Advanced Calculus: A Differential Forms Approach
This book is a highlevel 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.
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Book details
 PDF  508 pages
 Harold M. Edwards(Author)
 Birkhäuser; 1994 edition (January 5, 1994)
 English
 5
 Science & Math
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Review Text
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 discussedbut 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 kform defined as a member of the kth exterior power of the cotangent bundle of a manifold. Rather than such an abstract definition, this book is far more downtoearth 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 wellwritten 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.
This is a great text, maybe one of the best ever written. Selfcontained, non intimidating, accessible.C.H. Edwards also wrote on Advanced Calculus Book that is a masterpiece.
An excelent Book for Researchers
I heard that Richard Feynman used this calculus book. Really awesome !
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.
If you're looking for theorem, proof, exercises type of a textbook this book is not for you.However, you will love this book1. if you are looking for new insights into what you thought you already knew,2. if you want to learn multivariable calculus from a new, different, and deeper viewpoint.3. if you not only want to learn/teach multivariable calculus proofs but also want to understand/teach how the proofs were developed. (A typical proofs starts with rough geometric idea of a proof, then little more formal and finally the rigorous proof.)The book has number of exercises but unfortunately all of them have solutions at the end which makes them unusable for homework assignments.To teach from this book, try the flipped mode i.e. give reading assignments as homework and do problems in class.
Reading this book reminds me of "Feynman Lectures in Physics" : An extremely refreshing view of analysis.The author's point of view is that the theory of functions of multiple variables is very naturally understood if approached from the differential forms angle. And that the best public for that is the undergrad student. Well, he makes his case.The book is not written in the usual math style (theoreme,lemma,proof,...) and always exhibit the beauty behind the idea. In the first chapter, forms are introduced very naturally with example taken from work, flows and so on. Chapters 2 and 3 are devoted to integrals, integration and differentiation and that's where he unleashes all the power of forms before you notice it. From the fundamental theorem of calculus ($\int_a^b f(x)dx=F(b)F(a)$) he deduces the general stokes theorem on integration on manifolds and show why the exterior derivative is defined as it is. Chapter 4 talks about linear algebra, again demystifying the implicit function theorem when exetended to differential maps (chap5). Chapter 6 is where everything get prooved rigourously. Chapter 8 is a real gem, showing various application of forms. There are classical applications such as the integrability conditions, Maxwell Equations and special relativity. And very original ones such as revisiting harmonic functions and functions of complex variables.I wonder why this book is not taught as a classic textbook everywhere.
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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.