Advanced Calculus (Dover Books on Mathematics) by H.K Nickerson (PDF)

Description

“This book is a radical departure from all previous concepts of advanced calculus,” declared the Bulletin of the American Mathematics Society, “and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics.” Classroom-tested in a Princeton University honors course, it offers students a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives — gradient, divergent, curl, and exterior — are obtained from it by specialization. The corresponding theory of integration is likewise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes formula. The text concludes by applying these concepts to analytic functions of complex variables.

User’s Reviews

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

⭐Back in the 50’s and 60’s, it seems like math professors at the Ivies were competing with each other to design the most challenging undergraduate courses and textbooks. While Loomis and Sternberg wrote their “Advanced Calculus” for the infamous freshman course Math 55 at Harvard, Nickerson, Spencer, and Steenrod wrote “Advanced Calculus” for Math 303 and 304 at Princeton. While it’s unclear whether the audience for this course is composed of sophomores, or perhaps even freshmen, this text exposes them to a healthy mix of linear algebra, differential equations, vector calculus, topology, differential geometry, and algebraic topology.I would argue that Loomis and Sternberg (LS) still takes the cake for being the hardest textbook called “Advanced Calculus”, particularly in the differential geometry portion. However, Nickerson, Spencer and Steenrod (NSS) certainly presents the most “advanced” mathematics.However, though spoken of reverently in hushed tones, NSS should not be anyone’s primary textbook these days (or even back then): 1) The exposition is minimal — think Rudin, but with much more advanced and abstract subject matter. Only definitions, propositions, theorems and proofs, with only the rare remark peppered in. The motivation and background given for all these abstract definitions is appallingly little. 2) The whole text is written in typewriter. While hipster-chic, reading this text rapidly makes ones eyes tired, with all the awkwardly placed subscripts and subsubscripts, uniformly sized characters, and lack of italicized or boldface letters. 3) While explicitly claiming the opposite in the preface, the choice of topics and their order seems to lack unity and it’s unclear why some topics were chosen.It’s not the difficulty of the topics that makes this an unsuitable textbook per se, but the fact that this is really a bound set of lecture notes rather than an actual book. Thus, all the background and motivation is left to the instructors rather than the text. This is in contrast to LS, which is a wonderfully well-written book, difficult though it may be. If you’re choosing between LS and NSS, I would certainly get LS first.Still, if you’re a math-nerd like me, or a mathematician, this classic textbook occupies an important position in the history of mathematical pedagogy (like Spivak’s Calculus on Manifolds) and is a must-have, especially at Dover’s low, low prices. If one is already familiar or somewhat familiar with much of the topics, it’s really cool to see how so much math can be crammed into one textbook!

⭐I still have my original 1959 spiral bound copy and got the kindle for traveling. The number of errors in transcription is quite large. Some of the errors are unimportant, but most destroy the meaning and make it difficult for someone not familiar with the material. Also to keep the same spacing between lines, some of the equations, e.g., those with summation signs, are in tiny unreadable print.Kindle should be ashamed of the quality of this version

⭐Good product. Good deal.

