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# B S TRACI
## ALGEBRA
## THIRD EDITION
## DAVID S. DUMMIT
## RICHARD M. FOOTE
# Frequently Used Notation
| $f^{-1}(A)$ | the inverse image or preimage of $A$ under $f$ |
| :--: | :--: |
| $a \mid b$ | $a$ divides $b$ |
| $(a, b)$ | the greatest common divisor of $a, b$ also the ideal generated by $a, b$ |
| $\|A\|,\|x\|$ | the order of the set $A$, the order of the element $x$ |
| $\mathbb{Z}, \mathbb{Z}^{+}$ | the integers, the positive integers |
| $\mathbb{Q}, \mathbb{Q}^{+}$ | the rational numbers, the positive rational numbers |
| $\mathbb{R}, \mathbb{R}^{+}$ | the real numbers, the positive real numbers |
| $\mathbb{C}, \mathbb{C}^{\times}$ | the complex numbers, the nonzero complex numbers |
| $\mathbb{Z} / n \mathbb{Z}$ | the integers modulo $n$ |
| $(\mathbb{Z} / n \mathbb{Z})^{\times}$ | the (multiplicative group of) invertible integers modulo $n$ |
| $A \times B$ | the direct or Cartesian product of $A$ and $B$ |
| $H \leq G$ | $H$ is a subgroup of $G$ |
| $Z_{n}$ | the cyclic group of order $n$ |
| $D_{2 n}$ | the dihedral group of order $2 n$ |
| $S_{n}, S_{\Omega}$ | the symmetric group on $n$ letters, and on the set $\Omega$ |
| $A_{n}$ | the alternating group on $n$ letters |
| $Q_{8}$ | the quaternion group of order 8 |
| $V_{4}$ | the Klein 4-group |
| $\mathbb{F}_{N}$ | the finite field of $N$ elements |
| $G L_{n}(F), G L(V)$ | the general linear groups |
| $S L_{n}(F)$ | the special linear group |
| $A \cong B$ | $A$ is isomorphic to $B$ |
| $C_{G}(A), N_{G}(A)$ | the centralizer, and normalizer in $G$ of $A$ |
| $Z(G)$ | the center of the group $G$ |
| $G_{s}$ | the stabilizer in the group $G$ of $s$ |
| $\{A\},\{x\}$ | the group generated by the set $A$, and by the element $x$ |
| $G=\{\ldots \mid \ldots\}$ | generators and relations (a presentation) for $G$ |
| $\operatorname{ker} \varphi, \operatorname{im} \varphi$ | the kernel, and the image of the homomorphism $\varphi$ |
| $N \unlhd G$ | $N$ is a normal subgroup of $G$ |
| $g H, H g$ | the left coset, and right coset of $H$ with coset representative $g$ |
| $\|G: H\|$ | the index of the subgroup $H$ in the group $G$ |
| $\operatorname{Aut}(G)$ | the automorphism group of the group $G$ |
| $S y l_{p}(G)$ | the set of Sylow $p$-subgroups of $G$ |
| $n_{p}$ | the number of Sylow $p$-subgroups of $G$ |
| $[x, y]$ | the commutator of $x, y$ |
| $H \times K$ | the semidirect product of $H$ and $K$ |
| $\mathbb{H}$ | the real Hamilton Quaternions |
| $R^{\times}$ | the multiplicative group of units of the ring $R$ |
| $R[x], R\left[x_{1}, \ldots, x_{n}\right]$ | polynomials in $x$, and in $x_{1}, \ldots, x_{n}$ with coefficients in $R$ |
| $R G, F G$ | the group ring of the group $G$ over the ring $R$, and over the field $F$ |
| $O_{K}$ | the ring of integers in the number field $K$ |
| $\lim _{A} A_{i}, \underline{\lim } A_{i}$ | the direct, and the inverse limit of the family of groups $A_{i}$ |
| $\mathbb{Z}_{p}, \mathbb{Q}_{p}$ | the $p$-adic integers, and the $p$-adic rationals |
| $A \oplus B$ | the direct sum of $A$ and $B$ |
$L T(f), L T(I)$
$M_{n}(R), M_{n \times m}(R)$
$M_{\mathcal{B}}^{\mathcal{E}}(\varphi)$
$\operatorname{tr}(A)$
$\operatorname{Hom}_{R}(A, B)$
$\operatorname{End}(M)$
$\operatorname{Tor}(M)$
$\operatorname{Ann}(M)$
$M \otimes_{R} N$
$\mathcal{T}^{k}(M), \mathcal{T}(M)$
$\mathcal{S}^{k}(M), \mathcal{S}(M)$
$\bigwedge^{k}(M), \bigwedge(M)$
$m_{T}(x), c_{T}(x)$
$\operatorname{ch}(F)$
$K / F$
$[K: F]$
$F(\alpha), F(\alpha, \beta)$, etc.
$m_{\alpha, F}(x)$
$\operatorname{Aut}(K)$
$\operatorname{Aut}(K / F)$
$\operatorname{Gal}(K / F)$
$\mathbb{A}^{n}$
$k\left[\mathbb{A}^{n}\right], k[V]$
$\mathcal{Z}(I), \mathcal{Z}(f)$
$\mathcal{I}(A)$
$\operatorname{rad} I$
$\operatorname{Ass}_{R}(M)$
$\operatorname{Supp}(M)$
$D^{-1} R$
$R_{P}, R_{f}$
$\mathcal{O}_{v, V}, \mathbb{T}_{v, V}$
$\mathrm{m}_{v, V}$
$\operatorname{Spec} R, \operatorname{mSpec} R$
$\mathcal{O}_{X}$
$\mathcal{O}(U)$
$\mathcal{O}_{P}$
$\operatorname{Jac} R$
$\operatorname{Ext}_{R}^{n}(A, B)$
$\operatorname{Tor}_{n}^{R}(A, B)$
$A^{G}$
$H^{n}(G, A)$
Res, Cor
$\operatorname{Stab}(1 \unlhd A \unlhd G)$
$\|\theta\|$
$\operatorname{Ind}_{H}^{G}(\psi)$
the leading term of the polynomial $f$, the ideal of leading terms the $n \times n$, and the $n \times m$ matrices over $R$ the matrix of the linear transformation $\varphi$ with respect to bases $\mathcal{B}$ (domain) and $\mathcal{E}$ (range) the trace of the matrix $A$ the $R$-module homomorphisms from $A$ to $B$ the endomorphism ring of the module $M$ the torsion submodule of $M$ the annihilator of the module $M$ the tensor product of modules $M$ and $N$ over $R$ the $k^{\text {th }}$ tensor power, and the tensor algebra of $M$ the $k^{\text {th }}$ symmetric power, and the symmetric algebra of $M$ the $k^{\text {th }}$ exterior power, and the exterior algebra of $M$ the minimal, and characteristic polynomial of $T$ the characteristic of the field $F$ the field $K$ is an extension of the field $F$ the degree of the field extension $K / F$ the field generated over $F$ by $\alpha$ or $\alpha, \beta$, etc. the minimal polynomal of $\alpha$ over the field $F$ the group of automorphisms of a field $K$ the group of automorphisms of a field $K$ fixing the field $F$ the Galois group of the extension $K / F$ affine $n$-space the coordinate ring of $\mathbb{A}^{n}$, and of the affine algebraic set $V$ the locus or zero set of $I$, the locus of an element $f$ the ideal of functions that vanish on $A$ the radical of the ideal $I$ the associated primes for the module $M$ the support of the module $M$ the ring of fractions (localization) of $R$ with respect to $D$ the localization of $R$ at the prime ideal $P$, and at the element $f$ the local ring, and the tangent space of the variety $V$ at the point $v$ the unique maximal ideal of $\mathcal{O}_{v, V}$ the prime spectrum, and the maximal spectrum of $R$ the structure sheaf of $X=\operatorname{Spec} R$ the ring of sections on an open set $U$ in $\operatorname{Spec} R$ the stalk of the structure sheaf at $P$ the Jacobson radical of the ring $R$ the $n^{\text {th }}$ cohomology group derived from $\operatorname{Hom}_{R}$ the $n^{\text {th }}$ cohomology group derived from the tensor product over $R$ the fixed points of $G$ acting on the $G$-module $A$ the $n^{\text {th }}$ cohomology group of $G$ with coefficients in $A$ the restriction, and corestriction maps on cohomology the stability group of the series $1 \unlhd A \unlhd G$ the norm of the character $\theta$ the character of the representation $\psi$ induced from $H$ to $G$
# ABSTRACT ALGEBRA <br> Third Edition
David S. Dummit<br>University of Vermont
Richard M. Foote<br>University of Vermont
John Wiley \& Sons, Inc.
