (essentially nonlinear) cases, in chapter 3. Excitation of inertial waves (and of their limiting case, the Taylor columns) is the main theme of chapter 4. To balance the concentration on essentially laminar flow in most of the book, chapter 6 gives a good account of criteria for instability, including instability of Taylor vortices, Ekman layers, and vertical shear layers, as well as "baroclinic instability."

Discussions, in chapter 5 and elsewhere, concerning extensions of the book's central ideas into the enormous field of oceanography are for reasons of space somewhat slender. Stratification introduces more complexities even than those that are mentioned: it profoundly alters inertial waves; on a rotating spherical earth, it especially permits extra "baroclinic" modes, and does not bring about the "loss of effective stretching of vortex lines" that (in the context of flow in a cylinder) is noted on page 127. Again, "inertial boundary layers" are, possibly, of uncertain application to the depth-averaged equations for a real ocean, because the depth-averaged inertial terms are approximated in the theory in such a manner that their inaccuracy is both large and variable in form (depending on current distribution in depth).

The main section of the book, chapters 2 and 3 on contained rotating fluids, has many outstanding features. which include the making of several important distinctions. With the convention that the axis of rotation is vertical, containers whose curves of constant "height" (from top to bottom) are a set of closed contours behave in one way: "geostrophic" motions outside the Ekman boundary layers tend to be excited with flow mound those curves. When, however, those curves are not closed but terminate on a vertical boundary, it is waves of generalised Rossby type that are excited in forced motions or disturbance decay. A quite different degenerate case is that of a container of uniform height, which would permit 'any geostrophic motion. Then it is the mass balance of flow in and out of Ekman layers that forces the geostrophic motion to be irrotational, and a further crucial distinction then depends on whether the fluid region is or is not simply connected. In these degenerate cases vertical shear layers are often present. The clear account of this complex topic is one more excellent feature of this permanently important book.

1.3. Review by: Alan Faller.
Bulletin of the American Meteorological Society 50 (12) (1969), 990-991.

The theory of rotating fluids has blossomed over the past 15 or so years as a nearly separate branch of fluid dynamics. Greenspan's survey of the fundamental concepts and the current state of this field of knowledge is an important mile stone in the general development of this subject, one that marks a broader awareness of the special problems and the unique characteristics of rotating flows. Prior to 1950, and excepting direct geophysical studies, only a few individual scientists, among the foremost, G I Taylor, had devoted special effort to the study of rotating fluids per se. While meteorologists, oceanographers, and a few astrophysicists were concerned with the complexities of their respective rotating fluid problems, engineers with fluid dynamical interests considered rotating flows merely as extensions of non-rotating systems with secondary effects due to rotation. There was little communication between the two groups. By way of example, the pioneering work of Ekman in boundary layer studies as early as 1905 and well known to oceanographers and meteorologists is not recognised at all in Schlichting's "Boundary Layer Theory" whereas the rather similar boundary layers of von Karman and Bodewadt, published in 1921 and 1940, respectively, are discussed in detail. Thus the many unique aspects of fluid flows in rotating systems have not been generally recognised by hydrodynamicists. The Theory of Rotating Fluids will be an important bridge between the diverse fields of geophysics, engineering and applied mathematics.

