The study of the basic elements of smooth manifolds is one of the most important courses for mathematics and physics graduate students. Inexpensively priced and quality textbooks on the subject are currently particularly scarce. Matshushima's book is a welcome addition to the literature in a very low priced edition. The prerequisites for the course are solid undergraduate courses in real analysis of several variables, linear and abstract algebra and point-set topology. A previous classical differential geometry course on curve and surface theory isn't really necessary, but will greatly enhance a first course in manifolds by supplying many low-dimensional examples in &#8477;n . The standard topics for such a course are all covered masterfully and concisely: Differentiable manifolds and their atlases,smooth mappings,immersions and embeddings, submanifolds, multilinear algebra, Lie groups and algebras, integration of differential forms and much more. This book is remarkable in it's clarity and range, more so then most other introductions of the subject. Not only does it cover more material then most introductions to manifolds in a concise but readable manner, but it covers in detail several topics most introductions do not, such as homogeneous spaces and Lie subgroups. Most significantly, it covers a major topic that most books at this level avoid: complex and almost complex manifolds. Despite the fact complex and almost complex manifolds are incredibly important in both pure mathematics and mathematical physics-they play important roles in both differential and algebraic geometry, as well as in the modern formulation of geometry in general relativity, particularly in modeling spacetime curvature near conditions of extreme gravitational force such as neutron stars and black holes -almost all introductory textbooks on differentiable manifolds vehemently avoid both. Part of the reason is the subject's difficulty once one gets past the most basic elements, which is considerable and requires sophisticated machinery from algebra and topology such as sheaves and cohomology. Another reason is that complex manifolds are important in both differential geometry and its' sister subject, algebraic geometry-and it's difficult sometimes to separate these aspects. By discussing only the barest essentials of complex manifolds,Mashushima avoids both these problems. This unique content usually absent in introductory texts and presented by a master makes the book far more valuable as a supplementary and reference text. Blue Collar Scholar is now proud to republish this lost classic in an inexpensive new edition for strong undergraduates and first year graduate students of both mathematics and the physical sciences.BCS founder Karo Maestro has added his usual personal touch with a preface introducing the student to smooth manifolds and a recommended reading list for further study. Matsushima's book is a wonderful, self contained and inexpensive basis for a first course on the subject that will provide a strong foundation for either subsequent courses in differential geometry or advanced courses on smooth manifold theory. This revised printing has several minor but important revisions in responding to several users' criticisms. Primary among these is significantly enlarged print-many readers complained about the tiny print size in the original. This has been rectified; the entire interior has been enlarged and carefully recropped. Also, several additional inexpensive references have been added to the Recommended Reading section. It is sincerely hoped these revisions will make this new printing of this edition of Matshushima's classic more user friendly and therefore an even better choice for an introductory graduate course in differential manifolds.

Differentiable Manifolds

Science & Math

Maestro, Karo

Matsushima, Yozo

Mathematics

Independently published

This classic and long out of print text by the famous French mathematician Henri Cartan, has finally been retitled and reissued as an unabridged reprint of the Kershaw Publishing Company 1971 edition at remarkably low price for a new generation of university students and teachers. It provides a concise and beautifully written course on rigorous analysis. Unlike most similar texts, which usually develop the theory in either metric or Euclidean spaces, Cartan's text is set entirely in normed vector spaces, particularly Banach spaces. This not only allows the author to develop carefully the concepts of calculus in a setting of maximal generality, it allows him to unify both single and multivariable calculus over either the real or complex scalar fields by considering derivatives of nth orders as linear transformations. This prepares the student for the subsequent study of differentiable manifolds modeled on Banach spaces as well as graduate analysis courses, where normed spaces and their isomorphisms play a central role. More importantly, it’s republication in an inexpensive edition finally makes available again the English translations of both long separated halves of Cartan’s famous 1965-6 analysis course at the University of Paris: The second half has been in print for over a decade as Differential Forms , published by Dover Books. Without the first half, it has been very difficult for readers of that second half text to be prepared with the proper prerequisites as Cartan originally intended. With both texts now available at very affordable prices, the entire course can now be easily obtained and studied as it was originally intended. The book is divided into two chapters. The first develops the abstract differential calculus. After an introductory section providing the necessary background on the elements of Banach spaces, the Frechet derivative is defined, and proofs are given of the two basic theorems of differential calculus: The mean value theorem and the inverse function theorem. The chapter proceeds with the introduction and study of higher order derivatives and a proof of Taylor's formula. It closes with a study of local maxima and minima including both necessary and sufficient conditions for the existence of such minima. The second chapter is devoted to differential equations. Then the general existence and uniqueness theorems for ordinary differential equations on Banach spaces are proved. Applications of this material to linear equations and to obtaining various properties of solutions of differential equations are then given. Finally the relation between partial differential equations of the first order and ordinary differential equations is discussed. The prerequisites are rigorous first courses in calculus on the real line (elementary analysis), linear algebra on abstract vectors spaces with linear transformations and the basic definitions of topology (metric spaces, topology,etc.) A basic course in differential equations is advised as well. Together with its’ sequel, Differential Calculus On Normed Spaces forms the basis for an outstanding advanced undergraduate/first year graduate analysis course in the Bourbakian French tradition of Jean Dieudonné’s Foundations of Modern Analysis, but a more accessible level and much more affordable then that classic.

