1.1. Review by: Masayoshi Nagata.
Mathematical Reviews MR0100591 (20 #7021).

This is, as the title shows, an introduction to algebraic geometry, covering the following topics: Places, algebraic sets, $k$-varieties, (affine, projective, abstract) varieties, Zariski topology, correspondences, product varieties, derived normal varieties, linear systems of divisors, differential forms, simple points, algebraic groups, Riemann-Roch theorem on curves.

This book does not cover the intersection theory, Grassman varieties, Chow varieties nor equivalence relations of cycles.

Chapter I contains some basic theorems on valuation rings (places). In chapter II, algebraic sets (in affine spaces), $k$-varieties (affine), homogeneous varieties (= affine cones; it seems to the reviewer that this use of the term "homogeneous variety" is not suitable because of the possibility of confusing it with the concept of homogeneous spaces) and product varieties (affine; over an algebraically closed field) are defined. In Ch. III, the notion of varieties (= absolutely irreducible affine varieties) is defined and the Zariski topology is introduced. In Ch. IV, correspondences (which may be called "geometric correspondences" in order to distinguish them from correspondences in the sense of cycles, which are not covered by this book) are treated and then the notion of abstract varieties is defined.

Ch. V concerns the normality of a variety, including Zariski's main theorem for birational correspondences, derived normal varieties, and projective normality. Divisors and linear systems of divisors are the topics of Ch. VI. Ch. VII contains results due to Koizumi on differential forms. Ch. VIII concerns simple points and Ch. IX defines the notions of algebraic groups and abelian varieties.

The Riemann-Roch theorem on algebraic curves is proved in Ch. X, where the fact that the sum of residues of a differential form on an algebraic curve is zero and Harnack's theorem (which asserts that the number of components of the real part of an irreducible curve is at most one more that its genus) are proved.

Each chapter, except for Ch. VII, is accompanied by some comments and references to literature which are useful for students.

1.2. Review by: Alice T Schafer.
The American Mathematical Monthly 67 (6) (1960), 603-604.

This book offers much more than its modest title would lead one to expect. Although it is not intended to be a comprehensive account of classical differential geometry, a sizeable amount of the metric theory of curves and surfaces in three-dimensional Euclidean space is packed into Part 1, all done very neatly by use of vector methods. Following three chapters on curves and intrinsic and extrinsic properties of surfaces, Part 1 is concluded by a fourth chapter on geometry of surfaces in the large which includes material heretofore not seen in an English text. For an understanding of this material on global geometry the reader needs some familiarity with such concepts as compactness, Hausdorff space, homeomorphism, and covering space (which are not defined or explained here). However, this chapter is independent of the remainder of the book and may be omitted. In the author's view the material of Part 1 should be covered before the reader encounters the more general tensor theory in Part 2.

Under the general heading of differential geometry in $n$-dimensional space in Part 2, there appears a chapter each on tensor algebra, tensor calculus, Riemannian geometry, and applications of tensor methods to surface theory. Vector spaces and the dual space are introduced and then follows the tensor product of vector spaces. Differentiable manifolds, connections, and generalised covariant differentiation constitute part of the chapter on tensor calculus. The following chapter on Riemannian geometry portrays the tensor analysis of a special object that of a field of symmetric, second-order, covariant tensors. In the concluding chapter Riemannian geometry is specialised to a very brief account of the use of tensors in ordinary surface theory. In the progressive specialisation of the latter part of the book, the author appears to reverse his earlier conviction on order of presentation.

The book is admirably written by one who is familiar with current research in the field. Mention is made of various unsolved problems. Useful references are given for further reading. Much material is covered in the rich lists of exercises. Most of the sixty-six exercises in the final list are taken from examination papers of the University of Liverpool. The book is highly recommended as a textbook for a two-semester sequence on the essential ideas and methods of differential geometry.

2.1. Review by: Edward H Batho.
The American Mathematical Monthly 67 (5) (1960), 484.

It has been over ten years since Weil's definitive Variétés Abéliennes et Courbes Algébriques first appeared. During this period a great deal of work has been done on algebraic groups and, in particular, on abelian varieties. Many facets of abelian varieties only mentioned in Weil's treatise have now reached a state of substantial development. This development has been the combined work of the French, Italian, Japanese and American schools. It has therefore become desirable to have a book available which incorporates the results of Weil's treatise with the newer developments in the field of abelian varieties.

Lang has attempted to do just this in the present book and, for the most part, succeeds admirably. Even though - as Lang points out - several special topics are not touched upon, an amazing amount of material is covered. The topics dealt with are the Jacobian, Picard, and Albanese varieties; the $l$-adic representations; algebraic systems of abelian varieties. The emphasis is very definitely and very heavily on the "abstract." More examples drawn from "classical" algebraic geometry would help to illuminate and clarify the ideas developed-particularly for the beginner in algebraic geometry. This, however, is a minor flaw in an otherwise excellent book. The reader who is willing to give his time and close attention to this book will be amply rewarded by the insights he will receive into one of the most beautiful and fruitful branches of modern mathematics.

2.2. Review by: Teruhisa Matsusaka.
Mathematical Reviews MR0106225 (21 #4959).

This book deals with the theory of general Abelian varieties and also that of Albanese and Picard varieties of given varieties. It is chiefly based on the lectures given by A Weil during 1954-1955 (together with the author's own contribution). Also Chow's work on the trace and image is treated in the last chapter.

Generally speaking, this book is very well written and would give investigators an excellent account of what has been done, without going through many papers. Also it is convenient that this book contains all results in Weil's book on Abelian varieties [Variétés abéliennes et courbes algébriques] (except possibly a construction of a group variety from a variety having a normal law of composition); some proofs are simplified and are made lucid. However, it is regrettable that some of the basic and useful theorems on Abelian varieties (such as duality, absence of torsion, Riemann-Roch theorem, theorem of Frobenius, etc.) had to be omitted (partly because these were not available at the time when the book was being prepared). The following list of chapter headings and comments will make the scope of this book clear.

3.1. Review by: John W Gray.
The American Mathematical Monthly 71 (5) (1964), 582-584.

There are two aspects of this book that strike the reader immediately. The first is that the book begins with the basic definitions of categories, functors, and natural transformations. The second is that manifolds as here defined are locally homeomorphic to Banach spaces. We would like to comment on both of these aspects before turning to a discussion of the specific contents of the book.