There can be little adverse criticism of Greenspan's monograph either in its scope or in its manner of presentation. The limitations of the material are clearly outlined in the preface where the discussion is restricted to a "contained, incompressible, viscous fluid such as water in a simple controlled environment," and to "primary phenomena, those that occur only in a rotating medium." The text contains six chapters including a short but valuable introductory section. Chapter II includes the basic theories of rotating boundary layers, spin-up of a contained fluid, various oscillatory motions in cylindrical and spherical containers, and other linear theories. Under non linear theories in Chapter III emphasis is placed upon similarity theories such as the flow due to a rotating plate and extensions of this topic to the sphere and to turbulent flows. A brief discussion of the flow over obstacles and precessional flow in a sphere are included, and a perturbation expansion theory for the precessional flow is considered in detail. The theory of vortex flows is compressed to approximately three pages, somewhat surprisingly, but this abbreviation of a subject that has a long history of theoretical and experimental study is prefaced by remarks that, justifiably, cast considerable doubt upon the relevance of much of the earlier research. Chapter IV is primarily concerned with the excitation and propagation of inertial waves by various means, a subject where the special features of rotational effects are clearly evident. The depth-averaged equations considered in Chapter V emphasise the connections with theoretical physical oceanography and laboratory simulation of large-scale oceanic, and to some extent atmospheric, circulations. The stability of fluid flows in rotating systems is considered in Chapter VI but is limited to the Taylor cylinder experiment, the Ekman boundary layer, vertical shear layers, and a qualitative discussion of thermal convection in a differentially heated rotating annulus. An analysis of the stability of vertical shear layers, with and without depth variations is given, but it is unfortunate that the close connection of this topic to barotropic instability (as it is known to geophysicists) is not discussed. Basically the problem is one of inflectional instability where a depth variation coupled with rotation, analogous to the variation of the Coriolis parameter, may affect the stability criterion. The discussion of the annulus experiment is limited to qualitative remarks with comments about the inherent non-linearities of the problem and the difficulties of mathematical analysis. This section would have been an ideal setting for a basic treatment of the problem of baroclinic instability. Another conspicuous omission is the effect of rotation upon thermal convection, but this subject is extensively covered in other texts. In general rotating fluids with density stratification are sparsely treated and will be missed by those with primarily geophysical orientation. But as before the author is well aware of these limitations and gives suitable references to other texts and source material. The close connection of rotating fluid theory with its experimental background is emphasised by many excellent photographs of laboratory studies from various sources. These give aura of reality to the mathematical developments as well providing illustrative material for a clear discussion many of the problems

The Theory of Rotating Fluids will not answer all your problems connected with geophysical circulations, but it provide a valuable background and a coherent compilation of current theory for those primarily interested in dynamics. Much of the material originated from the consideration of geophysical problems, e.g., the Ekman boundary layer, Rossby waves, oceanic models, etc., so that this volume could well serve as a basic text for a course in geophysical fluid dynamics in preparation for advanced studies in dynamic meteorology or dynamic oceanography. The mathematics should be comprehendible to those who have studied vector analysis and advanced calculus, and who some background in fluid dynamics.

1.4. Review by: Richard Siegmund Lindzen.
The Journal of Geology 77 (2) (1969), 247.

This eminently readable volume concerns itself with the simplest, experimentally realisable, fluid problems for which rotation is of essential importance. Thus, the bulk of the book deals with the motions of contained, incompressible viscous fluids. Both linear and non-linear theory for such flows comprise the bulk of the volume. However, additional chapters deal with unbounded fluids, depth-averaged models (and models for oceanic circulation in particular), and the stability of rotating flows. The last item is treated in a rather sketchy fashion in contrast to the detailed presentations characterising most of the book.

Professor Greenspan's book can be recommended unreservedly to anyone with an interest in theoretical hydrodynamics. It is the only available book devoted to the interesting and burgeoning field of rotating fluids. The sections on contained flows may prove of value to those interested in the interior of the earth. However, the results are unlikely to prove immediately applicable since the important effects of stratification and electromagnetic processes have been ignored. Oceanographers, on the other hand, will find much of immediate interest.

