Demonstrating analytical and numerical techniques for attacking problems in the application of mathematics, this well-organized, clearly written text presents the logical relationship and fundamental notations of analysis. Buck discusses analysis not solely as a tool, but as a subject in its own right. This skill-building volume familiarizes students with the language, concepts, and standard theorems of analysis, preparing them to read the mathematical literature on their own. The text revisits certain portions of elementary calculus and gives a systematic, modern approach to the differential and integral calculus of functions and transformations in several variables, including an introduction to the theory of differential forms. The material is structured to benefit those students whose interests lean toward either research in mathematics or its applications.
Reviews from Amazon users which were colected at the time this book was published on the website:
⭐This book is one of the better written books on advanced calculus. The diagrams and figures are well done and specific to the examples or concepts being presented. However, this is not for a college student who is trying learn how to solve typical engineering problems. This is a theoretical math book of proofs and identifying limiting conditions, etc. The Exercise problems involve proving bounding characteristics of functions, showing where there is convergence or divergence, and discussing the “whys and hows” of non-specific surfaces or multiple integrals. You will not be reading about or solving problems involving rolling spheres, projectiles, fluids in motion, planetary motion or the conservation of energy. To paraphrase Jeff Foxworthy, “If you can solve the exercise problems based on what you read in the previous section, then you are probably a mathematician.” It is still a very well written and organized book – but definitely for the theoretically minded student.
⭐This book is a very good classic introduction to real analysis. It very clearly presents the usual sequence in one-variable analysis and does also good job with multivariable results. Concerning the great theorems of vector analysis (Green, Gauss, Stokes) it follows the classical approach but introduces differential forms without all the technical apparatus which is to be pursued further in books like Spivak’s Calculus on Manifolds. The treatment is rigourous but it doesn’t neglect computational and numerical aspects. Of course, I intend to proceed up to Rudin’s “hard stone” after Buck. The new edition is a very attractive and not so expensive in comparison to other books on the subject.
⭐The book arrived on time and as described.
⭐One of the strong point of this book is that its very easy to follow. The reason for four stars is because I feel as if it lacks organization.
⭐One of the best introduction to analysis. Especially good for those are interested in geometry. Some applications for differential geometry are hard to find in other books
⭐this was the Advanced Calculus text I had as an undergraduate and now I have it back. It brings backmany pleasant memories not the least of which is my introduction to differential forms.
⭐Another fine textbook providing for another viewpoint of Advanced Calculus. Not only is the topic of elementary calculus well-served by a plethora of great mathematics textbooks, the same can be had for advanced calculus textbooks ! No one has an excuse not to get a first-rate education in elementary and advanced calculus !Caveat: You must first acquire a first-rate mathematical foundation of the prerequisites to learning Calculus.Enough of my diatribe, onward to R. Creighton Buck’s excellent exposition:(1) I suppose ‘new’ in this endeavor (at least for 1956) is the introduction to differential forms. But, you have to wait until the final chapter to get at it. (And, if it was ‘new’ then, it was still ‘new’ in 1969, when Edward’s Advanced Calculus book was published, quickly falling into obscurity. Also, Loomis and Sternberg came out one year earlier and is pitched at a much higher level.) Buck is truly as elementary as that term implies, an accessible approach within an advanced calculus setting.(2) Now, of Topology. My preference (within an advanced calculus text) lies with the sprawling, spiral approach taken by Angus Taylor (an introductory setting). Even so, the approach here taken is worthy of consideration. Although brief (fifteen pages), it is lucid. As but one question, we are asked: “Does an infinite set which is unbounded have to have a cluster point ? ” (problem #3, page 11). Then, to sequences, then to functions.(3) If your epsilons and deltas did not get what they deserved in an elementary course, what Buck has to offer is here is crystal clear (pages 22-30). If anything is wanting, I can only refer the interested reader–again–to Angus Taylor.(4) Topology (first chapter) led us to functions, limits and continuity (second chapter). These lead us to Integration. That’s right, Integration. If expecting another long detour on derivatives, take your time and absorb Buck’s excellent third chapter. I highlight page 82, an interlude on the Heine-Borel theorem.The chapter will conclude with material ” considered as introductory to measure theory.” (pages 99-104). Very nice !(5) Next, infinite series, uniform convergence, and power series. Concluding this chapter: “a number of examples which illustrate these theorems and the manner in which they may be used in the evaluation of certain definite integrals.” This, pages 153-163. Of course, we meet Gamma. So, it is, that Chapters three and four (from integrals to series) form a well-rounded doublet. A fine exposition.(6) Next, a return to differentiation. Or, rather Linear Transformations (again, much here is replicated in Angus Taylor). Especially to be noted is the discussion of Inverse Functions (beginning page 200). From there (inverse of one-variable) to here (inverse of transformations), concluding with the implicit function theorem. The reader is delighted at the clarity of exposition. The exercises–following each section–are straightforward.(7) Sixth chapter, multiple integrals and elementary differential geometry. Study Goursat’s Volume One for more !We begin with basic survey of determinants. Why ? Because, in short order, we need it to define multiple integrals and transformations thereof. Curves and Arc-Length follow. Why ? Because Buck will define a curve in terms of transformations (page 251). This segues to Equivalence Classes ! We read: “all the curves in any one equivalence class share other geometric properties” (page 260). Beautiful introductory material. Another highlight will be surfaces.This, at intuitive vantage. Why ? Because, as Buck writes, surface theory is difficult without more topology.And, so, the approach is analogous to that of curves. Again, determinant, transformations, parametric equivalence (page 278). A very pretty discussion of extremal properties of several variables concludes the chapter (fifteen pages).(8) As previously noted, the final chapter continues with elementary differential geometry and presents differential forms. A quick and painless introduction to the important theorems (divergence, Gauss, Stokes) plus a survey of Maxwell’s equations is offered. A highlight is the lucid exposition in deriving solution to Poisson’s equation(pages 372-375) and another go-around with calculus of variations. The Appendix (Real Number System) should be absorbed. Especially the chart on page 391: Here we have offered a visual: Dedekind,” every cut is generated by an element” to Cauchy, “any Cauchy sequence is convergent.” We absorb six equivalent assertions, and thosesix assertions imply two further assertions, which themselves are subsumed within the “Archimedian” definition.(Read pages 387-393). Creighton Buck has presented another pathway to topics in Advanced Calculus.This is a valuable addition to the literature of calculus. Technically, lying between Angus Taylor’s (elementary) Advanced Calculus and Loomis and Sternberg’s (abstract) Advanced Calculus.As a stepping-stone to other (more advanced) tomes, Buck is priceless.
⭐I knew this book in the reference of “Analysis I & II” written by Mitsuo Sugiura which are the most standard and famous calculus books in Japan.Advanced Calculus by Buck is very reader-friendly book.This book covers mainly muti-variable calculus.I like the handwritten pictures drawn by artists.
⭐Fantástico. El cálculo en su estado puro y con una didáctica impresionante. Los fundamentos del cálculo al alcanze de los que lo amamos.Concepts are clearly explained. Useful for mathematicians and mathematically oriented applied scientists and engineers.
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