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ISBN 0-471-43334-9
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Printed in the United States of America.
# Dedicated to our families
especially
Janice, Evan, and Krysta
and
Zsuzsanna, Peter, Karoline, and Alexandra
# Contents
Preface ..... xi
Preliminaries ..... 1
0.1 Basics ..... 1
0.2 Properties of the Integers ..... 4
0.3 $\mathbb{Z} / n \mathbb{Z}$ : The Integers Modulo $n$ ..... 8
Part I - GROUP THEORY ..... 13
Chapter 1 Introduction to Groups ..... 16
1.1 Basic Axioms and Examples ..... 16
1.2 Dihedral Groups ..... 23
1.3 Symmetric Groups ..... 29
1.4 Matrix Groups ..... 34
1.5 The Quaternion Group ..... 36
1.6 Homomorphisms and Isomorphisms ..... 36
1.7 Group Actions ..... 41
Chapter 2 Subgroups ..... 46
2.1 Definition and Examples ..... 46
2.2 Centralizers and Normalizers, Stabilizers and Kernels ..... 49
2.3 Cyclic Groups and Cyclic Subgroups ..... 54
2.4 Subgroups Generated by Subsets of a Group ..... 61
2.5 The Lattice of Subgroups of a Group ..... 66
Chapter 3 Quotient Groups and Homomorphisms ..... 73
3.1 Definitions and Examples ..... 73
3.2 More on Cosets and Lagrange's Theorem ..... 89
3.3 The Isomorphism Theorems ..... 97
3.4 Composition Series and the Hölder Program ..... 101
3.5 Transpositions and the Alternating Group ..... 106
Chapter 4 Group Actions ..... 112
4.1 Group Actions and Permutation Representations ..... 112
4.2 Groups Acting on Themselves by Left Multiplication-Cayley's Theorem ..... 118
4.3 Groups Acting on Themselves by Conjugation-The Class Equation ..... 122
4.4 Automorphisms ..... 133
4.5 The Sylow Theorems ..... 139
4.6 The Simplicity of $A_{n}$ ..... 149
Chapter 5 Direct and Semidirect Products and Abelian Groups ..... 152
5.1 Direct Products ..... 152
5.2 The Fundamental Theorem of Finitely Generated Abelian Groups ..... 158
5.3 Table of Groups of Small Order ..... 167
5.4 Recognizing Direct Products ..... 169
5.5 Semidirect Products ..... 175
Chapter 6 Further Topics in Group Theory ..... 188
6.1 p-groups, Nilpotent Groups, and Solvable Groups ..... 188
6.2 Applications in Groups of Medium Order ..... 201
6.3 A Word on Free Groups ..... 215
Part II - RING THEORY ..... 222
Chapter 7 Introduction to Rings ..... 223
7.1 Basic Definitions and Examples ..... 223
7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings ..... 233
7.3 Ring Homomorphisms an Quotient Rings ..... 239
7.4 Properties of Ideals ..... 251
7.5 Rings of Fractions ..... 260
7.6 The Chinese Remainder Theorem ..... 265
Chapter 8 Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains ..... 270
8.1 Euclidean Domains ..... 270
8.2 Principal Ideal Domains (P.I.D.s) ..... 279
8.3 Unique Factorization Domains (U.F.D.s) ..... 283
Chapter 9 Polynomial Rings ..... 295
9.1 Definitions and Basic Properties ..... 295
9.2 Polynomial Rings over Fields I ..... 299
9.3 Polynomial Rings that are Unique Factorization Domains ..... 303
9.4 Irreducibility Criteria ..... 307
9.5 Polynomial Rings over Fields II ..... 313
9.6 Polynomials in Several Variables over a Field and Gröbner Bases ..... 315
Part III - MODULES AND VECTOR SPACES ..... 336
Chapter 10 Introduction to Module Theory ..... 337
10.1 Basic Definitions and Examples ..... 337
10.2 Quotient Modules and Module Homomorphisms ..... 345
10.3 Generation of Modules, Direct Sums, and Free Modules ..... 351
10.4 Tensor Products of Modules ..... 359
10.5 Exact Sequences-Projective, Injective, and Flat Modules ..... 378
Chapter 11 Vector Spaces ..... 408
11.1 Definitions and Basic Theory ..... 408
11.2 The Matrix of a Linear Transformation ..... 415
11.3 Dual Vector Spaces ..... 431
11.4 Determinants ..... 435
11.5 Tensor Algebras, Symmetric and Exterior Algebras ..... 441
Chapter 12 Modules over Principal Ideal Domains ..... 456
12.1 The Basic Theory ..... 458
12.2 The Rational Canonical Form ..... 472
12.3 The Jordan Canonical Form ..... 491
Chapter 13 Field Theory ..... 510
13.1 Basic Theory of Field Extensions ..... 510
13.2 Algebraic Extensions ..... 520
13.3 Classical Straightedge and Compass Constructions ..... 531
13.4 Splitting Fields and Algebraic Closures ..... 536
13.5 Separable and Inseparable Extensions ..... 545
13.6 Cyclotomic Polynomials and Extensions ..... 552
Chapter 14 Galois Theory ..... 558
14.1 Basic Definitions ..... 558
14.2 The Fundamental Theorem of Galois Theory ..... 567
14.3 Finite Fields ..... 585
14.4 Composite Extensions and Simple Extensions ..... 591
14.5 Cyclotomic Extensions and Abelian Extensions over $\mathbb{Q}$ ..... 596
14.6 Galois Groups of Polynomials ..... 606
14.7 Solvable and Radical Extensions: Insolvability of the Quintic ..... 625
14.8 Computation of Galois Groups over $\mathbb{Q}$ ..... 640
14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups ..... 645
Part V - AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA ..... 655
Chapter 15 Commutative Rings and Algebraic Geometry ..... 656
15.1 Noetherian Rings and Affine Algebraic Sets ..... 656
15.2 Radicals and Affine Varieties ..... 673
15.3 Integral Extensions and Hilbert's Nullstellensatz ..... 691
15.4 Localization ..... 706
15.5 The Prime Spectrum of a Ring ..... 731
Chapter 16 Artinian Rings, Discrete Valuation Rings, and Dedekind Domains ..... 750
16.1 Artinian Rings ..... 750
16.2 Discrete Valuation Rings ..... 755
16.3 Dedekind Domains ..... 764Chapter 17 Introduction to Homological Algebra and Group Cohomology ..... 776
17.1 Introduction to Homological Algebra-Ext and Tor ..... 777
17.2 The Cohomology of Groups ..... 798
17.3 Crossed Homomorphisms and $H^{7}(G, A)$ ..... 814
17.4 Group Extensions, Factor Sets and $H^{2}(G, A)$ ..... 824
Part VI - INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS ..... 839
Chapter 18 Representation Theory and Character Theory ..... 840
18.1 Linear Actions and Modules over Group Rings ..... 840
18.2 Wedderburn's Theorem and Some Consequences ..... 854
18.3 Character Theory and the Orthogonality Relations ..... 864
Chapter 19 Examples and Applications of Character Theory ..... 880
19.1 Characters of Groups of Small Order ..... 880
19.2 Theorems of Burnside and Hall ..... 886
19.3 Introduction to the Theory of Induced Characters ..... 892
Appendix I: Cartesian Products and Zorn's Lemma ..... 905
Appendix II: Category Theory ..... 911
Index ..... 919
