Calculus and analysis

A book in 22 chapters about change and accumulation: from the speed of a falling stone and the area under a curve to derivatives, integrals, series, differential equations, calculus in several variables, and then the careful analysis underneath it all, the real numbers, uniform convergence, metric spaces, Lebesgue integration and Fourier series. Each chapter starts from a question, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Algebra first. You need school algebra and the idea of a function; the Algebra book covers functions, exponentials and series if you want them first.
• Slower help. The first chapters point to the matching lessons on the Topics shelf ("Change"), which teach the same ideas more gently, for younger readers.
• Two passes. Parts I to V are calculus: how to use it and why it works. Part VI goes back and builds it properly from the real numbers up, which is what mathematicians call analysis. Some proofs promised early are completed there.

Contents

Part	Chapter	Level
I. The ideas	1. Rates and areas	beginner
	2. Limits	beginner to intermediate
	3. Continuity	beginner to intermediate
II. Derivatives	4. The derivative	intermediate
	5. The chain rule and special functions	intermediate
	6. The mean value theorem	intermediate
	7. Applications of derivatives	intermediate
III. Integrals	8. The integral	intermediate
	9. The fundamental theorem	intermediate
	10. Techniques of integration	intermediate
	11. Applications of integrals	intermediate
IV. Series and differential equations	12. Sequences and series	intermediate to advanced
	13. Power series and Taylor	intermediate to advanced
	14. Differential equations	intermediate to advanced
V. Several variables	15. Functions of several variables	advanced
	16. Multiple integrals	advanced
	17. Vector calculus	advanced
VI. Analysis	18. The real numbers	advanced
	19. Uniform convergence	advanced to expert
	20. Metric spaces and the topology of R	advanced to expert
	21. Lebesgue integration and Fourier series	expert
	22. Where analysis goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• Starting out: chapters 1 to 11. They are a full first course in calculus, with the reasons written out.
• You know school calculus: skim 1 to 5, then read 6, 12 to 14 and Part V.
• You want the rigour: Part VI, chapters 18 to 22, going back to earlier chapters for the examples.