Calculus and Analysis in Euclidean Space

The graceful analysis that underpins calculus is often lost in the delineation of curriculum between the two areas. Intertwining the two, this text provides a conceptual approach to multivariable calculus, motivated by the ideas and reasoning of analysis. This is a text for students who wish to explore concepts: more mathematically mature than most calculus books, yet less technical than a typical introductory analysis text. This balance makes the material accessible to those outside a traditional math major, while also offering an alternative perspective for those who have encountered calculus and analysis separately.

The structure of the text is aimed to communicate intent early in the discussion. The weight of the narrative is spread among figures, formulas, and prose to guide 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).

Engaging from various perspectives creates a backdrop in which the mathematics becomes coherent, inevitable, and fluent.

The prerequisite is single-variable calculus, including familiarity with its foundational theorems and some 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 ofgeometric 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.