1.5. Review by: Philip Gerald Drazin.
Science Progress (1933-) 57 (225) (1969), 116-117.

This is the first book specifically devoted to rotational flow. Until 30 years ago, rotational flows were allotted no more than a few pages in books on fluid dynamics. A look at the sixth and last edition of Lamb's Hydrodynamics the great classic of fluid dynamics, reveals the prevailing neglect of rotational flows in 1932. The fundamentals are there: Kelvin's theorem and the motion of vortex lines with the fluid. So are a few isolated solutions such as Gerstner's trochoidal waves and some exact solutions of the Navier-Stokes equations. There is also a short account of boundary layers. The equilibrium and stability of rotating masses of fluid stand apart in a short final chapter. The rule was to regard irrotational flow as the fundamental phenomenon, and rotational flow as the special case. Vorticity was manifest as isolated line vortices amid regions of irrotational flow; it appeared that the vorticity in Gerstner's trochoidal wave 'detracts somewhat from the physical interest of the results'; boundary layers were seen as the margins of regions of irrotational flow; special results on rotating fluid masses were found usually with the aid of a gravitational or velocity potential. The bulk of the theory of fluid dynamics rested firmly on the assumption that there is a velocity potential. This was in part necessary because of the mathematical difficulties in the absence of a velocity potential, in part supported by the successes of aerofoil theory and the theory of sound and water waves.

But before the last edition of Lamb there had been a growing appreciation of the importance of rotational flows. Understanding of the role of viscosity in the motion of air and water diminished the practical importance of the elegant mathematical theory of irrotational flow. Also the effects of rotation of the whole system in which a fluid moves were studied. The dynamics of the ocean and atmosphere was related to the vorticity imparted by the rotation of the earth. It was appreciated that the Coriolis force was dominant in the dynamics of current and wind systems of a large scale. The theory of geostrophic motion was developed, whereby winds blow along isobars, not perpendicular to them as one's intuition might at first suggest. Taylor demonstrated geostrophic flow in the laboratory, in accord with his and Proudman's theory. The extensive research on rotational flow, indicated in this paragraph, has increased rapidly since 1945. It has been already recognised by teachers and authors of books of fluid dynamics, who devote a diminishing proportion of their attention to irrotational flows. Indeed, of the 600 pages of Batchelor's An Introduction to Fluid Dynamics, published in 1967, only 130 are devoted to irrotational flow.

Now Professor Greenspan writes a research monograph specifically on those flows that are characteristic of rotation. He takes the special view of phenomena characteristic of rotation of the whole system in which the fluid moves rather than the more general one of rotational flow. He considers flows in which the Coriolis force dominates the dynamics by balancing with the inertia or viscous forces of the fluid. In general the book is a specialised one, using advanced mathematical techniques and elaborate notation suitable only for research workers in the field. The theory of basic phenomena and their relation to carefully controlled laboratory experiments is emphasised rather than applications to oceanography or meteorology.

The introductory chapter is less specialised, describing first some basic phenomena, and then the equations of motion and basic theory. It opens with accounts of three simple experiments, of a Taylor column, forced inertial oscillations, and spin-up. These experiments, and many later ones as well, are illustrated by striking photographs (which no doubt account for the high price of the book). Flows in rapidly rotating systems are indeed surprising and dramatic, as the author enthuses. A small obstacle can affect the flow at a great distance in a Taylor column parallel to the axis of rotation. The influence of an obstacle is carried by the fluid freely parallel to the axis but hardly at all perpendicular. Two-dimensional flows often result. Forced oscillations of a rapidly-rotating fluid illustrate this idea of domain of influence, with phenomena 'reminiscent of Mach cones in compressible aerodynamics'. Spin-up, which occurs when the rotation of a system is suddenly changed, is used to illustrate the important role that modern boundary-layer theory plays for rotating flows.

Later chapters give detailed accounts of 'Contained rotating fluid motion', 'Motion in an unbounded rotating fluid', 'Depth-averaged equations: models for oceanic circulation' and 'Stability'. Those are at a high professional level for the specialist, not for the general reader in search of a quick explanation of the jet stream, the Gulf Stream or the general circulation of the atmosphere. But scientists in other fields can enjoyably savour the essence of rotating fluids by reading the first few pages. The basic experiments are clearly presented as simple, but dramatic and provocative.

