Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition by James J. Callahan (PDF)

Description

With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones–the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes–the text treats other important topics in differential analysis, such as Morse’s lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. The book incorporates modern computational tools to give visualization real power. Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books. This is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses–including topics in geometry. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.

User’s Reviews

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

⭐As I continue on the tour of “five star” books I purchased recently for my research library, some texts really stand out above others: some for their mileage (usuage) compared with peers, like Rudin; some for their prose, like Janich; some for their organization, like Strang; and some for their originality of approach, like this text.In mathematics, you see things like “A Geometric Approach” so often without any real reason to back up these texts’ named approaches any more than the generic takes of their peers. This is not the case with Callahan.Encyclopedic in depth and approaching senior-level, if not graduate leveling work in some optional places like Morse’s Lemma, Advanced Calculus is the flagship to the modern approach to the topic in the vein of those old Ad Cal texts by Sternberg and the Edwardses. Plenty of pictures accompany the illustrations, which serve the reader as much as the well-written (and relatively bug free for a Springer first) text.Advanced ideas not appearing in some Ad Cal texts, like Morse’s and the Implicit/Inverse Function Theorems, abound, as do neat “tricks” that don’t even show up in some well-regarded baby real analysis texts; check out his slick insertion of push-forwards into change of variables for integration, for instance. This part, when I finally got to it, at last paid the price of admission.I feel like Shurman’s Ad Cal book (Calculus in Euclidean Space) has a better section on Differential Forms, despite its dismissive take on the topic from the outset. Bressoud’s “Second Year Calculus” is more suitable for introductory self-study for those utterly unfamiliar with differential geometry, for an honors Calculus III course, or for an Intro to Proofs course. Bruono is superior with descriptions of the tangent plane, Walschap is more oriented toward pure differential geometry at this level, Hubbard-Hubbard or John is better for review, and O’Niell or Munkres serve best as more to-the-point second semester baby analysis based in manifolds, or Thorpe for an analysis style based in differential geometry in the old sense (everyone can feel free to stop Rudin prior to his Differential Forms chapter, though, as is Pugh’s rushed approach to Differential Forms). The recurring nightmare of Jordan measure is present here for no reason other than time-saving as is par for the course.This is not a perfect book by any means, but it tops all the rest in its clarity and breadth. It belongs on eveery starting mathematician’s bookshelf, in my opinion, even if a trimmer book was used for your own Ad Cal or Analysis II course. Pick up Callahan for the coverage and excellent, unique summary; choose whatever you need beyond that for your depth, or if you’re interested in Algebra or some other non-geometric area, feel free to just stop here and enjoy.

⭐Professor Callahan makes a point of emphasizing the vastness of Advanced Calc as a subject. You could easily spend two solid years studying it (Advanced vector calc, Differential geometry, Manifolds, Differential forms ) and not pass the same way twice.He has a made a lovely book that snags your attention with clever integrals and leads you onward. Great illustrations are provided that are essential to this subject and really help to make it great for autodidacts (which we all are at some point). He must be a superb teacher. Prof. Munkres has written a very good book: Analysis on Manifolds that is worth reading after this one. Enjoy.

⭐Extremely interesting but not for the faint of heart. Great visual insights into calc theorems.