Category theory is playing a role in modern mathematics analogous to that played by set theory fifty to a hundred years ago. Roughly speaking, set theory began as a descriptive theory. Then, as its universal applicability was realised, and as more powerful set theoretical methods developed, it became a prescriptive theory, until, by now, its prescriptives have become so intrinsic to mathematics that they tend to be accepted merely as descriptions.

Category theory is just beginning to enter into the prescriptive stage of this sequence. When it was invented some eighteen years ago by Eilenberg and Mac Lane, it was almost purely descriptive, and its usefulness seemed restricted to algebraic topology and a few special topics in algebra. However, about ten years ago, it was found that there were quite reasonable ways to introduce additional structure into general categories, principally through the use of related notions of universal mapping problems, adjoint functors and representable functors. These ideas have been most dramatically exploited by Grothendieck in a series of papers mainly related to algebraic geometry. The methods and prescriptions laid down by him seem, how-ever, to be universally applicable, and one can expect that in the near future they will completely revolutionise the presentation of abstract mathematics, even of set theory itself. In the terminology of Kuhn (The Structure of Scientific Revolution, University of Chicago Press, 1962), there has been a change in the paradigms of mathematics.

Lang's book is the first in differential geometry to take advantage of these developments. One finds them in the universal mapping properties characterising submanifolds, kernels and cokernels of vector bundle morphisms, and fibre products or pullbacks. The only really significant use of functors is in the construction of associated bundles. The use of category theory, then, is rather minimal but it is probably as much as-if not more than contemporary differential geometric traffic will bear. A systematic study of differentiable manifolds from the standpoint of category theory will include a good deal more than is found in this slim volume. For example, what about (generalised) direct limits of manifolds?

We now turn to the second innovation of the book, that of considering locally Banachian manifolds. From this standpoint the book can be considered, in terminology and spirit, as a continuation of Chapters 8 and 10 of Dieudonné, Foundations of Modern Analysis (Academic Press, 1960). Chapter 8 is essentially the prerequisite for Lang and, in fact, both books are enriched by a concurrent reading.

Infinite dimensional manifolds were first formally considered by J Eells (see references in Lang), and the observation that most of the elementary theory generalises easily is due to him. He is also responsible for the only significant examples, function spaces, and the only deep theorem, a generalisation of Alexander-Poincaré duality. These results probably provide a sufficient motivation for treating the infinite dimensional case ab initio, aside from the fact that such a treatment forces proofs to be as natural and coordinate free as possible - a desirable end in itself. Also, it is possible that an infinite dimensional manifold with a distinguished two-form (or perhaps a one-form) is the natural habitat of a quantum field theory.
...

One is brought to the threshold of differential geometry, but unfortunately, except for a short appendix on Hilbert spaces, the book ends here. It would have been nice to have a discussion of some examples of manifolds in this sense or a proof, for example, that the geodesics constructed from the geodesic spray have stationary length. The style of the book is as condensed as a research paper and it does not even contain problems hinting at all of the auxiliary results that should appear in a complete discussion of the subject, even within its own framework. It seems highly unlikely that anyone who is not already familiar with differential geometry will get much from reading this book. Nevertheless, the paradigms have changed and the subject will never be the same again. No course in differential geometry can afford to neglect the viewpoint and the material of this book and, in fact, with sufficient geometrical padding it could be the basis for a very interesting series of lectures.

3.2. Review by: Per Holm.
Nordisk Matematisk Tidskrift 11 (3) (1963), 125-126.

The present book is an interesting newcomer, so far quite unique in its kind. The author announces that its purpose is to fill the gap in the literature that exists between the three differential theories: differential topology, differential geometry and the theory of ordinary differential equations. More precisely, we could say that the author wants to clarify the natural area of ​​relevance for these theories and to specify the apparatus common to them in its most general form. From this it should be clear that the book is not an introduction to differential geometry in the usual sense. One does not find mentioned concepts such as curvature, torsion, affine connection, etc. We clarify this to avoid possible misunderstanding regarding the content.

As the title suggests, it is differentiable manifolds that are the central subject of the presentation. In contrast to usual practice, however, differentiable manifolds in this case also mean infinite-dimensional differentiable manifolds, i.e. those where the tangent space at each point is a Banach space. The fact that the theory of differentiable manifolds can be developed for such manifolds is essentially due to the fact that the existence theorem for ordinary differential equations and the theorem for implicit functions from analysis also apply in Banach space. Other features of the book that catch the eye are the consistent use of concepts and techniques taken from differential topology and homological algebra. We mention in abundance categories and functors, morphisms and exact sequences, vector bundles and operations on such, pull backs, etc. Far from seeming burdensome (strangely enough), these have given the presentation extraordinary clarity. Otherwise, we mention that the integration theory used is limited to integration of regulated functions (uniform limits of step functions) and is as such very simple and clear (simpler than ordinary Riemann integration).

Let's say a little about the level of the book. According to the author's intentions, the book, together with a basic course in differentiation and integration of vector-valued functions, should constitute a 1-semester course for first-year graduate students or gifted undergraduate students. The reviewer finds such a structure somewhat harsh. It assumes in any case that the students are fairly solid in linear algebra and have a good knowledge of the simpler parts of general topology. In addition, one should probably be motivated for the material from at least one of the three disciplines mentioned first in the review. Otherwise, the book is self-sufficient, and the material has been given an elegant and secure design by a competent mathematician.

3.3. Review by: Shingo Murakami.
Mathematical Reviews MR0155257 (27 #5192).

This book offers a quick introduction to basic parts of the theory of differentiable manifolds. As a specific feature of this book, the whole theory is developed in a purely intrinsic way, that is, without any use of local coordinates. Indeed, dimensions of manifolds are not assumed to be finite and all manifolds are modelled on Banach spaces (including, of course, finite-dimensional ones); this means that the usual definition of differentiable manifolds is here generalised so that each point of a manifold has an open neighbourhood which is homeomorphic to an open set of a Banach space. With such a presentation that distinguishes this textbook from others on differentiable manifolds, the book is written in a quite self-contained manner and, in fact, the author derives each basic result after careful examination of its analytical and topological background; for example, the existence theorem of a flow corresponding to a vector field is preceded by several propositions about the existence and differentiability of the solutions of ordinary differential equations concerning functions with values in a Banach space. In this sense the book will serve as a concise foundation for the modern theory of differentiable manifolds. The subjects are carefully selected in order that the readers may quickly approach recent topics in differential geometry and differential topology. It should also be noted that this is the first book which exhibits the usefulness of sprays for the foundations of the theory of manifolds, a treatment the most of which the author attributes to R S Palais.

