Calculus and Analysis in Euclidean Space

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

  Autorentext
  Jerry Shurman is professor of mathematics at Reed College, Portland, OR, USA. He is also the co-author of GTM 228 "A First Course in Modular Forms."

  Inhalt
  Preface.- 1 Results from One-Variable Calculus.- Part I Multivariable Differential Calculus.- 2 Euclidean Space.- 3 Linear Mappings and Their Matrices.- 4 The Derivative.- 5 Inverse and Implicit Functions.- Part II Multivariable Integral Calculus.- 6 Integration.- 7 Approximation by Smooth Functions.- 8 Parameterized Curves.- 9 Integration of Differential Forms.- Index.