Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

505 Pages·2005·30.973 MB·English

by  Theodore Shifrin

MULTIVARIABLE
MATHEMATICS
Linear Algebra, Multivariable
Calculus, and Manifolds

MULTIVARIABLE
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Linear Algebra, Multivariable
Calculus, and Manifolds
THEODORE SHIFRIN
University of Georgia
WILEY
John Wiley & Sons, Inc.
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ISBN 978-0-471-52638-4
CONTENTS
Preface vii ► CHAPTERS
EXTREMUM PROBLEMS 196
> CHAPTER 1
VECTORS AND MATRICES 1 1 Compactness and the Maximum Value
Theorem 196
1 Vectors in R" 1
2 Maximum/Minimum Problems 202
2 Dot Product 8
3 Quadratic Forms and the Second Derivative
3 Subspaces of Rn 16
Test 208
4 Linear Transformations and Matrix Algebra 23
4 Lagrange Multipliers 216
5 Introduction to Determinants and die Cross
5 Projections, Least Squares, and Inner Product
Product 43
Spaces 225
► CHAPTER 2
► CHAPTER6
FUNCTIONS, LIMITS, AND CONTINUITY 53
SOLVING NONLINEAR PROBLEMS 244
1 Scalar- and Vector-Valued Functions 53
1 The Contraction Mapping Principle 244
2 A Bit of Topology in R" 64
2 The Inverse and Implicit Function Theorems 251
3 Limits and Continuity 72
3 Manifolds Revisited 261
► CHAPTERS
► CHAPTER 7
THE DERIVATIVE 81
INTEGRATION 267
1 Partial Derivatives and Directional Derivatives 81
1 Multiple Integrals 267
2 Differentiability 87
2 Iterated Integrals and Fubini’s Theorem 276
3 Differentiation Rules 97
3 Polar, Cylindrical, and Spherical Coordinates 288
4 The Gradient 104
4 Physical Applications 298
5 Curves 109
5 Determinants and n-Dimensional Volume 309
6 Higher-Order Partial Derivatives 120
6 Change of Variables Theorem 324
> CHAPTER 4
► CHAPTER 8
IMPLICIT AND EXPLICIT SOLUTIONS OF LINEAR
DIFFERENTIAL FORMS AND INTEGRATION ON
SYSTEMS 127
MANIFOLDS 333
1 Gaussian Elimination and the Theory of Linear
1 Motivation 333
Systems 127
2 Differential Forms 335
2 Elementary Matrices and Calculating Inverse
3 Line Integrals and Green’s Theorem 348
Matrices 147
4 Surface Integrals and Flux 367
3 Linear Independence, Basis, and Dimension 156
5 Stokes’s Theorem 379
4 The Four Fundamental Subspaces 171
6 Applications to Physics 393
5 The Nonlinear Case: Introduction to
7 Applications to Topology 403
Manifolds 186
vi ► Contents
► CHAPTER 9 GLOSSARY OF NOTATIONS AND RESULTS FROM
EIGENVALUES, EIGENVECTORS, AND SINGLE-VARIABLE CALCULUS 467
APPLICATIONS 413
FOR FURTHER READING 473
1 Linear Transformations and Change of Basis 413
2 Eigenvalues, Eigenvectors, and ANSWERS TO SELECTED EXERCISES 474
Diagonalizability 422
INDEX 488
3 Difference Equations and Ordinary Differential
Equations 436
4 The Spectral Theorem 455
PREFACE
I began writing this text as I taught a brand-new course combining linear algebra and a
rigorous approach to multivariable calculus. Some of the students had already taken a
proof-oriented single-variable calculus course (using Spivak’s beautiful book, Calculus),
but many had not: There were sophomores who wanted a more challenging entree to
higher-level mathematics, as well as freshmen who’d scored a 5 on the Advanced Placement
Calculus BC exam. My goal was to include all the standard computational material found
in the usual linear algebra and multivariable calculus courses and more, interweaving the
material as effectively as possible, and include complete proofs.