# Preface to the Third Edition
The principal change from the second edition is the addition of Gröbner bases to this edition. The basic theory is introduced in a new Section 9.6. Applications to solving systems of polynomial equations (elimination theory) appear at the end of this section, rounding it out as a self-contained foundation in the topic. Additional applications and examples are then woven into the treatment of affine algebraic sets and $k$-algebra homomorphisms in Chapter 15. Although the theory in the latter chapter remains independent of Gröbner bases, the new applications, examples and computational techniques significantly enhance the development, and we recommend that Section 9.6 be read either as a segue to or in parallel with Chapter 15. A wealth of exercises involving Gröbner bases, both computational and theoretical in nature, have been added in Section 9.6 and Chapter 15. Preliminary exercises on Gröbner bases can (and should, as an aid to understanding the algorithms) be done by hand, but more extensive computations, and in particular most of the use of Gröbner bases in the exercises in Chapter 15, will likely require computer assisted computation.
Other changes include a streamlining of the classification of simple groups of order 168 (Section 6.2), with the addition of a uniqueness proof via the projective plane of order 2. Some other proofs or portions of the text have been revised slightly. A number of new exercises have been added throughout the book, primarily at the ends of sections in order to preserve as much as possible the numbering schemes of earlier editions. In particular, exercises have been added on free modules over noncommutative rings (10.3), on Krull dimension (15.3), and on flat modules (10.5 and 17.1).
As with previous editions, the text contains substantially more than can normally be covered in a one year course. A basic introductory (one year) course should probably include Part I up through Section 5.3, Part II through Section 9.5, Sections 10.1, 10.2, 10.3, 11.1, 11.2 and Part IV. Chapter 12 should also be covered, either before or after Part IV. Additional topics from Chapters 5, 6, 9, 10 and 11 may be interspersed in such a course, or covered at the end as time permits.
Sections 10.4 and 10.5 are at a slightly higher level of difficulty than the initial sections of Chapter 10, and can be deferred on a first reading for those following the text sequentially. The latter section on properties of exact sequences, although quite long, maintains coherence through a parallel treatment of three basic functors in respective subsections.
Beyond the core material, the third edition provides significant flexibility for students and instructors wishing to pursue a number of important areas of modern algebra,
either in the form of independent study or courses. For example, well integrated onesemester courses for students with some prior algebra background might include the following: Section 9.6 and Chapters 15 and 16; or Chapters 10 and 17; or Chapters 5, 6 and Part VI. Each of these would also provide a solid background for a follow-up course delving more deeply into one of many possible areas: algebraic number theory, algebraic topology, algebraic geometry, representation theory, Lie groups, etc.
The choice of new material and the style for developing and integrating it into the text are in consonance with a basic theme in the book: the power and beauty that accrues from a rich interplay between different areas of mathematics. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible. We have not attempted to be encyclopedic, but have tried to touch on many of the central themes in elementary algebra in a manner suggesting the very natural development of these ideas.
A number of important ideas and results appear in the exercises. This is not because they are not significant, rather because they did not fit easily into the flow of the text but were too important to leave out entirely. Sequences of exercises on one topic are prefaced with some remarks and are structured so that they may be read without actually doing the exercises. In some instances, new material is introduced first in the exercises-often a few sections before it appears in the text-so that students may obtain an easier introduction to it by doing these exercises (e.g., Lagrange's Theorem appears in the exercises in Section 1.7 and in the text in Section 3.2). All the exercises are within the scope of the text and hints are given [in brackets] where we felt they were needed. Exercises we felt might be less straightforward are usually phrased so as to provide the answer to the exercise; as well many exercises have been broken down into a sequence of more routine exercises in order to make them more accessible.
We have also purposely minimized the functorial language in the text in order to keep the presentation as elementary as possible. We have refrained from providing specific references for additional reading when there are many fine choices readily available. Also, while we have endeavored to include as many fundamental topics as possible, we apologize if for reasons of space or personal taste we have neglected any of the reader's particular favorites.
We are deeply grateful to and would like here to thank the many students and colleagues around the world who, over more than 15 years, have offered valuable comments, insights and encouragement-their continuing support and interest have motivated our writing of this third edition.
David Dummit
Richard Foote
June, 2003
# Preliminaries
Some results and notation that are used throughout the text are collected in this chapter for convenience. Students may wish to review this chapter quickly at first and then read each section more carefully again as the concepts appear in the course of the text.
### 0.1 BASICS
The basics of set theory: sets, $\cap, \cup, \epsilon$, etc. should be familiar to the reader. Our notation for subsets of a given set $A$ will be
$$
B=\{a \in A \mid \ldots \text { (conditions on a) } \ldots\} .
$$
The order or cardinality of a set $A$ will be denoted by $|A|$. If $A$ is a finite set the order of $A$ is simply the number of elements of $A$.
It is important to understand how to test whether a particular $x \in A$ lies in a subset $B$ of $A$ (cf. Exercises 1-4). The Cartesian product of two sets $A$ and $B$ is the collection $A \times B=\{(a, b) \mid a \in A, b \in B\}$, of ordered pairs of elements from $A$ and $B$.
We shall use the following notation for some common sets of numbers:
(1) $\mathbb{Z}=\{0, \pm 1, \pm 2, \pm 3, \ldots\}$ denotes the integers (the $\mathbb{Z}$ is for the German word for numbers: "Zahlen").
