﻿ ﻿Two-Dimensional Calculus

## Two-Dimensional Calculus (2011)

Preface

Notation

Introduction

Chapter 1. Background

1. Vectors in the plane

2. Plane curves

3. Functions of two variables

Chapter 2. Differentiation

4. Partial derivatives and plane domains

5. Continuity

6. Directional derivatives and gradient

7. The chain rule

8. Level curves and the implicit function theorem

9. Maximum and minimum along a curve

10. Quadratic forms

11. Higher order derivatives

12. Taylor’s theorem; maxima and minima

Chapter 3. Transformations

13. Pairs of functions; geometric representation

14. Linear transformations

15. Special linear transformations; composition

16. Differentiable transformations

17. Composition; inverse mappings

18. Functional dependence; conformal mapping

19. Vector fields

20. Change of coordinates

Chapter 4. Integration

21. Line integrals

22. Potential functions; independence of path

23. Area

24. The double integral

25. Green’s theorem for a rectangle

26. Green’s theorem for arbitrary figures; applications

27. Mean value theorem for the double integral; vector forms of Green’s theorem

28. Transformation rule for double integrals

References

Answers to Selected Exercises

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