1.6. Review by: Raymond Hide.
Quarterly of Applied Mathematics 27 (4) (1970), 557-558.

Laboratory experiments and experience with large-scale flow in the oceans and atmosphere indicate that slow hydrodynamical motions in a rapidly rotating fluid are such that Coriolis forces approximately balance pressure forces nearly everywhere; when the balance is perfect the flow is said to be "geostrophic" (a term coined by Napier Shaw in 1916. "Geostrophic motion" satisfies the meteorologist's "thermal wind" equation when the fluid is baroclinic, which reduces to the celebrated Proudman-Taylor theorem when the fluid is barotropic (i.e. when the density is uniform for a liquid, or dependent only on pressure for a gas). Unfortunately, the equations of geostrophic motion are mathematically degenerate and ageostrophic effects must, therefore, be taken into account in any acceptable theoretical procedure.

Four types of ageostrophic effect occur in general, associated respectively with time-variations in flow pattern, with the acceleration of fluid particles relative to the rotating frame, with viscous friction, and with magneto-hydrodynamic processes if we include electrically-conducting fluids, such as the Earth's core. The first step in any theoretical study is usually the ranking of ageostrophic effects in order of importance. If either friction or time-variations in the flow pattern predominate then the mathematical problem can usually be linearised and treated analytically. More commonly, however, particle accelerations or magnetohydrodynamic effects predominate and the resulting mathematical equations are non-linear and generally intractable, except by numerical methods.

Several years ago, largely through pioneering studies by meteorologists and oceanographers, the number and variety of available solutions to "linear problems" in the hydrodynamics of rotating fluids reached the level at which it became possible to offer a course in the subject that would satisfy graduate students in applied mathematics, and the most successful part of Professor Greenspan's book is the outcome of such a course. The book attempts to give a comprehensive account of the theory of the hydrodynamics of rotating fluids (excluding magnetohydrodynamic effects).

In the first chapter (27 pages), a few simple laboratory demonstrations are described, the equations of motion are presented and the rudiments of vorticity theory and boundary-layer theory are outlined. In chapter 2, the theory of the Ekmen boundary layer is treated at some length. As meteorologists were probably the first to discover, Ekman-layer suction is the dominant viscous process in the vorticity-balance equation, a process brilliantly analysed by Professor Greenspan under the title of "spin-up". According to the theory, the typical interval of time required for friction to bring about substantial changes in vorticity in a barotropic fluid is the geometric mean of the viscous diffusion time (typically several months at least for the Earth's atmosphere) and the period of rotation. "Spin-up" is a recurring theme throughout the book.

A rotating barotropic fluid is capable of supporting inertial oscillations with frequencies less than twice the angular speed of basic rotation. "Rossby-Haurwitz" waves (Laplace's second-class tidal oscillations) comprise that class of inertial oscillation with which the meteorologist and oceanographer are particularly familiar. These waves are but one phenomenon due essentially to the spatial variation of the axial distance between the upper and lower surfaces of the fluid, western boundary currents (e.g. the Gulf Stream) and Taylor columns being others. Linear theories of these phenomena are treated in detail in chapter 2.

Chapters 3 (52 pages), 4 (40 pages) and 5 (46 pages) include further discussions of some of the phenomena. introduced in chapter 2, taking into account, where possible, non-linear effects and the absence of bounding surfaces. In chapter 6 (29 pages), experimental and theoretical investigations of the various barotropic and baroclinic flows are reviewed, with the greatest emphasis on the onset of instabilities in Ekman layers. Baroclinc instability, the process responsible for the conversion of potential energy due to solar heating of the atmosphere into the kinetic energy of large-scale motions, is treated in outline. Bénard convection in a rotating fluid is not included, presumably because it has been thoroughly treated elsewhere (by Chandrasekhar) fairly recently.

The book carries a detailed notation guide, a bibliography and author index and a subject index, and is fairly lavishly illustrated, which may account for its high price. This reviewer noticed only one or two minor errors. The book will constitute a useful addition to the libraries of applied mathematicians and fluid dynamicists and of mathematically inclined astronomers, geophysicists, meteorologists, oceanographers and engineers.