(2) $\mathbb{Q}=\{a / b \mid a, b \in \mathbb{Z}, b \neq 0\}$ denotes the rational numbers (or rationals).
(3) $\mathbb{R}=\left\{\right.$ all decimal expansions $\pm d_{1} d_{2} \ldots d_{n} \cdot a_{1} a_{2} a_{3} \ldots\}$ denotes the real numbers (or reals).
(4) $\mathbb{C}=\left\{a+b i \mid a, b \in \mathbb{R}, i^{2}=-1\right\}$ denotes the complex numbers.
(5) $\mathbb{Z}^{+}, \mathbb{Q}^{+}$and $\mathbb{R}^{+}$will denote the positive (nonzero) elements in $\mathbb{Z}, \mathbb{Q}$ and $\mathbb{R}$, respectively.
We shall use the notation $f: A \rightarrow B$ or $A \xrightarrow{f} B$ to denote a function $f$ from $A$ to $B$ and the value of $f$ at $a$ is denoted $f(a)$ (i.e., we shall apply all our functions on the left). We use the words function and map interchangeably. The set $A$ is called the domain of $f$ and $B$ is called the codomain of $f$. The notation $f: a \mapsto b$ or $a \mapsto b$ if $f$ is understood indicates that $f(a)=b$, i.e., the function is being specified on elements.
If the function $f$ is not specified on elements it is important in general to check that $f$ is well defined, i.e., is unambiguously determined. For example, if the set $A$ is the union of two subsets $A_{1}$ and $A_{2}$ then one can try to specify a function from $A$
to the set $\{0,1\}$ by declaring that $f$ is to map everything in $A_{1}$ to 0 and is to map everything in $A_{2}$ to 1 . This unambiguously defines $f$ unless $A_{1}$ and $A_{2}$ have elements in common (in which case it is not clear whether these elements should map to 0 or to 1). Checking that this $f$ is well defined therefore amounts to checking that $A_{1}$ and $A_{2}$ have no intersection.
The set
$$
f(A)=\{b \in B \mid b=f(a), \text { for some } a \in A\}
$$
is a subset of $B$, called the range or image of $f$ (or the image of $A$ under $f$ ). For each subset $C$ of $B$ the set
$$
f^{-1}(C)=\{a \in A \mid f(a) \in C\}
$$
consisting of the elements of $A$ mapping into $C$ under $f$ is called the preimage or inverse image of $C$ under $f$. For each $b \in B$, the preimage of $\{b\}$ under $f$ is called the fiber of $f$ over $b$. Note that $f^{-1}$ is not in general a function and that the fibers of $f$ generally contain many elements since there may be many elements of $A$ mapping to the element $b$.
If $f: A \rightarrow B$ and $g: B \rightarrow C$, then the composite map $g \circ f: A \rightarrow C$ is defined by
$$
(g \circ f)(a)=g(f(a))
$$
Let $f: A \rightarrow B$.
(1) $f$ is injective or is an injection if whenever $a_{1} \neq a_{2}$, then $f\left(a_{1}\right) \neq f\left(a_{2}\right)$.
(2) $f$ is surjective or is a surjection if for all $b \in B$ there is some $a \in A$ such that $f(a)=b$, i.e., the image of $f$ is all of $B$. Note that since a function always maps onto its range (by definition) it is necessary to specify the codomain $B$ in order for the question of surjectivity to be meaningful.
(3) $f$ is bijective or is a bijection if it is both injective and surjective. If such a bijection $f$ exists from $A$ to $B$, we say $A$ and $B$ are in bijective correspondence.
(4) $f$ has a left inverse if there is a function $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is the identity map on $A$, i.e., $(g \circ f)(a)=a$, for all $a \in A$.
(5) $f$ has a right inverse if there is a function $h: B \rightarrow A$ such that $f \circ h: B \rightarrow B$ is the identity map on $B$.
Proposition 1. Let $f: A \rightarrow B$.
(1) The map $f$ is injective if and only if $f$ has a left inverse.
(2) The map $f$ is surjective if and only if $f$ has a right inverse.
(3) The map $f$ is a bijection if and only if there exists $g: B \rightarrow A$ such that $f \circ g$ is the identity map on $B$ and $g \circ f$ is the identity map on $A$.
(4) If $A$ and $B$ are finite sets with the same number of elements (i.e., $|A|=|B|$ ), then $f: A \rightarrow B$ is bijective if and only if $f$ is injective if and only if $f$ is surjective.
# Proof: Exercise.
In the situation of part (3) of the proposition above the map $g$ is necessarily unique and we shall say $g$ is the 2 -sided inverse (or simply the inverse) of $f$.
A permutation of a set $A$ is simply a bijection from $A$ to itself.
If $A \subseteq B$ and $f: B \rightarrow C$, we denote the restriction of $f$ to $A$ by $\left.f\right|_{A}$. When the domain we are considering is understood we shall occasionally denote $\left.f\right|_{A}$ again simply as $f$ even though these are formally different functions (their domains are different).
If $A \subseteq B$ and $g: A \rightarrow C$ and there is a function $f: B \rightarrow C$ such that $\left.f\right|_{A}=g$, we shall say $f$ is an extension of $g$ to $B$ (such a map $f$ need not exist nor be unique).
Let $A$ be a nonempty set.
(1) A binary relation on a set $A$ is a subset $R$ of $A \times A$ and we write $a \sim b$ if $(a, b) \in R$.
(2) The relation $\sim$ on $A$ is said to be:
(a) reflexive if $a \sim a$, for all $a \in A$,
(b) symmetric if $a \sim b$ implies $b \sim a$ for all $a, b \in A$,
(c) transitive if $a \sim b$ and $b \sim c$ implies $a \sim c$ for all $a, b, c \in A$.
A relation is an equivalence relation if it is reflexive, symmetric and transitive.
(3) If $\sim$ defines an equivalence relation on $A$, then the equivalence class of $a \in A$ is defined to be $\{x \in A \mid x \sim a\}$. Elements of the equivalence class of $a$ are said to be equivalent to $a$. If $C$ is an equivalence class, any element of $C$ is called a representative of the class $C$.
(4) A partition of $A$ is any collection $\left\{A_{i} \mid i \in I\right\}$ of nonempty subsets of $A$ ( $I$ some indexing set) such that
(a) $A=\cup_{i \in I} A_{i}$, and
(b) $A_{i} \cap A_{j}=\emptyset$, for all $i, j \in I$ with $i \neq j$
i.e., $A$ is the disjoint union of the sets in the partition.
The notions of an equivalence relation on $A$ and a partition of $A$ are the same:
Proposition 2. Let $A$ be a nonempty set.
(1) If $\sim$ defines an equivalence relation on $A$ then the set of equivalence classes of $\sim$ form a partition of $A$.
(2) If $\left\{A_{i} \mid i \in I\right\}$ is a partition of $A$ then there is an equivalence relation on $A$ whose equivalence classes are precisely the sets $A_{i}, i \in I$.
Proof: Omitted.
Finally, we shall assume the reader is familiar with proofs by induction.
