Advanced Calculus (Revised Edition) by Lynn Harold Loomis (PDF)

Description

An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960’s. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year’s course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.

User’s Reviews

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

⭐Of the countless number of texts entitled “Advanced Calculus”, this is probably the most mathematically rigorous and challenging. Despite the title, this is a classic textbook written by two of the 20th century’s preeminent mathematicians on the topic of abstract analysis in Banach spaces for differential calculus (the so-called Frechet derivative) and smooth manifolds for integral calculus (as the setting for the generalized Stokes theorem). Moreover, the authors provide a large amount of abstract linear and multilinear algebraic background, in order to fully develop these concepts, so this text serves as an general introduction to many important concepts and techniques of modern mathematics like duality and natural isomorphisms. That said, the writing is engaging and clear, and the text is self-contained and assumes only undergraduate calculus and linear algebra. However, only a reader with a strong ability to appreciate mathematical abstraction will enjoy and benefit from this text, and the level of mathematical maturity assumed is much higher than the formal prerequisites suggest. Unless one is exceptionally gifted and able to quickly assimilate abstract concepts, a solid course in one-variable real analysis is probably the minimal requirement for using this text (for example, the first 8 chapters of Rudin’s Principles of Mathematical Analysis); the level of mathematical sophistication reached by the end of the book is such that it puts the student at a good spot to pursue graduate courses in modern analysis. Indeed, the text was written by Loomis and Sternberg for an “experimental” two-semester course aimed at extremely capable and motivated freshmen (yes, all male at the time; Harvard became co-ed in the late 70s) to allow them to quickly approach graduate level courses after finishing the class. The result was Harvard’s Math 55, a course that has now become famous for eating USAMO and IMO winners and spitting out wall street bankers mostly but quite a few preeminent math professors as well.Tl; dr: don’t buy this book if you just wanted to learn some “practical” (in the sense of a career in STEM) multivariable calculus, as the title might misleadingly suggest is covered in this book; this book would be useless and baffling. However, if you have the gift to be a mathematician, whether you’re an undergraduate or even a high school student “with the right stuff”, this book is strongly recommended to test and strengthen your ability to grasp abstract mathematical concepts in a context that does not initially assume much knowledge but ends up teaching a significant amount of 20th century analysis (although again, to gain the most out of it, I would suggest learning some one-variable real analysis first or at least concurrently.)Comparison to other texts: Spivak’s “Calculus on Manifolds” is a significantly less sophisticated textbook when it comes to the differential geometry content, but its poor pedagogical qualities make it almost as challenging, despite its much shorter length and less ground covered. Loomis and Sternberg’s treatment is authoritative and is better worth the substantial effort required to understand either text. (Although for historical interest, every budding mathematician should be aware of Spivak’s treatment of Stokes’ theorem.) For a gentler introduction than either of these choices, Munkres’ text (“Analysis on Manifolds”) is probably the best, requiring less mathematical maturity / time and effort, while still allowing the development of some intuition on the topic (albeit mostly confined to R^n).

⭐I don’t think that you need to be a “will-be” mathematician or some kind of genius to understand this book like some comment says, because I managed to get much out of this book. Although I did give up reading this book several years ago. Thanks to Russian analysis books, Especially Zorich’s book, I finally reach the math sophistication this book actually required. I gave Zorich’s book 5 stars, so I can only give this book 4 stars. Also the printing quality is not so great, on some pages the characters are not very clear.

⭐The contents is fine, although less mathematically inclined readers (like myself) will find the book by R. C. Buck,

⭐, a better option. What bugs me most is the horrible printing quality, with grayish letters and symbols and the failures in printing. I don’t recommend this edition; look for a used copy instead or just plain xerox copy the parts of the book that you want or need from the nearest library — the final quality of your copy will be superior.

⭐I only wish I had read this book early on in my undergraduate career. This book presents linear algebra (a chapter on generalities that hold for all vector spaces, one specializing to the finite dimensional case, later on one about scalar product spaces, and one for multilinear algebra) and multivariable calculus (the differential of maps between normed vector spaces, a quicky on existence and uniqueness theorems for differential equations, and integration of differential forms on manifolds) in a way that makes clear the underlying simplicity and coherence of the subjects. In addition there are two chapters on additional topics (potential theory and classical mechanics) that provide a valuable taste of these important topics for further study.

⭐Helpful

⭐good

⭐This is a beautifully done reprinting of a famous classic. It has been referred to time and again in many well known math and physics books, and its content is as modern and relevant as anything on the market. It is a must have. — Gary Walker, Ph.D-physics

⭐I’m falling in love

⭐In my efforts to improve my understanding of maths, I bought this book. It is well written and presented. I have to read a few pages at a time as the concepts are not simple, but I am slowly getting through it.This book cannot be classed as simple maths, but it sets out the foundations for getting into the deeper mathematics one needs to understand to appreciate the world of physics.

⭐1 star, wrt formatting of Kindle version only: equations can’t be scaled. they remain tiny no matter what you do. Content is probably great but the equations can’t be read

⭐tutto OK

⭐

⭐Excellent book on “Advanced Calculus” and very glad it is available in print.

⭐Poor quality and content ,,,confusing ,,,

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