Differential Calculus on Normed Spaces: A Course in Analysis

Cartan, Henri

CreateSpace Independent Publishing Platform

Blue Collar Scholar makes yet another wonderful out of print textbook available to everyone at a price everyone can afford with this outstanding linear algebra text. Murdoch's book was one of the first linear algebra textbooks that tried to give a broad presentation for well-prepared sophomores or juniors of average ability which was both mathematically careful and rich in applications. It's beautifully written, thorough and very well organized. This book emphasizes the relationship between linear algebra and the Euclidean and analytic geometry of the plane (and via isomorphism, the complex plane C) and three dimensional space.Murdoch completely avoids abstract vector spaces in the main text by limiting the scalar fields to R and C. It allows a careful presentation of the subject without recourse to abstract algebra concepts in a familiar context most students at this level should have at least a passing acquaintance with.Chapter 1 begins with vector spaces and their elementary properties, all strictly over the fieldsand C. No essential structure is lost in this manner that cannot be omitted in an introduction and the generalization to abstract fields is presented briefly in an appendix. Although the book's overall emphasis is on linear transformations, Chapters 2 and 3 give a virtually complete presentation of the theory and application of linear systems of n variables, n x n matrices and determinants. In a concrete presentation, this is the natural way to go because most applications and calculations in linear algebra are done via operations on matrices i.e. linear transformations with a choice of basis. However, this actual relationship between linear transformations and matrices is not explored explicitly until later, after a great deal of experience is obtained manipulating matrices. This furnishes a great many specific examples and provides experience with calculations. Chapters 4 and 5 develop classical geometry, inner products and orthogonality, relations (complements and projections) in real spaces .Chapter 6 finally defines and develops abstract linear transformations in low dimensions and the relationship to matrices under a change of basis. The critical theory of spectral analysis-i.e. the characteristic equation of a linear system and its solution producing eigenvalues as its roots and the resulting eigenvectors-is developed at length in chapter 7. It gives a detailed and very clear explication of the deduction of the Jordan form of a matrix-a critical topic many authors struggle to give an understandable but elementary discussion of. Chapter 8 gives applications of linear algebra to the analysis of quadratic forms, an old topic that usually isn't discussed much in basic courses anymore, but is quite important in both algebra and physics. The last chapter extends the previous results to complex vector spaces, where adjoints and Hermitian matrices come into play.In addition, Blue Collar Scholar founder and editor has added his usual personal 2018 edition preface, giving the historical background of this early textbook and the gradual emergence and evolution of the linear algebra course in American undergraduate mathematics programs. He has also added a substantial new Recommended Reading section, suggesting further reading in linear algebra for both teachers and students of this incredibly important subject.This careful, rich and inexpensive text is a fantastic choice for either a standard linear algebra text or self-study by undergraduates or graduate students in the physical or social sciences who need to learn linear algebra for either applications or the theory. The prerequisites are only high school algebra and geometry. A prior course in calculus, while helpful, is not really necessary. It should become a classic in rapid course and a popular selection for anyone and everyone interested in teaching or learning this beautiful and basic subject.