# EXERCISES
In Exercises 1 to 4 let $\mathcal{A}$ be the set of $2 \times 2$ matrices with real number entries. Recall that matrix multiplication is defined by
$$
\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\left(\begin{array}{ll}
p & q \\
r & s
\end{array}\right)=\left(\begin{array}{ll}
a p+b r & a q+b s \\
c p+d r & c q+d s
\end{array}\right)
$$
Let
$$
M=\left(\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right)
$$
and let
$$
\mathcal{B}=\{X \in \mathcal{A} \mid M X=X M\}
$$
1. Determine which of the following elements of $\mathcal{A}$ lie in $\mathcal{B}$ :
$$
\left(\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right),\left(\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right),\left(\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right),\left(\begin{array}{ll}
1 & 1 \\
1 & 0
\end{array}\right),\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right),\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right)
$$
2. Prove that if $P, Q \in \mathcal{B}$, then $P+Q \in \mathcal{B}$ (where + denotes the usual sum of two matrices).
3. Prove that if $P, Q \in \mathcal{B}$, then $P \cdot Q \in \mathcal{B}$ (where - denotes the usual product of two matrices).
4. Find conditions on $p, q, r, s$ which determine precisely when $\left(\begin{array}{ll}p & q \\ r & s\end{array}\right) \in \mathcal{B}$.
5. Determine whether the following functions $f$ are well defined:
(a) $f: \mathbb{Q} \rightarrow \mathbb{Z}$ defined by $f(a / b)=a$.
(b) $f: \mathbb{Q} \rightarrow \mathbb{Q}$ defined by $f(a / b)=a^{2} / b^{2}$.
6. Determine whether the function $f: \mathbb{R}^{+} \rightarrow \mathbb{Z}$ defined by mapping a real number $r$ to the first digit to the right of the decimal point in a decimal expansion of $r$ is well defined.
7. Let $f: A \rightarrow B$ be a surjective map of sets. Prove that the relation
$$
a \sim b \text { if and only if } f(a)=f(b)
$$
is an equivalence relation whose equivalence classes are the fibers of $f$.
# 0.2 PROPERTIES OF THE INTEGERS
The following properties of the integers $\mathbb{Z}$ (many familiar from elementary arithmetic) will be proved in a more general context in the ring theory of Chapter 8, but it will be necessary to use them in Part I (of course, none of the ring theory proofs of these properties will rely on the group theory).
(1) (Well Ordering of $\mathbb{Z}$ ) If $A$ is any nonempty subset of $\mathbb{Z}^{+}$, there is some element $m \in A$ such that $m \leq a$, for all $a \in A(m$ is called a minimal element of $A)$.
(2) If $a, b \in \mathbb{Z}$ with $a \neq 0$, we say $a$ divides $b$ if there is an element $c \in \mathbb{Z}$ such that $b=a c$. In this case we write $a \mid b$; if $a$ does not divide $b$ we write $a \nmid b$.
(3) If $a, b \in \mathbb{Z}-\{0\}$, there is a unique positive integer $d$, called the greatest common divisor of $a$ and $b$ (or g.c.d. of $a$ and $b$ ), satisfying:
(a) $d \mid a$ and $d \mid b$ (so $d$ is a common divisor of $a$ and $b$ ), and
(b) if $e \mid a$ and $e \mid b$, then $e \mid d$ (so $d$ is the greatest such divisor).
The g.c.d. of $a$ and $b$ will be denoted by $(a, b)$. If $(a, b)=1$, we say that $a$ and $b$ are relatively prime.
(4) If $a, b \in \mathbb{Z}-\{0\}$, there is a unique positive integer $l$, called the least common multiple of $a$ and $b$ (or l.c.m. of $a$ and $b$ ), satisfying:
(a) $a \mid l$ and $b \mid l$ (so $l$ is a common multiple of $a$ and $b$ ), and
(b) if $a \mid m$ and $b \mid m$, then $l \mid m$ (so $l$ is the least such multiple).
The connection between the greatest common divisor $d$ and the least common multiple $l$ of two integers $a$ and $b$ is given by $d l=a b$.
(5) The Division Algorithm: if $a, b \in \mathbb{Z}-\{0\}$, then there exist unique $q, r \in \mathbb{Z}$ such that
$$
a=q b+r \quad \text { and } \quad 0 \leq r<|b|
$$
where $q$ is the quotient and $r$ the remainder. This is the usual "long division" familiar from elementary arithmetic.
(6) The Euclidean Algorithm is an important procedure which produces a greatest common divisor of two integers $a$ and $b$ by iterating the Division Algorithm: if $a, b \in \mathbb{Z}-\{0\}$, then we obtain a sequence of quotients and remainders
$$
\begin{aligned}
a & =q_{0} b+r_{0} \\
b & =q_{1} r_{0}+r_{1} \\
r_{0} & =q_{2} r_{1}+r_{2} \\
r_{1} & =q_{3} r_{2}+r_{3} \\
\vdots & \\
r_{n-2} & =q_{n} r_{n-1}+r_{n} \\
r_{n-1} & =q_{n+1} r_{n}
\end{aligned}
$$
where $r_{n}$ is the last nonzero remainder. Such an $r_{n}$ exists since $|b|>\left|r_{0}\right|>\left|r_{1}\right|>$ $\cdots>\left|r_{n}\right|$ is a decreasing sequence of strictly positive integers if the remainders are nonzero and such a sequence cannot continue indefinitely. Then $r_{n}$ is the g.c.d. $(a, b)$ of $a$ and $b$.
# Example
Suppose $a=57970$ and $b=10353$. Then applying the Euclidean Algorithm we obtain:
$$
\begin{aligned}
57970 & =(5) 10353+6205 \\
10353 & =(1) 6205+4148 \\
6205 & =(1) 4148+2057 \\
4148 & =(2) 2057+34 \\
2057 & =(60) 34+17 \\
34 & =(2) 17
\end{aligned}
$$
which shows that $(57970,10353)=17$.
(7) One consequence of the Euclidean Algorithm which we shall use regularly is the following: if $a, b \in \mathbb{Z}-\{0\}$, then there exist $x, y \in \mathbb{Z}$ such that
$$
(a, b)=a x+b y
$$
that is, the g.c.d. of $a$ and $b$ is a $\mathbb{Z}$-linear combination of $a$ and $b$. This follows by recursively writing the element $r_{n}$ in the Euclidean Algorithm in terms of the previous remainders (namely, use equation ( $n$ ) above to solve for $r_{n}=r_{n-2}-q_{n} r_{n-1}$ in terms of the remainders $r_{n-1}$ and $r_{n-2}$, then use equation $(n-1)$ to write $r_{n}$ in terms of the remainders $r_{n-2}$ and $r_{n-3}$, etc., eventually writing $r_{n}$ in terms of $a$ and b).
# Example
Suppose $a=57970$ and $b=10353$, whose greatest common divisor we computed above to be 17. From the fifth equation (the next to last equation) in the Euclidean Algorithm applied to these two integers we solve for their greatest common divisor: $17=2057-(60) 34$. The fourth equation then shows that $34=4148-(2) 2057$, so substituting this expression for the previous remainder 34 gives the equation $17=2057-(60)[4148-(2) 2057]$, i.e., $17=(121) 2057-(60) 4148$. Solving the third equation for 2057 and substituting gives $17=(121)[6205-(1) 4148]-(60) 4148=(121) 6205-(181) 4148$. Using the second equation to solve for 4148 and then the first equation to solve for 6205 we finally obtain
$$
17=(302) 57970-(1691) 10353
$$
as can easily be checked directly. Hence the equation $a x+b y=(a, b)$ for the greatest common divisor of $a$ and $b$ in this example has the solution $x=302$ and $y=-1691$. Note that it is relatively unlikely that this relation would have been found simply by guessing.