1.7. Review by: V Barcilov.
Journal of Fluid Mechanics 52 (4) (1972), 794-795.

I undertook this review with some hesitation. First of all, this is a belated review, the book under consideration having been published in 1968. This is a considerable time-lag for any monograph in an active research field. Furthermore, the fact that I had the opportunity to watch at close range the actual writing of this book does not make me an ideal reviewer. In particular, having known the author's intentions and goals, I may perhaps have a tendency to read more in the text than is actually there. To that extent, it is a biased review.

The author's aims, which are stated in the preface, are "to provide a basic foundation for the support and promotion of research in rotating fluids". To that effect, he has focused his attention upon "the motion of a contained, incompressible, viscous fluid" and on phenomena which "occur only in a rotating medium". Indeed, the modifications brought, about by rotation on water waves, convection, MHD flows, etc. are not to be found in this book.
The material is presented in a highly organised manner. ...

... the presentation is more mathematical than phenomenological. This is necessary in view of the stated aims and is in keeping with the fact that the author was brought up on Cauchy and boundary-value problems. But, and this is the unique feature of the book, the mathematical sophistication of the treatment does not prevent the author from dealing with real fluid flows. Indeed, experiments both as illustration of and as motivation for analytical work are emphasised throughout.

The book could be used as a text for an advanced graduate course on the subject. However, it is primarily an excellent reference which makes readily accessible numerous results previously scattered in research papers. The book is in fact already widely referred to and the author has received the unwitting compliment of seeing some of his notations adopted. (This adoption would have been even more extensive were it not for the exotic symbols that are often used.)

Of course, one can always find faults. For instance, the oceanographers and meteorologists may not be entirely pleased with chapter 5 and some purists have objected to the dogmatic treatment of Stewartson layers. However, all the criticisms do not outweigh the order which the book has introduced in the field. All in all, this is an excellent book.

1.8. Review by: Keith Stewartson.
Nature 220 (1968), 203-204.

The mechanics of rotating fluids is currently attracting much interest from experimenters and theoreticians, partly because the subject is of great importance to many aspects of geophysics, including oceanography and geomagnetism, solar physics and astrophysics, and to turbomachinery and aerodynamics. In addition it contains many novel features, like the Taylor column and double boundary layers, and is a fine example of the mutual benefits which flow from a joint attack on difficult problems by both theoreticians and experimenters. Finally, it is a game which anyone can play: large experimental facilities and large computers are useful but not essential.

The publication of this book will be welcomed by large numbers of scientists for a number of reasons. First, it is up to date with a balanced account of the progress that has been made up to the end of 1966 and including much material, presumably due to the author, which has never been published before. It also includes a large number of photographs which help to make the theoretical arguments more convincing and demonstrate the reality of the surprising and fascinating phenomena under discussion. Second, the majority of topics covered have been brought together for the first time so that new workers will find it a gold-mine of information and opinion. This brings me to my third reason. The subject is full of pitfalls for the unwary and has led to a number of interesting controversies; for example, does a forward wake always precede a body moving up the axis of rotation? It is good to find that Professor Greenspan, himself a distinguished theoretical (and experimental) worker in the field, has brought most of these controversies out into the open and has been prepared to express firm opinions on them. (Fortunately his views usually coincide with mine.) Fourth, he makes it clear that there are many questions still awaiting answers so that the book will remain for long a source of theses and papers. Finally, the author's style of writing the book is entirely appropriate to the nature of the task before him. The basic ideas are explained in detail while the developments are discussed more briefly, the reader being referred for further information to a list of references which is complete, to my knowledge, at least as far as the West is concerned.