⭐It has been a while since I revisited these notes. The 1959 edition is typescript, published by Van Nostrand. The notes originate from an “honors” course at Princeton University. Totalling 540 pages, framed in “definition-theorem-proof” format. There is more: “remarks” are interjected into the discussion, highlighting various issues and exercises conclude each section. Mathematical proof, utilizing method of induction, is highlighted. Beware, no subject index. Read each page carefully and sequentially (that is linearly, avoid skipping around or ahead).(1) The final chapter is a big payoff: complex structure (pages 459-540). We read: “to show that the tensor calculus of the preceding chapters (1-12), especially the calculus of differential forms, has a complex analogue.” Recall “unitary as analogue of orthogonal” (page 466). A pedagogic device: analogy is invoked often. The material is complementary to Chern’s chapter seven (Lectures on Differential Geometry).(2) Chapter Eight (pages 169-189), vector-valued functions of a vector, required reading for mathematics and physics students. We read: “In the procedure we have followed, it is made clear that a vector-valued function has but one derivative, whose values are linear transformations.” (page 184).(3) Chapter ten will be challenging: topology. It is interesting to compare the topology content here to the topologic material in Advanced Calculus (by Loomis-Sternberg), as I consider it to be a companion text to this one (still, read Chinn and Steenrod !). You will find the notion of convexity introduced in an exercise #5 (page 323, it is a notion which is revisited later). Inverse function theorem is the goal of the chapter of topology, it is a sprawling, lucid, discussion (pages 302-322).(4) Chapter eleven will be differential calculus of forms. Learn of star-shaped region and its relation to convexity (page 377). The co-derivative defined (page 397): “the explicit computation of which, in terms of coordinates, is not so simple.” Spherical and cylindrical coordinates are utilized in multiple exercises (page 404). Integration, next: simplex, chain, boundary, cycle, homology. Duality is central (page 419). The material here is complementary to Chern’s chapter three (Lectures on Differential Geometry).(5) Chapter five and nine are somewhat interdependent. Begin with determinants for dimension less than or equal to three, then, generality –determinants “of an endomorphism” (chapter nine). First instance of the derivative as limit in chapter six (page 119). Chapters seven and eight generalize that usual derivative.(6) Some claim this book is merely formalism for the sake of formalism. I disagree. Some claim these notes are only beneficial for mathematicians. I disagree. Physicists will benefit from much here: simply ignore the mathematical proofs ! Read from Bulletin of the American Mathematical Society: “this book is a radical departure….” and “the authors have rethought the whole subject of vector analysis…” (1960, book review by Allendoerfer). Read that book- review for many insightful comments.(7) As with Loomis and Sternberg (another honors course), this book is available freely-online (archive). Unlike Loomis and Sternberg (chapter zero), you spend no time here discussing “logic.” Surprisingly, the list of references in Loomis and Sternberg neglects inclusion of this book ! Also, their index neglects inclusion of the words: endomorphism and homomorphism ! They are clearly defined here, those terms, page 25. You learn: definitions are paramount. Scalar and vector products are treated axiomatically (pages 63-94), then generalized. You do not get far without understanding the meaning of equivalence relation.(8) How does one approach this book ? That is, how best to learn what is encompassed ? I took three turns with the book: first pass, I neglected mathematical proofs and ignored any exercise asking for proof. In short, I approached it as a physicist would approach it. Second pass, I included a selection of proofs and a few of the more difficult exercises. Finally (third pass), include all content and (at least) read every exercise (the exercises include other material, so you do not want to skip anything, ideally). Then it all hangs together, it all makes sense. You really are “rethinking the whole subject of vector analysis.” (Allendoerfer). If you do not care to do that–to rethink vector analysis–then, you will not care for this book.(9) Concluding: If you have the inclination (mathematical maturity) this is a highly recommended discourse. Prerequisites: Flanigan & Kazdan’s Second -Year Calculus alongside Chinn & Steenrod’s First Concepts of Topology. After which, this book will be less frightful. A companion text is Loomis and Sternberg (it is more an ‘introductory analysis’ text) while Chern’s Lectures on Differential Geometry is another (being more inclined geometrical).Have some fun with this material.

⭐中級程度の微分積分学の教科書です。線型代数学を基礎として議論を展開しており、統一されていて、とてもすっきりした印象を持ちます。証明は、読者の演習として残してあるところが、けっこうあります。印刷が悪いので、星をひとつ減らしました。悪いというのは、なか見!検索でご覧になれるよう、タイプライタの原稿をそのまま印刷している体裁になっていることです。添え字の大きさが本体の文字と同じです。とても読みづらいです。索引が無いので、星をもうひとつ減らしました。500ページを超える数学の教科書で索引が無いのは、とても不便です。本書の邦訳「現代ベクトル解析―ベクトル解析から調和積分へ 」岩波書店は、印刷もきれいですし、索引もあり、訳文もこなれているので、こちらを読んだ方がよいかも知れません。わたくしは、邦訳を読んで、よい本だと感心して、原書を読みたくなり買ってみたのですが、がっかりしました。本質的でない部分への文句が多くなってしまいましたが、内容の美しさと価格の安さは魅力的で、お勧めします。

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