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Advanced Calculus: A Geometric View
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.
542 pages, Hardcover
First published September 1, 2010
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84 people want to read
About the author
James J. Callahan
7 books5 followersJames Callahan earned a B.A. from Marist College and a Ph.D. from New York University.
In 1975, Professor Callahan received the Lester R. Ford Award of the Mathematical Association of America and at Smith he has received the Faculty Distinguished Teaching Award and the Sears–Roebuck Foundation Award for Teaching Excellence and Campus Leadership. He has made research trips to England and France.
Callahan was the director of the Five College Calculus Project (funded by the National Science Foundation), and co–author of Calculus in Context (W. H. Freeman & Co, 1995). He wrote Geometry of Spacetime (Springer–Verlag, 2000) an undergraduate text in mathematics about relativity, as well as Advanced Calculus: a Geometric View.
Professor Callahan's interests include: geometry, dynamical systems, chaos and fractals, catastrophe theory, relativity, most areas of applied analysis and building things.
In 1975, Professor Callahan received the Lester R. Ford Award of the Mathematical Association of America and at Smith he has received the Faculty Distinguished Teaching Award and the Sears–Roebuck Foundation Award for Teaching Excellence and Campus Leadership. He has made research trips to England and France.
Callahan was the director of the Five College Calculus Project (funded by the National Science Foundation), and co–author of Calculus in Context (W. H. Freeman & Co, 1995). He wrote Geometry of Spacetime (Springer–Verlag, 2000) an undergraduate text in mathematics about relativity, as well as Advanced Calculus: a Geometric View.
Professor Callahan's interests include: geometry, dynamical systems, chaos and fractals, catastrophe theory, relativity, most areas of applied analysis and building things.
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Displaying 1 of 1 review
August 22, 2018
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 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.
Displaying 1 of 1 review