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.
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"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
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53 of 53 people found the following review helpful:
5.0 out of 5 stars
A GEM OF A BOOK ON DIFFERENTIAL FORMS,
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This review is from: Advanced Calculus: A Differential Forms Approach (Hardcover)
I can't believe that nonone has ever bothered to review this book. This is an outstanding book, filled with valuable insights. The author introduces differential forms right from the beginning. He provides associations with the utility of differential forms in Mathematical Physics through many examples. There is a lot of material in this book that cannot be covered easily, so the prospective reader is advised to be patient and initially skip sections when necessary. This is a book about advanced calculus via differential forms written with great care by someone who has thought things through very thoroughly. The book has all the attributes of a classic:1. Excellent explanations and plenty of examples. 2. Conceptual clarity of key ideas, a rare feature these days. 3. Solutions of all the exercises in the book (truly a lot). 4. Rigorous but not terse mathematics. Having read this book the reader can easily proceed to address more advanced topics without hesitation. I personally would have liked to have seen more applications in mathematical physics but this is by no means a criticism. The author wrote a book that is about the concept of differential forms in advanced calculus and in that he has succeeded admirably. Apparently this book was first published in 1969 and has gone in and out of print over the last three decades. So hurry up, go out and buy this book.
48 of 48 people found the following review helpful:
5.0 out of 5 stars
Best book on Differential Forms for an Undergraduate.,
By "himog" (Albion, PA United States) - See all my reviews
This review is from: Advanced Calculus: A Differential Forms Approach (Hardcover)
This book is very well written, and being a reprint of a second edition, has very few misprints. Forms and their calculus are introduced in the 3D Euclidean space familiar to students of Calclulus 3 or University Physics. Later in the book the fundamentals like Stokes' Theorem are generalized to higher dimensions and manifolds. This book is suitable for anyone who has had Calculus 3 and is interested in a better way to do multiple-variable calculus. (Just to whet your appetite: after reading this book, you won't have to remember Green's Theorem, Stokes' Theorem, or the Divergence Theorem separately, they are all one compact and simple theorem {the one on the cover} in the language of Differential Forms.) I, personally, had a lot of trouble with Flanders' and Darling's books on differential forms until I read some outside material on the subject. This book, however, starts off on a more basic level, and does not demand half the mathematical maturity that Flanders or Darling do. For more advanced topics, Flanders and Darling are fine books to go to, after getting the basics right here.
45 of 45 people found the following review helpful:
5.0 out of 5 stars
A singularity in books on differential forms,
By
This review is from: Advanced Calculus: A Differential Forms Approach (Hardcover)
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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Inside This Book
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First Sentence:
The purposes of this chapter are to introduce the notation and the algebraic operations of constant forms, and to illustrate the sorts of mathematical ideas which are described by forms. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
function assigning numbers, given affine map, magnified range, pullback operation, arbitrarily small total length, oriented triangle, constant force field, oriented polygon, composed map, pullback map, approximating sum, homology basis, affine manifold, trichotomy law, oriented area, elimination theorem, solution manifold, oriented volume, differentiable surface, implicit function theorem, oriented counterclockwise, oriented curve, affine maps, composed function, planar flow Key Phrases - Capitalized Phrases (CAPs): (learn more)
Implicit Function Theorem, Cauchy Criterion, Fundamental Theorem of Calculus, Theorem Let, Lebesgue Dominated Convergence Theorem, Cauchy Convergence Criterion, Cauchy Integral Formula, Cauchy's Theorem, Inverse Function Theorem, Oxford University Press New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The purposes of this chapter are to introduce the notation and the algebraic operations of constant forms, and to illustrate the sorts of mathematical ideas which are described by forms. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
function assigning numbers, given affine map, magnified range, pullback operation, arbitrarily small total length, oriented triangle, constant force field, oriented polygon, composed map, pullback map, approximating sum, homology basis, affine manifold, trichotomy law, oriented area, elimination theorem, solution manifold, oriented volume, differentiable surface, implicit function theorem, oriented counterclockwise, oriented curve, affine maps, composed function, planar flow Key Phrases - Capitalized Phrases (CAPs): (learn more)
Implicit Function Theorem, Cauchy Criterion, Fundamental Theorem of Calculus, Theorem Let, Lebesgue Dominated Convergence Theorem, Cauchy Convergence Criterion, Cauchy Integral Formula, Cauchy's Theorem, Inverse Function Theorem, Oxford University Press New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 3
books:
4
books
cite this book:
- Episodes from the Early History of Mathematics (Anneli Lax New Mathematical Library) by Asger Aaboe on page 380
- Continued Fractions (New Mathmatical Library, Number 9) by Carl D. Olds on page 378
- The Theory of Functions by Edward C. Titchmarsh on page 403
- Clifford Algebras and Their Applications in Mathematical Physics, Vol. 2: Clifford Analysis by John Ryan on page 284
- Fundamental Probability: A Computational Approach by Marc Paolella in Back Matter
- Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide by George B. Arfken on page 319
- Symmetry and Economic Invariance: An Introduction (Research Monographs in Japan-U.S. Business and Economics) by Ryuzo Sato in Back Matter
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