Karo Maestro

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.

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.

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.

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.

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.

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.