Calculus and Analysis in Euclidean Space
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Beschreibung
The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject.
The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skills of
- geometric intuition (the visual cortex being quickly instinctive)
- algebraic manipulation (symbol-patterns being precise and robust)
- incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).
Thinking in these ways renders mathematics coherent, inevitable, and fluid.
The prerequisite is single-variable calculus, including familiarity with the foundational theorems and some experience with proofs.The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum. This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis. The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus. More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject.
The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skills of
- geometric intuition (the visual cortex being quickly instinctive)
- algebraic manipulation (symbol-patterns being precise and robust)
- incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).
Thinking in these ways renders mathematics coherent, inevitable, and fluid.
The prerequisite is single-variable calculus, including familiarity with the foundational theorems and some experience with proofs.Concentrates on n-dimensional Euclidean space Uses multivariable calculus to teach mathematics as a blend of reasoning, computing, and problem-solving, doing justice to the structure, the details, and the scope of the ideas Contains figures, formulas, and words to guide the reader to do mathematics resourcefully by marshaling the skills of geometric intuition, algebraic manipulation, and incisive use of natural language Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material
Autor*in
Jerry Shurman
Themen in »Calculus and Analysis in Euclidean Space«
derivative implicit functions integration of differential forms linear mappings multivariable differentiable calculus multivariable integral calculus parameterized curves textbook adoption calculus textbook adoption analysis single-variable calculus Euclidean space inverse functions approximation smooth functions parametrized curves
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“Shurman (mathematics, Reed College) has succeeded in presenting a text that encompasses multivariable calculus, advanced calculus, and an introduction to point-set topology. In short, this book covers aspects one should know about the elementary analysis, geometry, and topology of Euclidean space. … Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty.” (J. T. Zerger, Choice, Vol. 54 (11), July, 2017)
“The author’s writing style is clear and easy to follow, but, more than that, it is exceptionally well-motivated and contains some useful pedagogical ideas. In addition, throughout the book, the author notes issues that are likely to cause trouble to beginning students, and takes the time and effort to single them out and discuss them thoroughly. There are lots of exercises, many of them quiteilluminating. … It is highly recommended.” (Mark Hunacek, MAA Reviews, maa.org, March, 2017)“This book contains a clear and well-planned lecture discussing the most important issues of differential and integral calculus. … The big advantage of this book are nice, transparent and often colourful drawings illustrating some considerations. A nice complement to mathematical statements are explanations and comments.” (Ryszard Pawlak, zbMATH 1357.26002, 2017)
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