The integers $x$ and $y$ in (7) above are not unique. In the example with $a=57970$ and $b=10353$ we determined one solution to be $x=302$ and $y=-1691$, for instance, and it is relatively simple to check that $x=-307$ and $y=1719$ also satisfy $57970 x+10353 y=17$. The general solution for $x$ and $y$ is known (cf. the exercises below and in Chapter 8).
(8) An element $p$ of $\mathbb{Z}^{+}$is called a prime if $p>1$ and the only positive divisors of $p$ are 1 and $p$ (initially, the word prime will refer only to positive integers). An integer $n>1$ which is not prime is called composite. For example, $2,3,5,7,11,13,17,19, \ldots$ are primes and $4,6,8,9,10,12,14,15,16,18, \ldots$ are composite.
An important property of primes (which in fact can be used to define the primes (cf. Exercise 3)) is the following: if $p$ is a prime and $p \mid a b$, for some $a, b \in \mathbb{Z}$, then either $p \mid a$ or $p \mid b$.
(9) The Fundamental Theorem of Arithmetic says: if $n \in \mathbb{Z}, n>1$, then $n$ can be factored uniquely into the product of primes, i.e., there are distinct primes $p_{1}, p_{2}, \ldots, p_{s}$ and positive integers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s}$ such that
$$
n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{s}^{\alpha_{s}}
$$
This factorization is unique in the sense that if $q_{1}, q_{2}, \ldots, q_{t}$ are any distinct primes and $\beta_{1}, \beta_{2}, \ldots, \beta_{t}$ positive integers such that
$$
n=q_{1}^{\beta_{1}} q_{2}^{\beta_{2}} \ldots q_{t}^{\beta_{t}}
$$
then $s=t$ and if we arrange the two sets of primes in increasing order, then $q_{i}=p_{i}$ and $\alpha_{i}=\beta_{i}, 1 \leq i \leq s$. For example, $n=1852423848=2^{3} 3^{2} 11^{2} 19^{3} 31$ and this decomposition into the product of primes is unique.
Suppose the positive integers $a$ and $b$ are expressed as products of prime powers:
$$
a=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{s}^{\alpha_{s}}, \quad b=p_{1}^{\beta_{1}} p_{2}^{\beta_{2}} \ldots p_{s}^{\beta_{s}}
$$
where $p_{1}, p_{2}, \ldots, p_{s}$ are distinct and the exponents are $\geq 0$ (we allow the exponents to be 0 here so that the products are taken over the same set of primes - the exponent will be 0 if that prime is not actually a divisor). Then the greatest common divisor of $a$ and $b$ is
$$
(a, b)=p_{1}^{\min \left(\alpha_{1}, \beta_{1}\right)} p_{2}^{\min \left(\alpha_{2}, \beta_{2}\right)} \ldots p_{s}^{\min \left(\alpha_{s}, \beta_{s}\right)}
$$
(and the least common multiple is obtained by instead taking the maximum of the $\alpha_{i}$ and $\beta_{i}$ instead of the minimum).
# Example
In the example above, $a=57970$ and $b=10353$ can be factored as $a=2 \cdot 5 \cdot 11 \cdot 17 \cdot 31$ and $b=3 \cdot 7 \cdot 17 \cdot 29$, from which we can immediately conclude that their greatest common divisor is 17 . Note, however, that for large integers it is extremely difficult to determine their prime factorizations (several common codes in current use are based on this difficulty, in fact), so that this is not an effective method to determine greatest common divisors in general. The Euclidean Algorithm will produce greatest common divisors quite rapidly without the need for the prime factorization of $a$ and $b$.
10) The Euler $\varphi$-function is defined as follows: for $n \in \mathbb{Z}^{+}$let $\varphi(n)$ be the number of positive integers $a \leq n$ with $a$ relatively prime to $n$, i.e., $(a, n)=1$. For example, $\varphi(12)=4$ since $1,5,7$ and 11 are the only positive integers less than or equal to 12 which have no factors in common with 12 . Similarly, $\varphi(1)=1, \varphi(2)=1$, $\varphi(3)=2, \varphi(4)=2, \varphi(5)=4, \varphi(6)=2$, etc. For primes $p, \varphi(p)=p-1$, and, more generally, for all $a \geq 1$ we have the formula
$$
\varphi\left(p^{a}\right)=p^{a}-p^{a-1}=p^{a-1}(p-1)
$$
The function $\varphi$ is multiplicative in the sense that
$$
\varphi(a b)=\varphi(a) \varphi(b) \quad \text { if }(a, b)=1
$$
(note that it is important here that $a$ and $b$ be relatively prime). Together with the formula above this gives a general formula for the values of $\varphi$ : if $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \ldots p_{s}^{\alpha_{s}}$, then
$$
\begin{aligned}
\varphi(n) & =\varphi\left(p_{1}^{\alpha_{1}}\right) \varphi\left(p_{2}^{\alpha_{2}}\right) \ldots \varphi\left(p_{s}^{\alpha_{s}}\right) \\
& =p_{1}^{\alpha_{1}-1}\left(p_{1}-1\right) p_{2}^{\alpha_{2}-1}\left(p_{2}-1\right) \ldots p_{s}^{\alpha_{s}-1}\left(p_{s}-1\right)
\end{aligned}
$$
For example, $\varphi(12)=\varphi\left(2^{2}\right) \varphi(3)=2^{1}(2-1) 3^{0}(3-1)=4$. The reader should note that we shall use the letter $\varphi$ for many different functions throughout the text so when we want this letter to denote Euler's function we shall be careful to indicate this explicitly.
## EXERCISES
1. For each of the following pairs of integers $a$ and $b$, determine their greatest common divisor, their least common multiple, and write their greatest common divisor in the form $a x+b y$ for some integers $x$ and $y$.
(a) $a=20, b=13$.
(b) $a=69, b=372$.
(c) $a=792, b=275$.
(d) $a=11391, b=5673$.
(e) $a=1761, b=1567$.
(f) $a=507885, b=60808$.