Linear Algebra For All Undergraduates

Murdoch, D C

This is a fairly standard level calculus textbook aimed at a first-year students. It was written by a master teacher at Massachusetts Institute of Technology whose calculus course there became nationally famous as a model for such courses before World War II. While this text focuses on applications and requires no more background then high school algebra and geometry, it differs from most standard textbooks, even of its contemporaries, in 2 major ways. Firstly, it’s clearly more comprehensive and sophisticated then most of those textbooks and covers a number of topics that are usually not present, such as basic vector algebra and geometry, conic sections, determinants, parametric equations, numerical integration and basic complex analysis of the plane. The 2 chapters on complex analysis in a basic calculus text are particularly noteworthy. The growing importance of complex variables in the physical sciences had become generally accepted during the early years of World War II due to its applications in hydrodynamics, engineering and electromagnetic theory. These additional topics are also indicative of the target audience, which were beginning mathematics and physical science majors at the Massachusetts Institute of Technology in the early 1940’s. Because they were preparing for careers in the technical fields, these students needed stronger and more diverse mathematical training for their future studies. Secondly, while not a rigorous mathematics textbook in the sense of real analysis or abstract algebra, it is certainly more careful then most calculus textbooks-either modern or classical-with many example calculations. For example, many limits and bounds are carefully computed with inequalities in the examples. Also, when available, Phillips gives a number of geometric proofs that are quite careful, particularly those with applications to physics and engineering. For example, a very clear geometric proof is given of the Squeeze Theorem. Indeed, in many ways, the working mathematical premises of the text appear to be a) focus on all tools and applications are that critical to the future training of physics and engineering students and b) Only give careful proofs of results when elementary methods using high school mathematics are available. No deep properties of the real numbers or topological properties are used beyond superficial use of the absolute value function. This outstanding textbook will help serious students of minimal background master calculus and lay the foundations for an in-depth study of the mathematical sciences.

Complete Calculus For Physics And Engineering (Old School Calculus)

Phillips, Henry B

For a number of reasons, the current basic calculus course is a mere shadow of what it was in previous generations. One of the main symptoms of this decline has been the essential automation of these courses where students no longer have to do any real thinking. This is because the course is built around exact answer problems in closed form, where calculators and computers can easily spit out answers. While a lot of mathematicians decry the loss of rigorous proofs in such courses, such as calculating limits precisely with inequalities, that's really just the most obvious loss. A more subtle loss has been the downplaying of the roles of algebra and elementary geometry that were so essential to pre-university training and played such critical roles not only in those calculus courses, but physics and engineering as well. Being able to analyze, set up and grind out lengthy computations forced students to not only understand calculus, but master the underlying preliminary skills they learned earlier far better. The republication of this wonderful collection by Petersen and Grasser, among other things, looks to expose today's students to the solutions of these kinds of problems to improve these skills. It also provides an excellent snapshot in a reasonably short volume of what the kinds of calculus courses we mourn the loss of consisted of and that the OSC Series is looking to resurrect. Petersen and Grasser isn't really a complete study guide to calculus, it's not complete enough for that. It doesn't have exercises for a student to work out nor detailed discussions of the topics each problem represents. What P&G does is provide a rapid overview of the subject through detailed solutions of problems representative of all the major topics that were typically covered in such a course when it was published. Richly diverse in its choice of problems with step by step detailed solutions and many visual illustrations, this inexpensive collection will become a treasured study supplement for both students and teachers of calculus at various levels, from introductory to advanced level.

Differential And Integral Calculus Through Problems and Solutions (Old School Calculus)

Petersen, G. M.

Graesser, R. F.