Although there have been a number of books that include both the linear algebra and
the calculus material, they have tended to segregate the material. Advanced calculus books
treat the rigorous multivariable calculus, but presume the students have already mastered
linear algebra. I wanted to integrate the material so as to emphasize the recurring theme of
implicit versus explicit that persists in linear algebra and analysis. In every linear algebra
course we should learn how to go back and forth between a system of equations and a
parametrization of its solution set. But the same problem occurs, in principle, in calculus: To
solve constrained maximum/minimum problems we must either parametrize the constraint
set or use Lagrange multipliers; to integrate over a curve or surface, we need a parametric
representation. Of course, in the linear case one can globally go back and forth; it’s not
so easy in the nonlinear case, but, as we’ll learn, it should at least be possible in principle
locally.
The prerequisites for this book are a solid background in single-variable calculus and, if
not some experience writing proofs, a strong interest in grappling with them. In presenting
the material, I have included plenty of examples, clear proofs, and significant motivation
for the crucial concepts. We all know that to learn (and enjoy?) mathematics one must
work lots of problems, from the routine to the more challenging. To this end, I have pro­
vided numerous exercises of varying levels of difficulty, both computational and more
proof-oriented. Some of the proof exercises require the student “merely” to understand and
modify a proof in the text; others may require a good deal of ingenuity. I also ask students
for lots of examples and counterexamples. Generally speaking, exercises are arranged in
order of increasing difficulty. To offer a bit more guidance, I have marked with an asterisk
(*) those problems to which short answers, hints, or—in some cases—complete solutions
are given at the back of the book. As a guide to the new teacher, I have marked with a sharp
(B) some important exercises to \vhich reference is made later. An Instructor’s Solutions
Manual (ISBN 0-471-64915-5) is available from the publisher.
► COMMENTS ON CONTENTS
The linear algebraic material with which we begin the course in Chapter 1 is concrete,
establishes the link with geometry, and is a good self-contained setting for working on
vii
viii ► Preface
proofs. We introduce vectors, dot products, subspaces, and linear transformations and
matrix computations. At this early stage we emphasize the two interpretations of multiplying
a matrix A by a vector x: the linear equations viewpoint (considering the dot products of the
rows of A with x) and the linear combinations viewpoint (taking the linear combination of
the columns of A weighted by the coordinates of x). We end the chapter with a discussion
of 2 x 2 and 3x3 determinants, area, volume, and the cross product.
In Chapter 2 we begin to make the transition to calculus, introducing scalar functions
of a vector variable—their graphs and their level sets—and vector-valued functions. We
introduce the requisite language of open and closed sets, sequences, and limits and conti­
nuity, including the proofs of the usual limit theorems. (Generally, however, I give these
short shrift in lecture, as I don’t have the time to emphasize 8-s arguments.)
We come to the concepts of differential calculus in Chapter 3. We quickly introduce
partial and directional derivatives as immediate to calculate, and then come to the definition
of differentiability, the characterization of differentiable functions, and the standard differ­
entiation rules. We give the gradient vector its own brief section, in which we emphasize
its geometric meaning. Then comes a section on curves, in which we mention Kepler’s
laws (the second is proved in the text and the other two are left as an exercise), arclength,
and curvature of a space curve.
In the first four sections of Chapter 4 we give an accelerated treatment of Gaussian
elimination (including a proof of uniqueness of reduced echelon form) and the theory of
linear systems, the standard material on linear independence and dimension (including a
brief mention of abstract vector spaces), and the four fundamental subspaces associated to
a matrix. In the last section, we begin our assault on the nonlinear case, introducing (with
no proofs) the implicit function theorem and the notion of a manifold.
Chapter 5 is a blend of topology, calculus, and linear algebra—quadratic forms and pro­
jections. We start with the topological notion of compactness and prove the maximum value
theorem in higher dimensions. We then turn to the calculus of applied maximum/minimum
problems and then to the analysis of the second-derivative test and the Hessian. Then
comes one of the most important topics in applications, Lagrange multipliers (with a rigor­
ous proof). In the last section, we return to linear algebra, to discuss projections (from both
the explicit and the implicit approaches), least-squares solutions of inconsistent systems, the
Gram-Schmidt process, and a brief discussion of abstract inner product spaces (including
a nice proof of Lagrange interpolation).