2. Prove that if the integer $k$ divides the integers $a$ and $b$ then $k$ divides $a s+b t$ for every pair of integers $s$ and $t$. | Abstract Algebra, 3rd Edition (David S. Dummit, Richard M. Foote) (Z-Library) (converted to markdown via Mistral OCR) |
## ALGEBRA
## Arithmetic Operations
$a(b+c)=a b+a c$
$\frac{a}{b}+\frac{c}{d}=\frac{a d+b c}{b d}$
$\frac{a+c}{b}=\frac{a}{b}+\frac{c}{b}$
$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \times \frac{d}{c}=\frac{a d}{b c}$
## Exponents and Radicals
$x^{m} x^{n}=x^{m+n}$
$\left(x^{m}\right)^{n}=x^{m n}$
$(x y)^{n}=x^{n} y^{n}$
$x^{1 / n}=\sqrt[n]{x}$
$x^{m / n}=\sqrt[n]{x}$
$\sqrt[n]{x y}=\sqrt[n]{x} \sqrt[n]{y}$
$$
\frac{x^{m}}{x^{n}}=x^{m-n}
$$
$x^{-n}=\frac{1}{x^{n}}$
$\left(\frac{x}{y}\right)^{n}=\frac{x^{n}}{y^{n}}$
$x^{m / n}=\sqrt[n]{x^{m}}=(\sqrt[n]{x})^{n}$
$\sqrt[n]{\sqrt{x y}}=\sqrt[n]{x} \sqrt[n]{y}$
$\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}$
## Factoring Special Polynomials
$x^{2}-y^{2}=(x+y)(x-y)$
$x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)$
$x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)$
## Binomial Theorem
$(x+y)^{2}=x^{2}+2 x y+y^{2} \quad(x-y)^{2}=x^{2}-2 x y+y^{2}$
$(x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}$
$(x-y)^{3}=x^{3}-3 x^{2} y+3 x y^{2}-y^{3}$
$(x+y)^{n}=x^{n}+n x^{n-1} y+\frac{n(n-1)}{2} x^{n-2} y^{2}$
$$
+\cdots+\binom{n}{k} x^{n-k} y^{k}+\cdots+n x y^{n-1}+y^{n}
$$
where $\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdot 3 \cdots \cdot k}$
## Quadratic Formula
If $a x^{2}+b x+c=0$, then $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$.
## Inequalities and Absolute Value
If $a<b$ and $b<c$, then $a<c$.
If $a<b$, then $a+c<b+c$.
If $a<b$ and $c>0$, then $c a<c b$.
If $a<b$ and $c<0$, then $c a>c b$.
If $a>0$, then
$$
\begin{array}{ll}
|x|=a \quad \text { means } \quad x=a \quad \text { or } \quad x=-a \\
|x|<a \quad \text { means } \quad-a<x<a \\
|x|>a \quad \text { means } \quad x>a \quad \text { or } \quad x<-a
\end{array}
$$
## GEOMETRY
## Geometric Formulas
Formulas for area $A$, circumference $C$, and volume $V$ :
Triangle
$A=\frac{1}{2} b h$
$=\frac{1}{2} a b \sin \theta$
Circle
$A=\pi r^{2}$
$C=2 \pi r$
Sector of Circle
$A=\frac{1}{2} r^{2} \theta$
$s=r \theta(\theta$ in radians $)$
Sphere
$V=\frac{4}{3} \pi r^{3}$
$A=4 \pi r^{2}$
Cylinder
$V=\pi r^{2} h$
Cone
$V=\frac{1}{3} \pi r^{2} h$
$A=\pi r \sqrt{r^{2}+h^{2}}$
## Distance and Midpoint Formulas
Distance between $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right)$ :
$$
d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}
$$
Midpoint of $\overline{P_{1} P_{2}} ;\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$
## Lines
Slope of line through $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right)$ :
$$
m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}
$$
Point-slope equation of line through $P_{1}\left(x_{1}, y_{1}\right)$ with slope $m$ :
$$
y-y_{1}=m\left(x-x_{1}\right)
$$
Slope-intercept equation of line with slope $m$ and $y$-intercept $b$ :
$$
y=m x+b
$$
## Circles
Equation of the circle with center $(h, k)$ and radius $r$ :
$$
(x-h)^{2}+(y-k)^{2}=r^{2}
$$
## TRIGONOMETRY
## Angle Measurement
$\pi$ radians $=180^{\circ}$
$1^{\circ}=\frac{\pi}{180}$ rad $1 \mathrm{rad}=\frac{180^{\circ}}{\pi}$
$s=r \theta$
( $\theta$ in radians)
## Right Angle Trigonometry
$\sin \theta=\frac{\text { opp }}{\text { hyp }} \quad \csc \theta=\frac{\text { hyp }}{\text { opp }}$
$\cos \theta=\frac{\text { adj }}{\text { hyp }} \quad \sec \theta=\frac{\text { hyp }}{\text { adj }}$
$\tan \theta=\frac{\text { opp }}{\text { adj }} \quad \cot \theta=\frac{\text { adj }}{\text { opp }}$
## Trigonometric Functions
$\sin \theta=\frac{y}{r} \quad \csc \theta=\frac{r}{y}$
$\cos \theta=\frac{x}{r} \quad \sec \theta=\frac{r}{x}$
$\tan \theta=\frac{y}{x} \quad \cot \theta=\frac{x}{y}$
## Graphs of Trigonometric Functions
Trigonometric Functions of Important Angles
| $\theta$ | radians | $\sin \theta$ | $\cos \theta$ | $\tan \theta$ |
| :--: | :--: | :--: | :--: | :--: |
| $0^{\circ}$ | 0 | 0 | 1 | 0 |
| $30^{\circ}$ | $\pi / 6$ | $1 / 2$ | $\sqrt{3} / 2$ | $\sqrt{3} / 3$ |
| $45^{\circ}$ | $\pi / 4$ | $\sqrt{2} / 2$ | $\sqrt{2} / 2$ | 1 |
| $60^{\circ}$ | $\pi / 3$ | $\sqrt{3} / 2$ | $1 / 2$ | $\sqrt{3}$ |
| $90^{\circ}$ | $\pi / 2$ | 1 | 0 | - |
## Fundamental Identities
$\csc \theta=\frac{1}{\sin \theta} \quad \sec \theta=\frac{1}{\cos \theta}$
$\tan \theta=\frac{\sin \theta}{\cos \theta} \quad \cot \theta=\frac{\cos \theta}{\sin \theta}$
$\cot \theta=\frac{1}{\tan \theta} \quad \sin ^{2} \theta+\cos ^{2} \theta=1$
$1+\tan ^{2} \theta=\sec ^{2} \theta \quad 1+\cot ^{2} \theta=\csc ^{2} \theta$
$\sin (-\theta)=-\sin \theta \quad \cos (-\theta)=\cos \theta$
$\tan (-\theta)=-\tan \theta \quad \sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta$
$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta \quad \tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta$
## The Law of Sines
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
## The Law of Cosines
$a^{2}=b^{2}+c^{2}-2 b c \cos A$
$b^{2}=a^{2}+c^{2}-2 a c \cos B$
$c^{2}=a^{2}+b^{2}-2 a b \cos C$
## Addition and Subtraction Formulas
$\sin (x+y)=\sin x \cos y+\cos x \sin y$
$\sin (x-y)=\sin x \cos y-\cos x \sin y$
$\cos (x+y)=\cos x \cos y-\sin x \sin y$
$\cos (x-y)=\cos x \cos y+\sin x \sin y$
$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$
$\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}$
## Double-Angle Formulas
$\sin 2 x=2 \sin x \cos x$
$\cos 2 x=\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1=1-2 \sin ^{2} x$
$\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}$
## Half-Angle Formulas
$\sin ^{2} x=\frac{1-\cos 2 x}{2} \quad \cos ^{2} x=\frac{1+\cos 2 x}{2}$
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# CALCULUS EARLY TRANSCENDENTALS
SEVENTH EDITION
## JAMES STEWART
McMASTER UNIVERSITY
AND
UNIVERSITY OF TORONTO