⭐The emphasis is on intuition but the author does not shy away from advanced concepts.The many diagrams and graphs have practical reasons to be there as they help with understanding. The examples that are given in order to motivate the reader do a great job at it.There are also some proofs whenever the proof is not very big. There are also proofs when the proof directly helps with the understanding.The writing style is also great, mixing a formal type of writing with a more conversational style.You can find this book at a very low price(and it’s also hardcover!). It is of great use to physicists and mathematicians in order to REALLY understand and reinforce your intuition behind these concepts.Also, the author states in the preface that the Feynman Lectures were a great influence due to the way that Feynman explained everything! So, this is a strong indication of an author trying(and succeeding) to write a book on mathematics using that very powerful way of thinking and explaining.Two things that I did not like were:1) The higher dimension derivative is just motivated through the Taylor series of many variables. Books like Marsden/Tromba’s don’t even try to motivate it. But, I expected more from this book. I mean, a student coming from only a one-variable calculus background would feel a bit overwhelmed while trying to figure out why the derivative in more dimensions is a matrix and why it is the way it is! I figured it out, but I only did so by hard thinking and referencing other books(like Colley’s books and Hubbard’s book). But, again, I have skipped some chapters, so I am not very sure that the author did not explain the derivative in higher dimensions.2)I certainly expected to gain more geometrical intuition from the section on the chain rule. Sure, no other book explains the geometrical meaning of the chain rule, but this book promises to give the geometrical meaning behind everything that it presents. Inexcusable.But, again, these points can’t make me give this book anything less than a 5-star rating because it’s so unique, helpful and well-written. Any book that concentrates on intuition and motivation without sacrificing essential things is a king in pedagogy.

⭐The emphasis is on intuition but the author does not shy away from advanced concepts.The many diagrams and graphs have practical reasons to be there as they help with understanding. The examples that are given in order to motivate the reader do a great job at it.There are also some proofs whenever the proof is not very big. There are also proofs when the proof directly helps with the understanding.The writing style is also great, mixing a formal type of writing with a more conversational style.You can find this book at a very low price(and it’s also hardcover!). It is of great use to physicists and mathematicians in order to REALLY understand and reinforce your intuition behind these concepts.Also, the author states in the preface that the Feynman Lectures were a great influence due to the way that Feynman explained everything! So, this is a strong indication of an author trying(and succeeding) to write a book on mathematics using that very powerful way of thinking and explaining.Two things that I did not like were:1) The higher dimension derivative is just motivated through the Taylor series of many variables. Books like Marsden/Tromba’s don’t even try to motivate it. But, I expected more from this book. I mean, a student coming from only a one-variable calculus background would feel a bit overwhelmed while trying to figure out why the derivative in more dimensions is a matrix and why it is the way it is! I figured it out, but I only did so by hard thinking and referencing other books(like Colley’s books and Hubbard’s book). But, again, I have skipped some chapters, so I am not very sure that the author did not explain the derivative in higher dimensions.2)I certainly expected to gain more geometrical intuition from the section on the chain rule. Sure, no other book explains the geometrical meaning of the chain rule, but this book promises to give the geometrical meaning behind everything that it presents. Inexcusable.But, again, these points can’t make me give this book anything less than a 5-star rating because it’s so unique, helpful and well-written. Any book that concentrates on intuition and motivation without sacrificing essential things is a king in pedagogy.

⭐This is one of my favourite books in mathematics. Not only does it present calculus and real analysis from a geometric point of view, but I think the book does an excellent job in constructing a natural bridge between basic courses such as Real Analysis, Multivariate Calculus and more advanced courses such as Topology, Differential Geometry, Calculus on Manifolds, etc. All students dealing with higher level mathematics will benefit from this book.

⭐The content is great, very well-written by the author. One star off because of the publisher, I agree with one previous comment about the binding. I wish there could be more space in the middle between the two pages. I have to press my fingers hard on the page in order to make the texts easy to read without bend over my head to see the full length of the line. It is a quite heavy book.

⭐O autor apresenta o assunto de uma maneira bastante original, sem dúvida é uma leitura bastante agradável. Os exercícios são bem dosados, os exemplos esclarecem a teoria exposta.Excellent book. It stands out from other calculus textbook by its more intuitive, geometric approach without lacking rigour. A good addition is the introduction of differential forms.

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Free Download Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition in PDF format
Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition PDF Free Download
Download Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition 2010 PDF Free
Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition 2010 PDF Free Download
Download Advanced Calculus: A Geometric View (Undergraduate Texts in Mathematics) 2010th Edition PDF
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