4.1. Review by: Paul T Bateman.
Pi Mu Epsilon Journal 3 (8) (1963), 428.

One of the most elegant, and at the same time most difficult, theorems on diophantine equations is the following one proved by Siegel in 1929: If $f(x, y)$ is a polynomial in two variables with integral coefficients such that the Riemann surface of the algebraic function defined by the equation $f(x, y) = 0$ has topological genus greater than zero, then the diophantine equation $f(x, y) = 0$ has at most a finite number of integral solutions.

Roth's theorem (1955) on rational approximations to algebraic numbers made it possible to simplify somewhat the logical structure of the proof of Siegel's theorem, and the advances in algebraic geometry of the last two decades have made it possible to generalise Siegel's theorem considerably. The present book is devoted to presenting these two developments. However, it does very little to make Siegel's theorem more accessible to the general mathematical public, for it is addressed to those who have scaled the Olympian heights of modern algebraic geometry.

4.2. Review by: Arthur Mattuck.
The American Mathematical Monthly 71 (9) (1964), 1060.

"Diophantine geometry," by abuse of language, is what an algebraic geometer sees when he looks at Diophantine equations: points on algebraic varieties whose coordinates are rational numbers or integers. The two big theorems of the subject - Siegel's proof that an algebraic curve of positive genus has only finitely many integral points, and the Mordell-Weil theorem that the rational points on an abelian variety form a finitely generated group - were proved thirty-odd years ago. The big intervening event has been Roth's "$2 + \epsilon$" theorem, which simplifies Siegel's proof. The author presents these results, dressed up and stylishly generalised, to an audience composed primarily of algebraic geometers. He gives these people in two introductory chapters a rapid account of what they need to know in algebraic number theory (and not one word more). The theorems alluded to above are then proved, and the book concludes on a simpler level with the Hilbert irreducibility theorem.

Granting the non-discursive style (the sort that can "do" algebraic number theory without ever mentioning cyclotomic fields), it's extremely well done. The author's taste is impeccable, and the cognoscenti - those who have gone to Andre Weil's school of algebraic geometry - will find this book deeply rewarding. On the other hand, it will probably be alien corn for the number theorists. They will certainly find no numbers here, and the two central proofs are so deeply imbedded in geometry that even to extract a proof of the relatively elementary theorem of Mordell would require heavy surgery. Unless they are willing to bone up on the author's two previous books on algebraic geometry, "Diophantine geometry" is likely to remain for them a closed book.

4.3. Review by: W E Jenner.
Mathematical Reviews MR0142550 (26 #119).

This book is concerned with the interactions of diophantine analysis and algebraic geometry. It is up to the author's usual high standard, and should do much to renew profitable interest in the subject it treats. The first two chapters are concerned with absolute values and the arithmetic of the classical fields, ideal theory, divisors, units and ideal classes. The account given of these matters is one of the cleanest in the current literature, and is beautifully done. ...
...
The historical notes throughout the book are of very high order. These include conjectures as to the future history as well, which possibly constitute one of the more valuable features of the book.

5.1. Review by: Thomas L Wade.
Science, New Series 144 (3614) (1964), 43.

In the foreword Lang states that this book "is written for the student, to give him an immediate, and pleasant, access to the subject. I hope that I have struck a proper compromise between dwelling too much on details, and not giving enough technical exercises. ..." It is readily discernible that the goal of avoiding too many details has been attained.

The author reflects a thorough knowledge of the subject about which he is writing and of related mathematical specialties. Many notions which are basic to the study of calculus are explained in an excellent manner - among these are the treatment of inequalities (in section 1.2) and that of the slope of a curve (in section 3.1). The concept of a derivative and the theorems on derivatives are clearly presented. Chapter 9, "Integration," and chapter 10, "Properties of the integral," are worthy of special commendation.
...
Antiderivatives, inflection points, and differentials are not mentioned, and relatively little attention is given to differentiation of composite functions. Chapters 9 and 10 (on integration) give practically no attention to integration of composite functions, but brief consideration is given to this in chapter 11. Integration by use of trigonometric substitution appears in one brief illustration, apparently without mention by name and without further discussion. In this respect one wonders if the student who uses this book may not be handicapped in subsequent study of intermediate calculus and differential equations.

The book has a pleasing, uncomplicated appearance and I noted only a few misprints. To those who are looking for a short calculus book, of limited coverage, written in an informal manner, I recommend Lang's book.

6.1. Review by: Sarvadaman Chowla.
Mathematical Reviews MR0160763 (28 #3974).

We list the titles of the ten chapters: Algebraic Integers, Completions, The Different and Discriminant, Cyclotomic Fields, Parallelotopes, Ideles and Adeles, Functional Equation, Density of Primes and Tauberian Theorem, The Brauer-Siegel Theorem, Explicit Formulas.

6.2. Review by: Basil Gordon.
The American Mathematical Monthly 74 (3) (1967), 345.

This volume is a most welcome addition to the literature of the subject. The interplay of algebra and analysis, together with the author's lively style and erudition, combine to produce a book which should excite and stimulate its readers. Lang was a student of Artin, and we are fortunate to find in him an able expositor of the wonderful tradition in number theory associated with his teacher.

The first half of the book presupposes something like volume I of van der Waerden's Algebra, the latter half, complex variables, elementary point set topology and a little distribution theory. The treatment strikes a balance between the global methods of earlier books such as Hecke's and the emphasis on the local theory to be found in some newer books such as that of Weiss. Beginning with a nicely arranged sketch of classical ideal theory in Dedekind domains, the author deals with algebraic integers, the different and discriminant, cyclotomic fields, parallelotopes, ideles and adeles, functional equation of ζ-functions, density of primes, the Brauer-Siegel theorem, and Weil's approach to explicit formulas.

7.1. Review by: K D Magill, Jr.
The American Mathematical Monthly 72 (9) (1965), 1048-1049.

This is a continuation of A First Course in Calculus by the same author and covers the work of approximately two semesters. The first half of the book deals with the calculus of functions of several variables and the last half is devoted to a treatment of linear algebra. This book differs in several respects from a number of texts which are intended for the same level of instruction. First of all, there is a complete absence of $\delta-\epsilon$ techniques in proofs. One should not conclude from this that there is a lack of rigour. Simple modifications result in standard $\delta-\epsilon$ proofs and any student who understands the discussions of $\delta-\epsilon$ techniques in the appendices of this and the preceding volume should be capable of making those modifications. Secondly, the topics in solid analytic geometry that are usually discussed in connection with functions of two variables are not considered in this book. We found few misprints and none of any consequence. Overall, we found the book to be well written with lucid treatments of topics which are important to any science or mathematics major. Any instructor wishing to combine linear algebra with the calculus should give it (along with first volume) special consideration.