After a short introduction the book has a long chapter on the linear theory of contained motions, that is, when the fluid is confined by rigid boundaries and is in a state of almost rigid rotation. The discussion largely concentrates on inertia waves, Ekman and vertical shear layers, all of which play important parts in the fluid motions. My only, and mild, complaint about the book occurs here. It seems to me that the double boundary layer (page 103) is made more mysterious by studying a special case for which a complete solution is available. A strictly boundary layer approach, such as that initiated by Jacobs, is more revealing and natural. The next chapter, on nonlinear theories, deals largely with viscous effects on fluid motions caused by rotating disks and spheres, and chapter four discusses exterior problems when the fluid is bounded internally, by a moving sphere, for example. Next, the simplifications which can arise by integrating the equations with respect to the depth are examined together with their relevance, after suitable approximations, to the problems of oceanic circulation. Finally, the stability of fluid motions dominated by rotation is reviewed, including valuable discussions of Ekman layers and the "dish-pan" experiment.

In brief, an excellent book and unreservedly recommended.

1.9. Review by: Nikolai Dmitrievich Kopachevskii.
Mathematical Reviews MR0639897 (83m:76077).

The author's monograph is part of the well-known Cambridge monograph series devoted to current problems of geophysics; it gives a survey of the current state of knowledge in a field that is growing rapidly. The book is intended, on the one hand, to serve as a textbook, and therefore contains general information from hydrodynamics and the theory of a viscous boundary layer; and on the other hand, it brings together original research by the author and his colleagues on the dynamics of rotating fluids enclosed in a reservoir, and the theory of wave motions.

The basic principle the author is guided by is the originality, interest, and completeness of the research. Some idea of the contents is given by the chapter headings: Motion of a rotating fluid in a reservoir (linear and nonlinear theories); Motion in an unbounded rotating fluid; Models of oceanic circulation; Stability. The book is written in a lively and clear manner, and the examples and illustrations are aesthetically pleasing. It will certainly be of use to all who are interested in the dynamics of rotating fluids, particularly specialists in oceanic and atmospheric dynamics. Applied mathematicians will find in it a fair number of practical problems for continuing research in this important area.
2. Calculus - An Introduction to Applied Mathematics (1973), by Harvey P Greenspan and David J Benney.
2.1. From the Preface.

In presenting calculus as an introduction to applied mathematics, we begin an educational curriculum whose basic objective is the twofold capability of formulating a problem mathematically and extracting, by any means, useful and desirable information. The selection of subject matter, the emphasis placed, and the priorities assigned are all predicated to this end, and, in a more personal sense, they reflect our experience and judgment as scientists. Accordingly, our values are quite different from those most commonly expressed in mathematics texts. But in a larger sense we really return to the traditional values that marked the vital development and vigorous application of calculus.

The magnitude and difficulty of the task - dealing with nature and the social sciences - requires its own approach, an attitude stressing intuition, versatility, a willingness to explore and to test, and a dedication which will not be subverted by comparatively minor issues. It is a no-holds-barred contest where the ends often justify the means, where the sources of motivation and intuition are usually experiment and observation, and where the final vindication of procedure is comparison with reality.

In order to convey the process of discovery realistically, this book is written in the discursive style of scientific literature. This form provides the vast majority of the calculus audience with a typical and most useful account of how mathematics is actually created and employed in practice. The intent here is a sound presentation that is sometimes intuitive, often physically inspired, but always relevant and utilitarian. Sound mathematics is basically correct mathematics, developed to the point of optimum return in a manner most easily understood, assimilated, and applied. We could have been precise without fault, had we so desired, but we deliberately chose not to spend precious time on that which is obvious, minor, strictly pedantic, or without interest or purpose. Complete rigour is often unnecessary from the scientific viewpoint and may even be counterproductive.

The conversion of this discursive exposition into the deductive and conservative mathematical format of axioms, theorems, and proofs is an appropriate enterprise at a more specialised and advanced level of study. In this spirit, the concept of a function is treated simply, befitting a first course, and the historic and extraordinarily useful notation y=f(x)y = f(x) is adopted rather than the "modern" nomenclature. The treatment of increments as infinitesimal elements, and the full use of illustrations in their explanation, is entirely consistent with our stated aims. The understanding and appreciation of how such approximations are and have been used is of paramount importance in learning calculus. A discussion of the subtleties involved is pedagogically sound after basic ideas are assimilated and practical methods are mastered.

