... The numerical interpretation ... is however necessary . ... So long
as it is not obtained, the solutions may be said to remain incomplete and
useless, and the truth which it is proposed to discover is no less hidden
in the formulae of analysis than it was in the physical problem itself.
Joseph Fourier, The Analytic Theory of Heat
Chapters 1 through 6 of this book cover most of the standard topics in multivariate calculus and a substantial part of a standard first course in linear algebra. If, in addition, one teaches the proofs in Appendix A, the book can be used as a textbook for a course in analysis.
The organization and selection of material differs from the standard approach in three ways.
First, we believe that at this level linear algebra should be more a convenient setting and language for multivariate calculus than a subject in its own right. The guiding principle of this unified approach is that locally, a nonlinear function behaves like its derivative. Thus whenever we have a question about a nonlinear function we will answer it by looking carefully at a linear transformation: its derivative. In this approach, everything learned about linear algebra pays off twice: first for understanding linear equations, then as a tool for understanding nonlinear equations.
We discuss abstract vector spaces in Section 2.6, but the emphasis is on R', as we believe that most students find it easiest to move from the concrete to the abstract.
Second, we emphasize computationally effective algorithms, and prove theorems by showing that those algorithms really work. We feel this better reflects the way this mathematics is used today, in both applied and in pure mathematics. Moreover, it can be done with no loss of rigor.
For linear equations, row reduction (the practical algorithm) is the central tool from which everything else follows, and we use row reduction to prove all the standard results about dimension and rank. For nonlinear equations, the cornerstone is Newton's method, the best and most widely used method for solving nonlinear equations; we use it both as a computational tool and in proving the inverse and implicit function theorems. We include a section on numerical methods of integration, and we encourage the use of computers both to reduce tedious calculations and as an aid in visualizing curves and surfaces.
Third, we use differential forms to generalize the fundamental theorem of calculus to higher dimensions. The great conceptual simplifications gained by doing electromagnetism in the language of forms is a central motivation for using forms, and we will apply the language of forms to electromagnetism in a subsequent volume.
In our experience, differential forms can be taught to freshmen and sophomores, if forms are presented geometrically, as integrands that take an oriented piece of a curve, surface, or manifold, and return a number. We are aware that students taking courses in other fields need to master the language of vector calculus, and we devote three sections of Chapter 6 to integrating the standard vector calculus into the language of forms.
Chapter 0 is intended as a resource. Students should not feel that they need to read it before beginning Chapter 1. Another resource is the inside back cover, which lists some useful formulas.
Numbering of theorems, examples, and equations
Theorems, lemmas, propositions, corollaries, and examples share the same numbering system: Proposition 2.3.8 is not the eighth proposition of Section 2.3; it is the eighth numbered item of that section. We often refer back to theorems, examples, and so on, and hope this numbering will make them easier to find.
Figures and tables share their own numbering system; Figure 4.5.2 is the second figure or table of Section 4.5. Virtually all displayed equations are numbered, with the numbers given at right; Equation 4.2.3 is the third equation of Section 4.2. When an equation is displayed a second time, it keeps its original number, but the number is in parentheses.
We use a symbol to mark the end of an example or remark, and to mark the end of a proof.
Mathematical notation is not always uniform. For example, A can mean the length of a matrix A or the determinant of A. Different notations for partial derivatives also exist. This should not pose a problem for readers who begin at the beginning and end at the end, but for those who are using only selected chapters, it could be confusing. Notations used in the book are listed on the front inside cover, along with an indication of where they are first introduced.
Exercises are given at the end of each section; chapter review exercises are given at the end of each chapter, except Chapter 0. Exercises range from very easy exercises intended to make the student familiar with vocabulary, to quite difficult exercises. The hardest exercises are marked with an asterisk (in rare cases, two asterisks). A student solution manual is planned, with complete solutions to odd-numbered exercises.
What's new in the second edition
While anyone familiar with the first edition will recognize the second, there have been many changes, large and small. Perhaps the biggest change is that we now treat Lebesgue integration (Section 4.11). Other major ones include:
- More exercises, more examples, and more figures, including approximately 270 additional exercises and 50 additional examples. We have also added pictures of mathematicians.
- There are now end-of-section exercises, as well as review exercises for Chapters 1-6.