7.2. Review by: Ivan Niven.
Mathematics Magazine 43 (5) (1970), 277-278.

The title of the book describes the contents rather well; it could not be called a book on advanced calculus because of the severe limitations on what is included. There are 306 pages of text, numbered from 317 to 622 as a continuation of the First Course in Calculus by the same author. The chapter numbers, however, are not continued from the first book.

Chapters 1-7 constitute a block of material, beginning with a chapter on vectors in $n$-space, followed by differentiation of vectors, partial derivatives, the chain rule, the gradient, potential functions, line integrals, higher derivatives and Taylor's formula, and problems in maxima and minima. Chapters 8-12 offer the basic material in linear algebra needed in the final chapters 13-17. These last five chapters are concerned with the Jacobian matrix, multiple integrals, Green's Theorem, Fourier series, and normed vector spaces. (The chapter on normed vector spaces appeared to this reviewer as an anomaly, more appropriately a first chapter of a more advanced book.) Many of the chapters are on the short side, offering just a bare-bones introduction to a subject.

For example, the work on vectors is light on the geometry of the subject, with virtually nothing on the scalar triple product and its application to volumes and to coplanar lines and vectors. The divergence and curl of a vector field are mentioned only in exercises, not in the text itself. Green's theorem is included, but not the divergence theorem or Stokes' theorem. Rotations in 3-space are not treated. The length of a curve and the area of a surface are given scant attention. These limitations are cited not as criticisms but as simple facts that a potential user should keep in mind. They are not criticisms because the author makes no claim to matching one of the 700 or 800 page treatises. But this reviewer found that he wanted to offer his class additional material in dittoed form to augment the text.

It was also necessary to give the class additional problems, not for every section in the book but only in some instances. Generally speaking, the book has substantial problem sets, but not uniformly. The last section in Chapter 5, on the question of the dependence of a line integral on the path of integration, is followed by one problem, and that is just an extension of the theory to take care of the case "when $h$ is negative" in the proof of the central theorem. A similar observation can be made about the balance of problems between applications of the theory and extensions of the theory, with the latter given almost exclusive attention in a few cases.

The book was well received by the students. They learned quickly that the answers in the back of the book are not completely reliable, so that the question "Is the answer given to problem so-and-so correct?" was often asked by a student. For a second edition the number of typographical errors seemed rather large, including a rather novel one: the title of Chapter 5 is a little different in the Table of Contents than in the text itself.

The book is contemporary in idiom and in spirit. The proofs, many of which are novel and clever, are arranged so that most of the arguments are incisive and brief. The vector approach is central, so that separate arguments for 2-space and 3-space are avoided except where concepts are tied to a specific dimension, such as the cross-product of two vectors. The notation is mostly chosen to facilitate the entire line of argument. However, the use of symbols like $A$ and $(a_{1}, a_{2}, ..., a_{n})$ to denote either a point or a vector was not a happy choice; this notational arrangement caused the students trouble when confronted with a vector from a point $A$ to a point $B$.

In summary, the book was found to be quite acceptable for classroom use. There are many excellent features and no serious flaws.

8.1. Review by: A G Brandstein.
The American Mathematical Monthly 74 (1, Part 1) (1967), 103.

Columbia's Professor Lang, in this first year graduate text, manages to achieve an admirable and needed blend of several seemingly divergent facets of algebra. The book roughly follows the "classical" Artin-van der Waerden order, but the inclusion of newer results, especially in the chapter on representations of finite groups, greatly extends and amplifies the presentation. Lang spices the book liberally with homological notation. Fortunately, the use of diagrams is not forced, and lengthy diagram chasing is avoided. The early introduction of categories and functors provides a good basis for the extension of theorems to various topics. As with most modern works, the influence of Bourbaki is also strongly felt. The word "warning" even replaces Bourbaki's "Z." Lang has done a presentable job of editing from Bourbaki's massive publications on linear algebra and representations. Many of the chapters are independent and the use of key repetitions allows the instructor a good deal of flexibility.

The book, however, may be a little too sophisticated for the audience for which it is intended. Perhaps in the interest of being somewhat encyclopaedic in a limited space, long lists of definitions are presented with few relevant examples or applications. Proofs are perhaps too abbreviated and often might be better motivated. A less parsimonious use of commas might make the text more readable. Nevertheless, all these peccadillos are amply offset by Professor Lang in a timely, complete, and illuminating work.

8.2. Review by: Richard Scott Pierce.
Mathematical Reviews MR0197234 (33 #5416).

The author states that his aim in writing this book was to produce a modern basic text for a first year graduate course in algebra. He has succeeded very well indeed. In spirit and content, his textbook is similar to van der Waerden's classical Moderne Algebra. It is, however, much more up-to-date and complete.
...
Without first-hand experience, it would be difficult to predict the success of this book in the classroom. Average students will find it hard going, and only the best will be able to master the immense amount of material that is laid out by the author. Some persons will find faults in the book. There are quite a few typographical slips, and a few careless errors and omissions. It is the reviewer's opinion that the exposition could be improved in places. Nevertheless, this book is a remarkable accomplishment. It covers all of the basic material of abstract algebra in the short span of 500 pages. The author has an impressive knack for presenting the important and interesting ideas of algebra in just the "right" way, and he never gets bogged down in the dry formalism which pervades some parts of algebra. His book is indeed a worthy successor to van der Waerden's classical transcription of the lectures of Artin and Noether.

9.1. Review by: C V R.
Current Science 35 (24) (1966), 633.

The present book is meant as a text for a one-year course in algebra at the undergraduate level, during the sophomore or junior year. It starts with the basic notion of vectors in real Euclidean space, which sets the pattern for much that follows. Examples and special cases are worked out in abundance, and abstractions are developed slowly.

Examples from calculus are given throughout, and in some sense, the book is a completion of the limited amount of algebra in the author's A Second Course in Calculus. A chapter treating the tensor product and the alternating product appears in the text.

9.2. Review by: Irvine Reiner.
Mathematical Reviews MR0204434 (34 #4276).

This book is intended as a text for a one-year under-graduate algebra course, and would probably be suitable for the junior or senior year. It is well-written in a lively style, and abounds with examples and exercises.

