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Summary: At last - A great book on elementary mathematics
Review: This book is unique in several ways: it covers an immense amount of material, much of which is never presented in books aimed at this level. The underlying idea of the authors is to present constructive proofs, which has the great benefit of providing the reader with the ability to actually compute quantities appearing in the theorems. As an example, the Inverse Function Theorem is proved using Newton's method, which relies on Kantorovich's Theorem, and thus actually gives an explicit size of the domain on which the inverse exists. The book also contains a very nice section on Lebesgue integrals, a topic which is usually reserved for graduate level courses. The construction is to my knowledge completely new, and does not rely on sigma-algebras, but utilizes only elementary mathematics. Another nice feature is that the book considers abstract spaces at an early stage. Thus the reader is presented with the idea of computing derivatives of functions acting on e.g. matrix-spaces, as opposed to the usual Euclidian spaces. The concluding treatment on differential forms brings a lot of the introduced ideas together and completes the picture by a thorough treatment on integration over manifolds.This book can be studied at several levels. For a first year honours course, one may skip the trickiest proofs, which appear in the appendix. More advanced readers may choose to study constructions and details of selected theorems and proofs. Anyone who buys this book will have a solid companion for many years ahead.
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Summary: Hubbard's unified approach
Review: This book is very helpful in the way that it describes the jist of a theorem or idea in simple terms before stating the actual theorum. Then the theorem is usually stated in a generalized form, which would be difficult to understand without the introduction, but is ideal for people who really want to think about the topic in more depth. The book is a wonderful combination of explanations using simple terms and a presentation of the multivariable and linear algebra concepts in a more rigorous mathematical sense.
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Summary: Students Hate It
Review: This book, by the esteemed Cornell University professor John H. Hubbard, is nearly incomprehensible to the average Cornell student. There is heavy reliance on convoluted notation, there are no sample problems done for each major concept of the book. All in all, while Hubbard's method for teaching this course may be very strong, the textbook itself is a student's nightmare.
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Summary: notation
Review: this guy tells u a story before he gives you another story in his proof of theorem, he notation is very hard to follow. His proofs are ugly. is there a better advanced calculus book?
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Summary: amazing book!
Review: This is one of the best written Math books I have seen. The authors write in a clear and engaging style which makes the reader understand the beauty of Math. After you read this text you can put this on your bookcase beside other classics such as Spivak's Calculus. Read this book-study it- and enjoy. Let's hope that the sequel (Part 2) will appear in the near future.
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Summary: If only I had had this...
Review: when I took Hubbard's Math 223-224 classes. I was one of the original group of students when the manuscript and class were being tried out. I must say the class was excruciating, even more than the students (who wrote the reviews below) could imagine. The primary source of pain was the incomprehensible manuscript. There were not only typos all over, but the layout was not as nice as in the published version. Actually, the book was changed _a lot_. The harder proofs in the appendix used to be in the main body of the manuscript, and the original appendix had even harder proofs that were cut out eventually. The published version is great. I've looked through most of it over the last few years, admittedly from a more advanced viewpoint (no, I didn't buy it, Hubbard gave us free copies). It's very lucid, and the intros to new concepts provide good motivations. I suspect Barbara Hubbard had a great deal to do with how readable it is; she deserves a good deal of credit. I say this, because John Hubbard himself is incomprehensible. His lectures, while sometimes entertaining, were so dense that no one could follow them. The unique aspect of the book is the 'unified approach.' This works very well at showing the interconnectedness of mathematics. I also like the fact that it is a useful reference. Of course, most of the theorems are proved in the context of Euclidean space, but it is not hard to see how to generalize it. This is not an easy book though. I found a reviewer's comment that the book is 'incomprehensible to the average Cornell student' very funny. Any math book would be incomprehensible to the average student, whether at Cornell or not. But one should keep in mind that this book is used for the second year honors calc sequence. It is very 'meaty' and not to be delved into lightly. But compared to other books of the same standards, it holds up well. I give it four stars, or maybe up to four and a half.
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