Some candid and critical questions must be raised concerning the present college mathematics curricula. Is calculus presented as an impressive and meaningful intellectual achievement? Can it be said that the subject matter selected meets the real needs of the engineering and scientific community? Or has calculus become almost exclusively an introduction to analysis for that small minority destined for pure mathematics? Our answers culminated in this text.

Let us turn to more practical matters concerning the organisation and use of the material in this text. Chapter 1 is by design a rapid survey of calculus pivoted about the central concept of a limit, which is illustrated in a variety of contexts. It also provides time to establish a common foundation for a class despite the different backgrounds involved. All objectives are met if a good working knowledge of limit operations is acquired, as well as a review of certain prerequisites and an exposure to the principal themes of calculus. An overview of the subject gives an immediate appreciation of its content, structure, and purpose; goals can be identified and kept in sight from then on. But it is not necessary to achieve a deep understanding of any concept introduced here other than that of a limit, because each is considered anew in subsequent chapters. Even a modest familiarity with fundamentals and basic attitudes is an effective aid and stimulant for further study.

Of course, this survey can be modified, sections omitted, or deferred and integrated at a later time with related material in the general development. Reasonable facility is assumed with algebra and trigonometry, but very few basic facts about analytic geometry (given in Appendix 5) are actually used. It is neither essential nor desirable to have any prior knowledge of calculus. Approximate techniques are emphasised, and the basis is set for computer use. No specific programming language is advocated or adopted, nor have we included many associated topics which are best left as a separate option.

There are three kinds of exercises: problems to develop facility with techniques already  illustrated; elaborations on material in the text; applications. These are easily recognised, and, roughly speaking, they are listed consecutively. Several assigned problems per lecture, properly and systematically done, are an essential facet of the learning process. To learn calculus "in principle" only but to be unable to use it is to have learned nothing. The material content is adequate for a course of two, three, and even four terms. Sections are sufficiently independent to accommodate many arrangements, and reference to the numbers in the margin allows subdivisions into even smaller natural units. For example, the theoretical ϵ,δ\epsilon, \delta description of continuity can be deleted from Sec. 1.5 by simply omitting subsections 2 and 4, and this leaves a purely verbal and descriptive discussion. Most sections can be manipulated this way to satisfy particular requirements. Obviously, a very large number of schedules are possible.

2.2. Review by: Edward H Lipper.
The American Mathematical Monthly 82 (9) (1975), 949-950.

Does anyone remember Aristotle? In Book II of the Nicomachean Ethics he stated that, "Moral virtue is a disposition to choose the mean." Is there, or has there ever been, a more consistently ignored idea than the one contained in this statement? Today the dominant idea is excess. We move from extreme to opposite extreme, passing through moderation, like a pendulum, with maximum velocity. Education, alas, is not immune from the process. The overemphasis on abstraction in undergraduate mathematics of recent times has produced the inevitable "backlash" typified, unfortunately, by this text.

The book is intended as an introduction to the differential and integral calculus from an applied, "practical" point of view. Chapters on limits, differentiation , integration and infinite series are followed by an introduction to vectors (two or three dimensional 'arrows'), partial derivatives, multiple integration, and finally, the main integral theorems of vector calculus. In the preface the authors state that, "Complete rigour is often unnecessary from the scientific viewpoint and may even be counter-productive." Quite true, on this level, but a funny thing happened on our way to the text. We lost not only "complete rigour," but almost all rigour as well as a great deal of explanatory text. What remains is section after section consisting almost entirely of worked examples, leading even some of my engineering students to complain that there was insufficient "theory" and that they were simply being "fed formulas" as one student remarked on a final examination discuss-the-course question.