The first third of the book develops, in a leisurely manner, basic ideas of vectors in real n-space, vector spaces over subfields of the complex field, linear mappings and matrices, and determinants.

The material in the middle part is somewhat more condensed, as the author covers some of the standard theorems concerning vector spaces with inner products, orthogonality, bilinear forms, Hermitian matrices, eigenvectors and eigenvalues, and primary decomposition of vector spaces. Admittedly, an introductory book on linear algebra is not meant to be encyclopaedic, but it seems unusual that there is no discussion of rational canonical forms and Jordan forms of matrices, nor of invariant factor theory and minimal polynomials.

The final third of the book is devoted to material on tensor products and alternating products, followed by brief chapters on groups, rings and modules, and an appendix on convex sets.

The book would be especially valuable to students heading for graduate work in mathematics, since it touches on many important algebraic concepts. The emphasis is on the theoretical approach for the most part, although computational problems are occasionally considered in the text and in the exercises.

10.1. Review by: William Judson LeVeque.
Mathematical Reviews MR0209227 (35 #129).

According to the author's foreword, "The quantitative aspects of the theory of diophantine approximations are, at the moment, still not very far from where Euler and Lagrange left them. Very recent work seems to have opened some fruitful lines of research ...".

The first chapter presents the fundamental approximation properties of continued fractions: near the end of the chapter it is asserted that Perron made the first attempt to extend such results to linear forms in several variables.

The title of this book might lead the reader to expect somewhat broader coverage, and to feel some surprise at the names missing from the bibliography. However, it evidences the same care and scholarship as the other books of the author.

11.1. Review by: Bodo Volkmann.
Mathematical Reviews MR0214547 (35 #5397).

The theory of transcendental numbers is understood here as the task of "proving the transcendence and algebraic independence of values ​​of classical, suitably normed functions." From this perspective, some areas of the theory are left out of consideration - such as Diophantine approximations (which the author has treated elsewhere) or the metric theory of transcendental numbers, where a number of interesting results have recently been achieved. However, this approach makes the presentation coherent and clear.
...
Each chapter concludes with historical remarks, which are combined with heuristic explanations of the proof concepts and methodological difficulties, as well as with references to open problems.

The book's strength lies in its attempt to integrate many individual results into a unified framework by axiomatising the proofs, thus making them suitable for textbook use. All in all, it is a well-executed work that will undoubtedly greatly stimulate the further development of this field.

12.1. Review by: Seth L Warner.
Mathematical Reviews MR0210527 (35 #1419).

This test and the author's Linear algebra are designed for undergraduates. The present book is self-contained, but the author considers it preferable to study Linear algebra first. The style is very concise. Topics treated include the standard ones that should be taught in any undergraduate course. Titles and special features of chapters are as follows:
(1) The integers. A formal discussion of axioms for the natural numbers and the construction of the integers are deferred to an appendix.
(2) Groups.
(3) Rings. All rings are assumed to have an identity.
(4) Polynomials. First, a polynomial over an infinite field is defined to be a polynomial function. ...
(5) Vector spaces and modules. ...
(6) Field theory. All fields are subfields of the complex field, which here is assumed to be algebraically closed. Galois theory is presented in this context. ...
(7) The real and complex numbers. Real numbers are constructed by Cauchy sequences. ...
(8) Sets. Basic notions of set theory are discussed, including cardinal numbers. Zorn's lemma is stated as an axiom and various algebraic and set-theoretic consequences are drawn from it.

13.1. Review by: R A Rankin.
The Mathematical Gazette 53 (385) (1969), 339.

The first edition of this book appeared in 1963. The second edition contains additional exercises and new sections on convexity and complex numbers (transferred from Volume 2). The treatment is suitable for an introductory pre-university course. The $\delta, \epsilon$ definitions of limit are not used (although they appear in an appendix) and geometrical and intuitive arguments take their place. However, the limitations of the basic proofs given are carefully explained and an appropriate measure of rigour is used at later stages. Differentiation, integration, curve sketching, the elementary special functions, Taylor's theorem, infinite series and complex numbers are covered. Integration of continuous functions is based on a geometric notion of area and the usual non-analytic definitions of the trigonometric functions are assumed. The book is written for use in American schools and colleges and it is a little hard to see where it could fit in suitably in the English educational system, although it might be suitable as a text for some first year courses in Scottish universities.

13.2. Review by: C V R.
Current Science 37 (16) (1968), 479.

The purpose of this book is to teach the student the basic notions of derivative and integral, and the basic techniques and applications which accompany them. In this Second Edition, a chapter on complex numbers is included and the exercise sets have been greatly expanded.

The contents of this volume are: Numbers and Functions; Graphs and Curves; The Derivative; Sine and Cosine; The Mean Value Theorem; Sketching Curves; Inverse Functions; Exponents and Logarithms; Integration; Properties of the Integral; Techniques of Integration; Some Substantial Exercises; Applications of Integration; Taylor's Formula; Series; and Complex Numbers.

13.3. Review by: C R Miers.
Mathematics Magazine 43 (4) (1970), 221-223.

Professor Lang states in the foreword to the First Course that "This book is written for the student, to give him an immediate, and pleasant, access to the subject. I hope that I have struck a proper compromise between dwelling too much on special details, and not giving enough technical exercises, ... In any case, certain routine habits of sophisticated mathematicians are unsuitable for a first course."

One cannot complain about an author who writes expressly for the benefit of the student, or who attempts to make the subject as accessible and agreeable as possible. But "pleasant" seems to denote "easy" and it is always the case that learning mathematics, even in the beginning, requires hard work. It is the very "pleasantness" of the first part of the book which leaves the student unprepared to cope with the way integration theory is developed. In fact, Lang's development of this theory seems to be one of those "routine habits of sophisticated mathematicians" which he tried to avoid.
...
The actual exposition of the first eight chapters is as "pleasant" as any student could hope. Arguments are brief, often area-geometric in nature, and definitions of elementary functions are chosen to give the most economical development of their properties. Students readily appreciate the flavour of these sections, and the author does achieve "immediacy" in the sense that the student, early in the course, can work with some nontrivial tools of calculus. A minor annoyance in this part of the book is the inclusion of the second derivative test for convexity in a section on graphing instead of the earlier section on the mean value theorem which included the first derivative test.