This book is an introduction to point set topology for undergraduates. Many of the classic textbooks on the subject cover the subject exhaustively and at the highest possible level of generality. The result of using traditional textbooks has been that students spend 2 semesters learning far more general topology on abstract spaces then most of them will ever need to use or know. More importantly, students get the impression from geometers and topologists in later courses that they “wasted” a year of their studies learning material that most mathematicians don’t even consider topology anymore. This leaves many of them feeling deceived and frustrated. Unfortunately, the reaction has been in recent decades to write elementary topology textbooks that only present the barest minimum of point set topology needed for students in advanced geometry or algebraic topology. Indeed-some recent beginning textbooks in topology largely skip general topology altogether and jump straight into algebraic and geometric topology such as homotopy, curves and surfaces! We believe this ludicrous solution is essentially throwing the baby out with the bathwater. This reissued edition of Hall/ Spencer should seriously be considered by mathematicians as the benchmark for such a course. The book contains what we believe to be approximately the irreducible minimum of point set topology any student of mathematics needs to learn regardless of level or interest. The book is quite detailed, covering sufficient general topology of interest and use for analysts, geometers and topologists. The book falls into two rather distinct parts. The first half is concerned with an introductory study of topological and metric spaces. The basic operations with sets are introduced in Chapter I, relations and mappings are discussed, and an introduction to infinite and uncountable sets is given. Chapter 2 introduces the basic topological structure of the real numbers in a review of basic analysis. In Chapter 3, general topological and metric spaces are introduced and such topics as compactness, separation and continuous functions are discussed. Metric spaces are pursued further in Chapter 4, with discussions of local connectivity, countability, metrizability and completion being included. The second part is less elementary in character. The long Chapter 5 is concerned with giving topological characterizations of arcs, simple closed curves, and simple closed surfaces. Peano spaces are discussed and the Jordan curve theorem and Jordan-Schoenflies theorem are proved. Chapter 6 discusses partitionable spaces, a topic often missing from modern texts. Finally, Chapter 7 discusses the axiom of choice, Zorn's lemma (in the form commonly called the Hausdorff niaximality principle) and the Tychonoff product theorem. The book in particular will help students understand the deep connection between general topology and real and complex analysis. The most natural path towards understanding abstract topological spaces, general continuous mappings and topological invariants on families of open sets is to see how they directly generalize the usual structures of analysis on the real line. Also. Blue Collar Scholar founder/editor Karo Maestro has added his usual personal touch to the new edition, with a new preface on his own reflections on point set topology and recommendations for supplementary or subsequent study. The prerequisites for the text are very minimal-just calculus and some experience with rigorous proofs. This wonderful lost text in this new inexpensive edition will serve a new generation of mathematics students who need to learn this crucial foundational subject with a presentation that’s both detailed and informative without being exhaustive. It will indoctrinate students into the beauty and simplicity of point-set topology and convince them of its’ intrinsic importance-primarily to analysis, but also to other areas of mathematics.

Elementary Topology: A Course in Point Set Topology

Hall, Dick

Spencer, Guilford

This wonderful textbook, written by one of the preeminent teachers and researchers of analysis of the mid-20th century, gives a deep and comprehensive presentation of undergraduate real analysis of one and several variables that is accessible to any student with a good working knowledge of calculus and some experience with proofs, such as can be provided by a non-applied first linear algebra course or discrete mathematics course. The book lies midway in difficulty between the very basic analysis texts i.e. “baby real variables” texts that present a first course in rigorous single variable calculus and hard-edged real variables courses set in abstract metric spaces like Rudin and Pugh. It is also very broad in coverage. The republication of this book for the first time in nearly 50 years will provide an excellent choice for either a course text or self-study in undergraduate analysis.Several aspects of the book’s unusual organization and content make it very deserving of low cost republication. Firstly, while it covers just about all the usual topics in any undergraduate analysis text-number systems, functions, limits of functions and sequences of one and several variables in ℝn , continuity, differentiation and integration of functions in ℝ, bounded sequences, metric spaces, basic point set topology, infinite series, power series, convergence tests, improper integrals, partial and total derivatives and multiple integrals- it has a number of unique aspects to the presentation that distinguish it from other textbooks. For example, a number of important concepts of analysis are covered in the starred sections and exercises that are not usually covered in these courses, such as point set topology, Riemann-Steijles integration, vector analysis and differential forms. Another excellent innovation that an entire opening chapter giving a far more detailed axiomatic description of the number systems without explicitly constructing them. While most analysis texts have such an opening section, Olmstead’s is longer and more detailed then the ones found in most books with many substantial exercises. Another positive quality of the book is its’ unusual midway level of difficulty. Calculus courses today are far weaker than they were when the standard textbooks such as Walter Rudin’s Principles of Mathematical Analysis were published. As a result, a number of students beginning analysis today need a bit more foundational training in rigorous calculus before tackling functions in Euclidean spaces and abstract metric spaces. So usually students have to begin with a “baby real variables” text before moving on to analysis on metric spaces. Olmsted does a fine job in his early chapters of presenting the properties of the real numbers and a precise presentation of calculus on the real line. This allows the first half of the text to act as a “baby real variables” book i.e. a bridge between today’s calculus courses and hard-edged classical analysis courses on metric spaces. As a result, students will only need one inexpensive text rather than two. Lastly, Olmsted contains “pragmatic” sections that discuss classical, more computational aspects of analysis that would be of great interest to applied mathematics, physics and engineering students. It’s clear that Olmsted’s book is an extraordinarily versatile textbook for undergraduate analysis courses at all levels. It will make a strong addition to the undergraduate analysis textbook literature and will be immensely useful to students and teachers alike as either a low-priced main textbook or as a supplement.

