Katz N. Calculus for Cranks 2021

Textbook in PDF format

In mathematics, "cranks" are people who insist they understand something new about math even when the world tells them they are doing it wrong. This introduction to calculus is written with those cranks in mind, based on the foundational course that Nets Katz teaches at Caltech. It emphasizes the practical purposes of the foundations, such as tracking errors in calculations. In addition to covering the basics of single variable calculus, the book outlines the mathematical method-the ability to express oneself with absolute precision and then to use logical proofs to establish that certain statements are universally true. Katz emphasizes conceptual clarity, as well as testing hypotheses and writing complete proofs. The result is a rigorous calculus book of use not only to future mathematicians but also to scientists and engineers.
Induction and the Real Numbers
Sequences and Series 4
Functions and Derivatives
Integration
Convexity
Trigonometry, Complex Numbers, and Power Series
Complex Analysis
Appendix
How to Use This Book
Logic, Axioms, and Problem Solving
Further Reading