A real problem in teaching from this book arises in Chapter IX, Integration.
...
Finally, a lack of quality, and sometimes quantity, in exercises is a recurring problem. The exercises are sometimes inappropriate, out of order (with respect to the body of text preceding them), and there are many mistakes in the answers printed at the end of the book. There is also a lack of good worked examples to illustrate concepts. In a book dedicated to giving students immediate and intuitive access to calculus, special attention should have been given to these details.

14.1. Review by: Hugh Thurston.
Mathematics Magazine 44 (5) (1971), 283-284.

After (I) a review of elementary analysis (of functions in $\mathbb{R}$ into $\mathbb{R}$), in which an axiomatic treatment of the integral as an interval-function is an interesting touch, the book gets down to business (II) with a concise treatment of normed vector spaces, of continuous mappings of one normed vector space into another, and of integrals as uniform limits of step-mappings (of an interval into a normed vector space).

A section (III) on Dirac sequences, convolutions, and Fourier series is rather a bypath, as the author indeed warns us, and the real meat of the book starts in part IV: calculus on (complete normed) vector spaces, using the Fréchet derivative. The fifth and last part is on (Riemann-Darboux) integration of functions in $\mathbb{R}^{n}$ into $\mathbb{R}$, ending with a chapter on differential forms.

Although the selection of material in this text is attractive, its organisation and detailed treatment are disappointing. The chapter on differential forms seems to lead nowhere; and I wonder whether it should have been the first chapter in volume II rather than the last in volume I. Integration of functions in $\mathbb{R}^{n}$ comes as somewhat of an anti-climax after differentiation in the more general setting of normed vector spaces. Chapter IX consists almost entirely of a review of series of real numbers and of series of functions in $\mathbb{R}$ into $\mathbb{R}$ and would fit better in Chapter I: true, the theorem about rearrangement of an absolutely convergent series is stated and proved for series of elements of a normed vector space, but the proof is the same as for series of real numbers. In any case, the theorem seems out of place in a text of this sort: the important series for analysis are power-series, and the last thing we ever want to do to a power series is to rearrange it.
...
The course for which we used the book was taught in several sections, and my colleagues seem in general agreement that the book is in the general spirit of modern mathematics, but falls short in too many details for them to be willing to recommend its use again.

For these reasons, and because the students found the book's arguments substantially harder to follow than those in other similar books, I would regard this book as more suitable as background reading than as a text for a course of lectures.

14.2. Review by: C V R.
Current Science 37 (22) (1968), 655.

The present volume is a text designed for a first course in analysis. Although it is logically self-contained, it presupposes the mathematical maturity acquired by students who will ordinarily have had two years of calculus. When used in this context, most of the first part may be omitted or reviewed extremely rapidly. The course can proceed immediately into Part Two after covering Chapters 0 and I.

The chapters contained in this volume are: Part One: Sets and Mappings; Real Numbers; Limits and Continuous Functions; Differentiation; Elementary Functions; The Elementary Real Integral; Part Two: Convergence-Normed Vector Spaces; Limits; Compactness; Series; The Integral in One Variable. Part Three: Applications of the Integral; Approximation with Convolutions; Fourier Series; Improper Integrals; The Fourier Integral. Part Four: Calculus in Vector Spaces - Functions on $n$-space; Derivatives in Vector Spaces; Inverse Mapping Theorem; Ordinary Differential Equations; and Part Five: Multiple Integration-Multiple Integrals and Differential Forms.

14.3. Review by: R P Gillespie.
The Mathematical Gazette 54 (390) (1970), 425-426.

This book, by a very experienced expositor, will be of great use to university teachers and students as a textbook of analysis from a modern point of view. It is very clearly written and the author explains at each stage his method of approach to the subject. His style is light and attractive. In a sense the title of this volume is misleading, in that the subject is really advanced calculus (admittedly from a more abstract point of view than has been customary in traditional texts). The intention is to follow this book with a second volume and there are references to subjects to be treated later. For example, the Hahn-Banach theorem is used but its proof is postponed for discussion in Analysis II.

The present book is in two distinct parts. The first 90 pages (Part One) are devoted to a review of elementary calculus as applied to functions of a single variable, finishing with a discussion of integrals of continuous functions. This part provides an excellent revision course and a helpful introduction to the major part of the book (Parts Two-Five) which follows. The essential feature of this second part is that the fundamental concepts of limit and convergence are discussed in terms of normed vector spaces rather than in terms of real numbers. This enables the author to deal with n-space and with function spaces. The usual notions with regard to limits as applied to real numbers are generalised to normed vector spaces and it is soon clear that it is just as easy to work with normed vector spaces as with real numbers. Part Two deals with convergence in various contexts - maps, series, sequences and closes with a discussion of the integral in one variable. Part Three deals with various applications of the integral such as Fourier series and the Fourier integral. Part Four begins with the usual discussion of partial derivatives on $\mathbb{R}^{n}$ and this is followed by a general treatment of derivatives in vector spaces and a short discussion on differential equations. A similar pattern to that of Part Four is followed in Part Five which deals with multiple integration in that integrals in $\mathbb{R}$ are treated first and then integrals over parametrised sets are introduced.

There is a plentiful supply of exercises at the ends of sections with hints as to solution in many cases.

14.4. Review by: Donald Malm.
The American Mathematical Monthly 77 (8) (1970), 901.

The reviewer used this book as the text in a two-semester senior level course in analysis. There is much more material than can be covered in one year, which allows the teacher a great deal of choice of topics. As would be expected from this author, the selection of material is very good, presenting an interesting body of mathematics which forms a coherent whole.

The book starts with a review of calculus which, based upon our experience, would have been more effective if it were not so concise. After this review, the ideas of limit, continuity, compactness, series, etc., are developed in the context of normed vector spaces, which are allowed to be infinite dimensional. This is quite effective. The integral is defined for functions from the reals to a normed vector space, in the manner of Dieudonné, Foundations of Modern Analysis. This is the so-called "Cauchy integral". Functions which are uniform limits of step functions are integrable by this method, which is weaker than the Riemann integral. However, it is a very natural definition, and nicely prepares the students for the Lebesgue integral which they will meet later in their careers.

Unfortunately, these virtues are quite outweighed by a serious fault-seemingly, not enough care was taken with the writing and proofreading. Mis-prints are extremely abundant, some of them quite confusing to the students. In addition, there are a large number of errors. Often theorems and problems need more hypotheses than stated (for example Theorem 6, page 134, and problem 1, page 86). Occasionally a term is used which has not been defined (page 115 problem 12). A more interesting error occurs on page 35, where it is claimed that convergence of Cauchy sequences is equivalent to the least upper bound property.