- More difficult material from Chapter 0 was moved to the Appendix.
- Chapter 0 now includes a new section on functions; some material from Section 1.3 was moved to this section. The section on complex numbers was expanded.
- The inverse and implicit function theorems have been rewritten.
- In Chapter 3 we simplified the definition of a manifold, and we now begin with the general case and discuss curves and surfaces as examples, Similarly, in Chapter 5, we eliminated the separate sections on arc length and surface area; we now have one section on volume of manifolds.
- In Chapter 6, we rewrote the discussion of orientation and changed the definition of a piece-with-boundary of a manifold, to make it both simpler and more inclusive.
- Some useful formulas are listed on the back cover.
Different ways to use this book
This book can be used at different levels of rigor. Chapters 1 through 6 contain material appropriate for a course in linear algebra and multivariate calculus. Appendix A contains the technical, rigorous underpinnings appropriate for a course in analysis. It includes proofs of those statements not proved in the main text, and a painstaking justification of arithmetic.
Most of the proofs included in this appendix are more difficult than the proofs contained in the main text, but difficulty was not the only criterion; many students find the proof of the fundamental theorem of algebra (Section 1.6) quite difficult. But we find this proof qualitatively different from the proof of the Kantorovich theorem, for example. A professional mathematician who has understood the proof of the fundamental theorem of algebra should be able to reproduce it. A professional mathematician who has read through the proof of the Kantorovich theorem, and who agrees that each step is justified, might well want to refer to printed notes in order to reproduce it. In this sense, the first proof is more conceptual, the second more technical.
Following is a brief description of different courses that could be taught using this book.
At Cornell University this book is used for the honors courses Math 223 (fall semester) and 224 (spring semester), for students who have studied one-variable calculus. Students are expected to have a 5 on the Advanced Placement BC Calculus exam, or the equivalent. When John Hubbard teaches the course, he typically gets to the middle of Chapter 4 in the first semester, skipping Section 3.8 on the geometry of curves and surfaces and Section 4.4 on measure 0. In the second semester he gets to the end of Chapter 6 and goes on to teach some of the material that will appear in a sequel volume, in particular differential equations.
Eventually, he would like to take three semesters to cover Chapters 1-6 of the current book and the material of the second book (referred to as "Volume 2" throughout this text):
- Chapter 7: Differential Equations
- Chapter 8: Inner Products and Signal Processing
- Chapter 9: Linear Differential Equations with Constant Coefficients
- Chapter 10: Differential Forms: Advanced Topics
- Chapter 11: Inner Products and Forms
- Chapter 12: Electromagnetism
Another approach would be to spend a year on Chapters 1-6. This could be done at different levels of difficulty. Some students may need to review material in Chapter 0; others may be able to include some of the proofs in the appendix.
(1) A one-semester course for students who have studied neither linear algebra V nor multivariate calculus.
For such a course, we suggest covering only the first four chapters, omitting the parts marked "optional" (the part of Section 2.8 concerning a stronger version of the Kantorovich theorem, and Section 4.4 on measure 0). Other topics that can be omitted include
- The proof of the fundamental theorem of algebra in Section 1.6,
- The discussion of criteria for differentiability in Section 1.9,
- Section 3.8 on the geometry of curves and surfaces,
- The proof of theorem 4.9.1,
- The discussion in Section 4.11 on Fourier and Laplace transforms.
Sections 4.2 (integrals. and probability) and 4.6 (numerical methods of integration) could also be skipped, but we feel these topics are generally given too little attention. If Section 4.2 is skipped, then one should also skip the discussion of Monte Carlo methods in Section 4.6.
(2) A course for students who have had some exposure to either linear algebra or multivariate calculus, but who are not ready for a course in analysis.
We used an earlier version of this text with students who had taken a course in linear algebra, and feel they gained a great deal from seeing how linear algebra and multivariate calculus mesh. Such students could be expected to cover Chapters 1-6, possibly omitting some of material, as discussed above. For a less fast-paced course, the book could also be covered in a year, possibly including some proofs from the appendix.
(3) A one-semester analysis course.
In one semester one could hope to cover all six chapters and some or most of the proofs in Appendix A. This could be done at varying levels of difficulty; students might be expected to follow the proofs, for example, or they might be expected to understand them well enough to construct similar proofs.