Advanced Calculus: A Differential Forms Approach 1994th Edition by Harold M. Edwards (PDF)

Description

This book is a high-level introduction to vector calculus based solidly on differential forms. Informal but sophisticated, it is geometrically and physically intuitive yet mathematically rigorous. It offers remarkably diverse applications, physical and mathematical, and provides a firm foundation for further studies.

User’s Reviews

Editorial Reviews: Review “This book can serve as a delightful guide to advanced calculus, giving firm foundations to further studies.”–Acta Sci. Math”An inviting, unusual, high-level introduction to vector calculus, based solidly on differential forms. Superb exposition: informal but sophisticated, down-to-earth but general, geometrically and physically intuitive but mathematically rigorous, entertaining but serious. Remarkably diverse applications, physical and mathematical.” –The American Mathematical Monthly

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The first three chapters of this book are worthy of separate publication. They could be read by any bright undergraduate with full comprehension, and they introduce in a marvelously clear way the unifying power of forms defined on an ambient Euclidean space using basic examples from physics (work and flow).Throughout the author clearly demonstrates the need for mathematical rigor. Whenever he uses an informal example or argument, he will always conclude the section by analyzing why a rigorous argument is needed and often outlining how such an argument could be achieved. Later on in the book (in the sixth chapter) he will finally develop all the arguments rigorously in full depth.After this third chapter, however, the book starts becoming less elegant and more tedious. Linear algebra is discussed–but without any of the modern notation! Vectors are a rare character here, and matrices are scantly used other than to define ideas. Instead, you will be bombarded with a hoard of individual variable names. Keeping track of exactly what’s going on with all the variable names and summations becomes a task of mental endurance, not ingenuity or understanding. Some modern terminology is actually discussed, such as vector spaces and linear transformations, but not until the end of the chapter on linear algebra, effectively defeating the point. It is as if the material were tacked on just to make the book conform more to the standard content coverage.Note that here you will not find a k-form defined as a member of the k-th exterior power of the cotangent bundle of a manifold. Rather than such an abstract definition, this book is far more down-to-earth and hence will allow readers who do not have serious mathematical training to grasp the power and beauty of forms. Depending on your previous familiarity with forms and on your mathematical background, this is a plus or a minus. For me, it was certainly a plus because until recently the abstract definition I provided above was meaningless garbage to me.Overall, this is a book that would be best thumbed through at a book store so you can decide if it’s worth your time and if the author’s style meets your taste. It’s a very well-written book with plenty of fresh insights and a novel approach. Mistakes are nearly impossible to find. The author has a powerful and humbling command of mathematics. Unfortunately, the notation was often too outdated for my taste and hindered not only my enjoyment of the book but also my ability to fully understand concepts that appear difficult here because of the onslaught of symbols but which are really rather straightforward in modern notation. But I suppose some people may prefer the different notation.

⭐The differential forms approach has considerable intuitive appeal as well as capturing more useful math for the physics or engineering student than the conventional approach. Edwards is a little too much the mathematician. The text misses the mark for the typical physics or engineering student who has taken only the usual calculus sequence and needs a little more intuitive introduction and to be led into the abstraction more gently. A more geometric approach might have been useful. I would have introduced the wedge product explicitly with a geometric explanation in terms of vectors.My objective in purchasing the book was to fill in my background on the subject the easy way after pretty much figuring out what it is all about. For that the book is fine. But back in 1959 when I took advanced calculus, I think I would have found the books difficult without a good teacher to help me along. The book is probably not what I’ll use for a course.

⭐Well written from the author’s point of view. He also has a fabulous book on the Riemann Zeta function, very scholarly. This calculus book offers a certain consistent abstract approach suitable for honors courses.

⭐I bought this book based on some very positive reviews I read here on amazon.com. I regret it. Deeply. The book’s approach and notation are sorely outdated and, in order to avoid rigorous arguments, the author will make u wade through a bunch of ‘intuitive’ arguments that should supposedly enlighten you. They honestly do not do that. They are more likely to confuse you about the ideas which you may have mastered elsewhere in more rigorous books.Famous books are famous for a reason. This book wasn’t one and yet, following the few recommendations I read here, I invested both Time and money in it. Now I regret it. If there is a way for me to get my money back, I would not hesitate to go for it. Unfortunately that would probably mean I will have to sell this book to another fellow mathematician.I won’t do that though. For its unethical.

⭐This is a great text, maybe one of the best ever written. Self-contained, non- intimidating, accessible.C.H. Edwards also wrote on Advanced Calculus Book that is a masterpiece.

⭐An excelent Book for Researchers

⭐Edwards wählt in diesem Buch einen völlig unkonventionellen Einstieg in die Analysis.Ich habe dieses Buch neben einer speziellen Vorlesung (Analysis 1), die sich auf das Buch bezog, gelesen.Es ist angenehm geschrieben und auch für Erstsemester (trotz der englischen Sprache) problemlos zu verstehen. Die einzelnen Kapitel bauen logisch aufeinander auf und der Leser wird nicht “abgehängt”, wenn er das Buch konzentriert von Anfang an durcharbeitet.Hinweisen möchte ich allerdings, dass der Inhalt des Buches wohl mit den wenigsten Analysis-Einsteiger-Vorlesungen an deutschen Universitäten übereinstimmt. Als vorlesungsbegleitende Literatur sind da sicherlich andere Werke deutlich besser geeignet, zumal auch die typischen Arbeitsweisen (Beweise etc.), die gerade zu Beginn des Studiums unbedingt erlernt werden müssen, kaum thematisiert werden.Für jeden, der die Analysis aber mal auf einem anderen Weg kennenlernen möchte, ist das Buch sehr zu empfehlen. Man erlangt ein ganz anderes Verständnis, manche Problemstellungen werden auch deutlich eleganter gelöst, als dies normalerweise der Fall ist.Einen Punkt Abzug gibt es für die Übungsaufgaben: Es sind zwar einige am Ende der Kapitel zu finden (die sicherlich auch für den Anfang ausreichen), die Lösungen fallen aber äußerst knapp aus, sodass sie ohne entsprechende Hilfen (Vorlesung etc.) nur schwer zu verstehen sind.

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