The class felt frustrated by the large number of errors and misprints. If these were removed in a second edition the virtues of the book would have a chance to show themselves.

15.1. Review by: J S Fowlie.
The Mathematical Gazette 56 (395) (1972), 56.

The author has reorganised with additional exercises basic sections on linear algebra from some of his other books to fulfil the need for this subject at elementary levels. He intends this book to serve as a text for a short course after the freshman year in an American university following, or simultaneously with, the first course in calculus.

In the British context it could usefully be recommended to the mathematics specialist to be read before going to university. He will find much of the material familiar and will be ready for the generalisation to $n$ dimensions, and the treatment, supplemented by a plentiful supply of straight-forward exercises with answers, will prepare him for the university approach. The text is friendly and self-contained; starting with vectors and vector fields introduced in terms of coordinate geometry, matrices are then developed from properties of systems of simultaneous equations and related to linear mappings. A final independent chapter deals with determinants and inverse matrices.

For a paper-back this book is expensive; the undergraduate may find more economically produced British texts covering similar ground adequate.

16.1. Review by: A S G.
Current Science 40 (11) (1971), 306.

The present publication supersedes the author's previous book, Algebraic Numbers and includes a great deal of new material, especially the class field theory. The point of view in the treatment is principally global, and local fields are dealt with only incidentally. The subject-matter is in three parts: Part I: General Basic Theory Part II: Class Field Theory; and Part III: Analytic Theory.

The book will serve as a useful text on the subject for courses on modern algebra for graduate students.

16.2. Review by: A A Mullin.
Mathematics Magazine 44 (3) (1971), 163-164.

Modern algebra has two principal sources, both initiated during the first half of the 19th Century. One path leads through the Gaussian integers, cyclotomy, and the ideal theories of Kummer (Fermat's Last Theorem), Kronecker (his Jugendtraum), Dedekind, Hilbert, and E Artin, while the other path leads through the so-called double algebra (the complex field), W R Hamilton's quaternions and related hypercomplex systems, and nonnumerical systems such as Boolean algebras. The first path emphasises number theory with a transition from unique factorisation to non-unique factorisation and, finally, to some modified form of unique factorisation. The second path emphasises algebraic systems which do not satisfy various number axioms such as the commutative and associative laws. These two paths are not entirely distinct.

The book under review emphasises results on the first path including some recent developments. Clearly, this tome is not an undergraduate textbook; indeed, it will pro-vide challenging reading for "most" graduate students who fancy that they understand some Galois theory and some measure theory. Since exercises and problems are virtually non-existent, the monograph will be best utilised as (1) collateral reading in a graduate course on commutative ring theory and its applications, (2) a subject-matter guide for a graduate level seminar on global class field theory over number fields (rather than, say, over function fields), and (3) supplementary reading on the topics of functional equations and Tauberian theorems. The title of this concise little book is slightly mis-leading. A more accurate title might be Topics in algebraic and analytic number theory.

The book opens with discussions of prime ideal structure to include local rings (but without mention of the result that every regular local ring is a unique factorisation domain), integral closure (every unique factorisation domain is integrally closed), the Chinese remainder theorem, finite Galois extensions, and Noetherian rings and their relations to Dedekind rings. Completions are studied to include complete Dedekind rings. Attention is given to cyclotomic fields (including Gaussian sums and the law of quadratic reciprocity), since class field theory, which occupies the middle third of the book, has many of its proofs governed by the proof paradigms for cyclotomic fields. The basic propositions for ideles and adeles (multiplicative and additive constructions, respectively) are developed. As a preliminary to attacking class field theory, the author digresses to study elementary properties of Dirichlet series and $L$-series, going so far as the usual Eulerian product for Dedekind's zeta-function for a number field, but without mention of the analogous result for a function field in one variable. An interesting aspect of the author's development of class field theory is his use of little known results of Herbrand, who may be better known in mathematical logic for his deduction theorem. The last third of monograph is devoted to analytic number theory. Two proofs of the functional equation are given. One follows Hecke (the Poisson summation formula) and the other follows Tate (the adelic form of the Poisson formula). Convexity results are used to obtain a charming little proof of Hadamard's classical three circle theorem. Another novel feature of this Topica is its presentation of Tate's thesis, including a concise proof of the graduate student's nemesis, the Riemann-Roch theorem.

Whether the author is discussing the geometry of numbers, using the Cyrillic alphabet, or only punctuating a proof ("oh miracle!"), there is always a refreshing twist involved. To this reviewer, the author's use of analysis throughout the last third of the book seems to be a challenge to eliminate the need for analytical tools in otherwise algebraic results. In any case, this fine monograph is not for undergraduates unless they want sleepless nights.

16.3. Review by: George William Whaples.
Mathematical Reviews MR0282947 (44 #181).

This book should be of great value to modern students of the subject because it contains clear, complete, and readable expositions of important earlier results that are in some danger of being forgotten. The author includes an excellent brief history of class field theory and explains the various approaches to it.

17.1. Review by: Jack Price.
The Mathematics Teacher 65 (2) (1972), 146-147.

The "basic mathematics" referred to here is basic college mathematics, a necessary foundation for calculus, linear algebra, or other topics. Both manipulative and theoretical mathematics are dealt with through standard topics in algebra and geometry as well as through intuitive geometry (isometrics), coordinate geometry (some transformations), and miscellaneous units (mappings, and so on). The author has some interesting biases that show through in the treatment of most topics. Could be valuable for adults going back to school.

18.1. Review by: A Holme.
Nordisk Matematisk Tidskrift 21 (1) (1973), 43-44.

This book provides a first introduction to the theory of abelian functions and abelian manifolds. Here in Norway it would be excellent for a master's course in this subject. After reading this book, one can move on to more advanced works, the bibliography provides a selection of such. But in addition, one can also recommend this material to future mathematics teachers in higher education: the fruitful interaction found here between the theory of algebraic curves (or actually algebraic function fields in one variable) on the one hand, and complex function theory with Riemann surfaces and elements of algebraic topology on the other, is absolutely mind-bending, and provides a connection between different areas of mathematics.

The book opens with the Riemann-Roch theorem for an algebraic function field of one variable, including the proof of Hurwitz's formula. Then follows a short chapter on the Riemann surface of an algebraic function field, including a proof that algebraic and topological genus coincide. Without getting lost in detail, the author here gets the essentials across.