In past years at Stevens Institute the one-year sophomore vector calculus sequence was the object of much criticism. A great deal of it - understandable and even partially justified - was centred on what was considered to be an excess of linear algebra. We talked about abstract vector space theory to students who were seeing second and third degree matrix operations for the very first time. In the text being reviewed here, there isn't a matrix to be found anywhere (determinants used for Jacobians are defined without any mention of matrices). Is there anything too rigorous about a matrix, or a transformation of two dimensional space, say, to itself? Indeed, one would be hard pressed to find a more "applicable" mathematical object than a matrix.

It must be mentioned that the problem sets appearing at the end of each section are excellent. There is a large number of problems ranging from the routine to the extremely difficult and challenging. Many of them are quite fascinating to solve and have proved to be highly instructive to both teacher and student.

2.3. Review by: Stuart Jay Sidney.
American Scientist 62 (4) (1974), 498.

Although an elementary text cannot be definitively evaluated by one who has not taught from it, I must nevertheless credit the authors with producing a first-rate calculus-applied mathematics book for the student with a well-developed geometrical and physical intuition (which will grow as he progresses). Commonly, applications are merely a continuing postscript in calculus texts, even those which are considered application-oriented. Here, however, they are the raison d'être of the book, motivating the mathematics and then being motivated by the mathematics; further, the mathematical content is strongly influenced, with an abundance of approximate and formal methods and intuitive approaches and, in addition to the many routine exercises, myriads of problems which afford ample opportunity for exploration, experimentation, and formulation on the part of the student.

There is no systematic theoretical development of the calculus, a deliberate omission which (surprisingly) I found not at all disturbing (though even minor gaps here bother me in other texts); I can only conclude that this book provides much justification for the authors' contention that analysis should be left to analysis courses. Those teaching the usual three-semester calculus sequence either to science and engineering students or to honours students who like their mathematics related to other things will want to examine this book.

2.4. Review by: George Green.
The Mathematics Teacher 67 (5) (1974), 440.

The authors of this text have attempted to provide the student with opportunity to gain a sound intuition about the basic concepts of calculus while simultaneously developing computational skill. To these ends, many theorems are stated without proof, exposition is kept at a minimum, varied examples are given, numerous diagrams are provided, and abundant problems of varied difficulty appear. Answers to many exercises are provided within the text. Several problem sets are prefaced with suggestions or hints, and frequently the student is asked to go back over exercises and try to work them men tally after having worked them with paper and pencil.

The text has four parts: precalculus topics (functions, analytic geometry, trigonometry, and vectors), differential calculus, integral calculus, and multivariate calculus. The authors have endeavoured, both in the preface and throughout the body, to keep the teacher and student apprised of alternate uses of the text and of how various parts of the book relate to one another. For example, a note at the beginning of one section of chapter 9 indicates that the material will not be used until chapter 17 but that it is placed there for the benefit of physics students who may need to use it sooner. Another appealing feature of the text is an abundance of in formative historical notes and comments about the problems and difficulties faced by the founders of the subject ; although only one chapter (the introductory chapter on the derivative) is followed by a list of historical references, numerous informative anecdotes appear as footnotes.
3. Calculus - An Introduction to Applied Mathematics (Second Edition) (1987), by Harvey P Greenspan and David J Benney.
3.1. Review by: Stanley M Lucas.
The Mathematics Teacher 81 (1) (1988), 73.

This revised edition covers the full range of topics expected in a calculus text including vector calculus. It has been reorganised so as to start with the chapter on limits that includes an overview of the succeeding chapters. It is written in a discursive easy-to-follow style with various intuitive and analytical approaches. The rigorous proofs are left for later chapters. Engineering students would appreciate the many realistic examples throughout the text. Answers to odd-numbered problems are given, as well as more detailed hints and answers to more difficult problems. In addition to the table of integrals, the appendix is a ready reference to the commonly used algebraic, trigonometric, and analytic geometry formulas. On the basis of their experiences at MIT and McGill University, the authors feel that once the concept is understood, then the rigorous explanation and proof would follow. As the authors point out, to learn the calculus "in principle" but not be able to use it in practice is useless. It appears that their goal has been met.

Last Updated November 2020