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.