Next comes a chapter with proofs of the Abel-Jacobi's theorem, Riemann's relations, and the duality between the first homology group of the Riemann surface and the group of divisor classes of degree 0.

In the next chapter, we first get the basic definitions and properties of theta functions and abelian functions on a complex torus. Furthermore, the proof of the Riemann-Roch theorem, and projective embedding of a complex torus with a non-degenerate Riemannian form. For this, the correspondence between divisors and theta functions is not needed, so this can be proven in a final chapter, independent of the rest of the presentation.

But before this comes an introduction to duality theory, where one gets, among other things, the relations between complex, rational and $p$-adic representations of $End(A)$, where $A$ is an abelian manifold.

The book is highly recommended for all categories of mathematics enthusiasts, from graduate level onwards.

18.2. Review by: Gerhard Pfister.
Mathematical Reviews MR0327780 (48 #6122).

The book begins with a brief introduction to valuation theory and then presents Weil's proof of the Riemann-Roch theorem. The chapter concludes with Hurwitz's gender formula. The second chapter examines Riemann surfaces. First, the local uniformisation theorem is proven, followed by a discussion of the topology and analytic structure of Riemann surfaces. Among other things, it is shown that Riemann surfaces are connected, triangulable, and orientable. Finally, integration on Riemann surfaces is discussed, and Cauchy's theorem is proven. Building on this foundation, Chapter III deals with Abel-Jacobi's theorem (proof by Artin). Discussions of Riemann relations and Pontryagin duality complete the chapter. The next chapter presents the linear theory of theta functions (based on Weil's Bourbaki lecture from 1949). Among other things, the book shows that every equivalence class of theta functions contains a normalised theta function, introduces abelian functions, and proves the Riemann-Roch theorem for the torus. Furthermore, it presents Lefschetz's theorem on projective embedding by a non-degenerate theta function. An appendix illustrates the one-dimensional case. This chapter is followed by discussions of duality theory (dual abelian manifolds, the Tate group in connection with rational and p-adic representations, and Kummer pairing). Finally, the relationship between theta functions and divisors is examined. For example, it is shown that the positive divisors on the torus can be represented by theta functions on $\mathbb{C}^{n}$.

This book provides an excellent introduction to the theory of abelian functions. It requires little prior specialised knowledge, and the proofs are presented in a very detailed and easily understandable manner. This coherent presentation of the results is well suited for students or as a basis for lectures or seminars.

19.1. Review by: Noel J Hicks.
Mathematical Reviews MR0431240 (55 #4241).

This book supersedes another of the author's books [Introduction to differential manifolds, 1962] and is in the same spirit. The reader must be quite sophisticated, and, more specifically, comfortable with abstraction. The author's stated purpose is to fill a gap between advanced calculus and the theories of differential topology, differential geometry and differential equations. In the reviewer's opinion, one finds rather the foundations of global analysis. The chapter headings are (in order): Differential calculus, Manifolds, Vector bundles, Vector fields and differential equations, Differential forms, The theorem of Frobenius, Riemannian metrics, Integration of differential forms and Stokes's theorem. There is an appendix on the spectral theorem.

In case one is not familiar with the style of the author's earlier book, the first chapter of this new book is one the advanced calculus of Banach spaces and the first section is on categories, morphisms and functors. The treatment continues at this level, i.e., manifolds are modelled on Banach spaces, etc. There are no problems (other than filling in the things "left to the reader"), and few explicit examples.
...
Theoretically, the book is quite useful. Especially the chapter on vector fields and differential equations where the treatment runs cleanly and elegantly from local existence theorems through to global flows on manifolds and onto sprays, the exponential map of a spray and tubular neighbourhoods.

20.1. Review by: R P Gillespie.
The Mathematical Gazette 59 (407) (1975), 52.

This is designed by the author as a continuation of his well known text A first course in calculus. As might be expected from an experienced expositor like Serge Lang, it is clearly written and the style is informal and even colloquial, although the use of such expressions as "not very hip on theory" or "this jibes with the answer" may puzzle some readers on this side of the Atlantic. The author implies that this is not an advanced calculus course. Many important theorems, though very carefully stated, are given without proof. However, we have here a most useful account of mathematical methods for students who wish to use the techniques of such topics as partial differentiation, multiple integrals, Green's and Stokes' theorems and elementary Fourier series. Included are discussions on vectors, matrices, linear maps and determinants sufficient for the author's development of the calculus theory. At the end of each chapter there is an ample supply of illustrative exercises, and answers are provided at the end of the book.

20.2. Review by: V G Tikekar.
Current Science 43 (17) (1974), 565.

This book is a welcome arrival. It is nice to devote a separate book for calculus of several variables, because it is likely that this topic gets comparatively less space and hence cramped treatment when it is included in the same book in which calculus of one variable is presented.

This book is organised in four parts. Part one has 7 chapters and is devoted to mappings from numbers to vectors and vectors to numbers. It introduces the reader to vector preliminaries, differentiation of vectors, chain rule, gradient, potential function, curve integrals, higher derivatives, and maxima and minima including the Lagrange multipliers. Part two has 3 chapters, one each for matrices, linear maps and determinants. Part three has only one chapter that considers mappings from vectors to vectors discussing applications to functions of several variables. Multiple integrals, change of variables, Green's theorem, and surface integrals are studied in 4 chapters of the fourth part. This organisation helps the author to go little deeper, if necessary, in one part than is required, to read another part and similarly helps the reader to omit certain in-between portions without impairing his under-standing of later portions. For example, one can go to study the 12th chapter on multiple integrals immediately after the first chapter on vector algebra. The book has one appendix on Fourier series. Every section of each chapter has exercises and answers to selected exercises are included.

Study of vectors is basic to that of functions of several variables. The author has to be selective in the treatment of vector analysis as it is to be used only as a tool for the study of main concern of the book. As such scalar and vector triple products, reciprocal system of vectors do not find place in this book. The treatment of vectors is not geometry-dominant but lays algebraic foundations.
...
The treatment of topics in the second part is kept to the bare minimum. For example, only second and third order determinants are introduced. Part three is an illustration of the fact that "analysis profits from algebra, and conversely, the algebra of linear mappings finds a neat application which enhances its attractiveness". The chain rule for arbitrary compositions of mappings is established. Further the important Inverse mapping theorem, and Implicit function theorem are included. The treatment of part four is well structured.