Chapter 6 is a brief, but sophisticated, introduction to the inverse and implicit function
theorems. We present our favorite proof using the contraction mapping principle (which is
both more elegant and works just fine in the infinite-dimensional setting). In the last section
we prove that all three definitions of a manifold are (locally) equivalent: the implicit
representation, the parametric representation, and the representation as a graph. (In the
year-long course that I teach, I find I have time to treat this chapter only lightly.)
In Chapter 7 we study the multidimensional (Riemann) integral. In the first two sections
we deal predominantly with the theory of the multiple integral and, then, Fubini’s Theorem
and the computation of iterated integrals. Then we introduce (as is customary in a typical
multivariable calculus course) polar, cylindrical, and spherical coordinates and various
physical applications. We conclude the chapter with a careful treatment of determinants
(which will play a crucial role in Chapters 8 and 9) and a proof of the Change of Variables
Theorem.
Preface ◄ ix
In single-variable calculus, one of the truly impressive results is the Fundamental
Theorem of Calculus. In Chapter 8 we start by laying the groundwork for the analogous
multidimensional result, introducing differential forms in a very explicit fashion. We then
parallel a traditional vector calculus course, introducing line integrals and Green’s Theorem,
surface integrals and flux, and, then, finally stating and proving the general Stokes’s Theorem
for compact oriented manifolds. We do not skimp on concrete and nontrivial examples
throughout. In Section 8.6 we introduce the standard terminology of divergence and curl
and give the “classical” versions of Stokes’s and the Divergence Theorems, along with
some applications to physics. In Section 8.7 we begin to illustrate the power of Stokes’s
Theorem by proving the Fundamental Theorem of Algebra, a special case of the argument
principle, and the “hairy ball theorem” from topology.
In Chapter 9 we complete our study of linear algebra, including standard material
on change of basis (with a geometric slant), eigenvalues, eigenvectors and discussion of
diagonalizability. The remainder of the chapter is devoted to applications: difference and
differential equations, and a brief discussion of flows and their relation to the Divergence
Theorem of Chapter 8. We close with the Spectral Theorem. (With the exception of
Section 3.3, which relies on Chapter 8, and the proof of the Spectral Theorem, which relies
on Section 4 of Chapter 5, topics in this chapter can be covered at any time after completing
Chapter 4.)
We have included a glossary of notations and a quick compilation of relevant results
from trigonometry and single-variable calculus (including a short table of integrals), along
with a much-requested list of the Greek alphabet.
There are over 800 exercises in the text, many with multiple parts. Here are a few
particularly interesting (and somewhat unusual) exercises included in this text:
• Exercises 1.2.22-26 and Exercises 1.5.19 and 1.5.20 on the geometry of triangles,
and Exercise 1.5.17, a nice glimpse of affine geometry
• Exercise 2.1.12, a parametrization of a hyperboloid of one sheet in which the param­
eter curves are the two families of rulings
• Exercises 2.3.15-17, 3.1.10, and 3.2.18-19, exploring the infamous sorts of discon­
tinuous and nondifferentiable functions
• Example 3.4.3 introducing the reflectivity property of the ellipse via the gradient,
with follow-ups in Exercises 3.4.8, 3.4.9, and 3.4.13, and then Kepler’s first and
third laws in Exercise 3.5.15.
• Exercise 3.5.14, the famous fact (due to Huygens) that the evolute of a cycloid is a
congruent cycloid
• Exercise 4.5.13, in which we discover that the lines passing through three pairwise­
skew lines generate a saddle surface
• Exercises 5.1.5, 5.1.7, 9.4.11, exploring the (operator) norm of a matrix
• Exercise 5.2.15, introducing the Fermat/Steiner point of a triangle
• Exercises 5.3.2 and 5.3.4, pointing out a local minimum along every line need not
be a local minimum (an issue that is mishandled in surprisingly many multivariable
calculus texts) and that a lone critical point that is a local minimum may not be a
global minimum