Real Variables: An Introduction To The Theory of Functions

Olmsted, John

Old School Advanced Calculus is exactly what the title says it is: A full year course in advanced calculus the way it was offered at all American universities until the 1970’s saw the sundering of the sequence into various “analysis for mathematicians” and “analysis for physical science students” courses. With the republication of this comprehensive, long-out-of-print text by Fite in a wonderfully inexpensive edition, the hope is to bring the advanced calculus course as it was taught for nearly half a century back into the consciousness of the 21st century mathematics and physical science students and educators. The main advantage of the original AC course, as exemplified by Fite, is a unified presentation of mathematical analysis comprised of virtually all the main topics of undergraduate analysis needed by both mathematics and physical science majors, covered using a uniform terminology and level of rigor. Even if each semester was taught by a different faculty member, they were both bound by more or less the same syllabus, which limited their ability to diverge from it drastically. When the subject selection, notation and rigor level is consistent throughout like it is with books like Fine’s, then a balance that benefits all involved is achieved and maintained in the entire course. Pure mathematics students get exposed to important physical and geometric applications along with mathematical rigor. Physics and engineering students get exposed to pure mathematics and the abstract minimalist deductive skills it builds in them that will be invaluable when they begin research. Fite, in particular, does a terrific job of combining a careful “epsilon-delta’ presentation of calculus of one and several variables with many applications to classical physics, differential equations and geometry. This book can be used for a number of different courses, either a standard classical advanced calculus course, an honors calculus course for strong freshman or independent reading by students or professors of analysis. Requiring only a year-long basic single variable calculus course as prerequisite, a course based on this book will give both the beginning mathematics major and serious physics or engineering major a thorough grounding in classical analysis and it’s many applications in preparation for further research in either real variables or mathematical physics. A lengthy new preface has been added by Karo Maestro explaining the history of the advanced calculus course in America and where Fite's book was groundbreaking as one of the first standard such texts. He has also added a recommended reading section reviewing many of the other standard classical analysis texts for additional reading.

Old School Advanced Calculus

Fite, William Benjamin

This brief and inexpensive text is intended to provide a modern introduction to vector analysis analysis in R2 and R3 to complement the very rigorous and wonderfully written presentation of classical analysis in my soon-to-be-published book, Old School Advanced Calculus by W'illiam Benjamin Fite. While this book is otherwise very comprehensive, the presentation of functions of several variables in it is purely analytic and rather archaic in nature. Fite is intended as a model of what the standard year-long advanced calculus course-which has largely been abandoned at most universities since the l980’s-would look like. Such courses were intended not onlv for mathematics majors, but serious physical science majors, for whom of course vector analysis is a necessary part of their mathematical training. Therefore, the absence of the differential and integral calculus of vector valued functions in low dimensional Euclidean spaces is a highly problematic lacuna in the book. The concurrent republication of this book by Miller is intended the rectify this. While the language of the book is classical in many regards, Miller is careful when possible to connect the material to modern formulations so he doesn’t alienate mathematics majors reading the book. The best examples are in the first chapter, where he carefully lays out century vector algebra using “arrows” while detailing their algebraic structure as a vector space over the real or complex numbers. This keeps the book’s intended audience very general, inviting not only mathematics majors, but physics, engineering and professionals in other fields that need to either review or learn this material. Also, most of the current standard books on vector analysis are rather expensive and lengthy. While Dover Books has made available a number of classical books on vector analysis at a very affordable price, many of these are quite old fashioned and may be difficult for students to read -either by itself or used in conjunction with another text or the instructor’s notes-will give students a very affordable option that’s still presented in a full modern context. The hope is that although the book is intended to supplement Fite, it can and should be used as a vector analysis text in its’ own right Indeed, the hope is that because of the book's brevity and low cost, it will become an indispensable study aid for students who need to either learn or review this material quickly and accurately.

Vector Analysis: A Supplement to Old School Advanced Calculus

Miller, Kenneth S.

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