## Calculus: Early Transcendentals, Seventh Edition James Stewart
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# Contents
Preface ..... xi
To the Student ..... xxiii
Diagnostic Tests ..... xxiv
A PREVIEW OF CALCULUS ..... 1
1 Functions and Models ..... 9
1.1 Four Ways to Represent a Function ..... 10
1.2 Mathematical Models: A Catalog of Essential Functions ..... 23
1.3 New Functions from Old Functions ..... 36
1.4 Graphing Calculators and Computers ..... 44
1.5 Exponential Functions ..... 51
1.6 Inverse Functions and Logarithms ..... 58
Review ..... 72
Principles of Problem Solving ..... 75
2 Limits and Derivatives ..... 81
2.1 The Tangent and Velocity Problems ..... 82
2.2 The Limit of a Function ..... 87
2.3 Calculating Limits Using the Limit Laws ..... 99
2.4 The Precise Definition of a Limit ..... 108
2.5 Continuity ..... 118
2.6 Limits at Infinity; Horizontal Asymptotes ..... 130
2.7 Derivatives and Rates of Change ..... 143
Writing Project - Early Methods for Finding Tangents ..... 153
2.8 The Derivative as a Function ..... 154
Review ..... 165
Problems Plus ..... 170
3.1 Derivatives of Polynomials and Exponential Functions ..... 174
Applied Project $\cdot$ Building a Better Roller Coaster ..... 184
3.2 The Product and Quotient Rules ..... 184
3.3 Derivatives of Trigonometric Functions ..... 191
3.4 The Chain Rule ..... 198
Applied Project $\cdot$ Where Should a Pilot Start Descent? ..... 208
3.5 Implicit Differentiation ..... 209
Laboratory Project $\cdot$ Families of Implicit Curves ..... 217
3.6 Derivatives of Logarithmic Functions ..... 218
3.7 Rates of Change in the Natural and Social Sciences ..... 224
3.8 Exponential Growth and Decay ..... 237
3.9 Related Rates ..... 244
3.10 Linear Approximations and Differentials ..... 250
Laboratory Project $\cdot$ Taylor Polynomials ..... 256
3.11 Hyperbolic Functions ..... 257
Review ..... 264
Problems Plus ..... 268
4 Applications of Differentiation ..... 273
4.1 Maximum and Minimum Values ..... 274
Applied Project $\cdot$ The Calculus of Rainbows ..... 282
4.2 The Mean Value Theorem ..... 284
4.3 How Derivatives Affect the Shape of a Graph ..... 290
4.4 Indeterminate Forms and l'Hospital's Rule ..... 301
Writing Project $\cdot$ The Origins of I'Hospital's Rule ..... 310
4.5 Summary of Curve Sketching ..... 310
4.6 Graphing with Calculus and Calculators ..... 318
4.7 Optimization Problems ..... 325
Applied Project $\cdot$ The Shape of a Can ..... 337
4.8 Newton's Method ..... 338
4.9 Antiderivatives ..... 344
Review ..... 351
Problems Plus ..... 355
5.1 Areas and Distances 360
5.2 The Definite Integral 371 Discovery Project $\cdot$ Area Functions 385
5.3 The Fundamental Theorem of Calculus 386
5.4 Indefinite Integrals and the Net Change Theorem 397 Writing Project - Newton, Leibniz, and the Invention of Calculus 406
5.5 The Substitution Rule 407
Review 415
Problems Plus 419
# 6 Applications of Integration 421
6.1 Areas Between Curves 422
Applied Project - The Gini Index 429
6.2 Volumes 430
6.3 Volumes by Cylindrical Shells 441
6.4 Work 446
6.5 Average Value of a Function 451
Applied Project - Calculus and Baseball 455
Applied Project - Where to Sit at the Movies 456
Review 457
Problems Plus 459
## 7 Techniques of Integration 463
7.1 Integration by Parts 464
7.2 Trigonometric Integrals 471
7.3 Trigonometric Substitution 478
7.4 Integration of Rational Functions by Partial Fractions 484
7.5 Strategy for Integration 494
7.6 Integration Using Tables and Computer Algebra Systems 500 Discovery Project - Patterns in Integrals 505
7.7 Approximate Integration ..... 506
7.8 Improper Integrals ..... 519
Review ..... 529
Problems Plus ..... 533
8 Further Applications of Integration ..... 537
8.1 Arc Length ..... 538
Discovery Project $\cdot$ Arc Length Contest ..... 545
8.2 Area of a Surface of Revolution ..... 545
Discovery Project $\cdot$ Rotating on a Slant ..... 551
8.3 Applications to Physics and Engineering ..... 552
Discovery Project $\cdot$ Complementary Coffee Cups ..... 562
8.4 Applications to Economics and Biology ..... 563
8.5 Probability ..... 568
Review ..... 575
Problems Plus ..... 577
9 Differential Equations ..... 579
9.1 Modeling with Differential Equations ..... 580
9.2 Direction Fields and Euler's Method ..... 585
9.3 Separable Equations ..... 594
Applied Project $\cdot$ How Fast Does a Tank Drain? ..... 603
Applied Project $\cdot$ Which Is Faster, Going Up or Coming Down? ..... 604
9.4 Models for Population Growth ..... 605
9.5 Linear Equations ..... 616
9.6 Predator-Prey Systems ..... 622
Review ..... 629
Problems Plus ..... 633
10.1 Curves Defined by Parametric Equations ..... 636
Laboratory Project - Running Circles around Circles ..... 644
10.2 Calculus with Parametric Curves ..... 645
Laboratory Project - Bézier Curves ..... 653
10.3 Polar Coordinates ..... 654
Laboratory Project - Families of Polar Curves ..... 664
10.4 Areas and Lengths in Polar Coordinates ..... 665
10.5 Conic Sections ..... 670
10.6 Conic Sections in Polar Coordinates ..... 678
Review ..... 685
Problems Plus ..... 688
11 Infinite Sequences and Series ..... 689
11.1 Sequences ..... 690
Laboratory Project - Logistic Sequences ..... 703
11.2 Series ..... 703
11.3 The Integral Test and Estimates of Sums ..... 714
11.4 The Comparison Tests ..... 722
11.5 Alternating Series ..... 727
11.6 Absolute Convergence and the Ratio and Root Tests ..... 732
11.7 Strategy for Testing Series ..... 739
11.8 Power Series ..... 741
11.9 Representations of Functions as Power Series ..... 746
11.10 Taylor and Maclaurin Series ..... 753
Laboratory Project - An Elusive Limit ..... 767
Writing Project - How Newton Discovered the Binomial Series ..... 767
11.11 Applications of Taylor Polynomials ..... 768
Applied Project - Radiation from the Stars ..... 777
Review ..... 778
Problems Plus ..... 781
12.1 Three-Dimensional Coordinate Systems ..... 786
12.2 Vectors ..... 791
12.3 The Dot Product ..... 800
12.4 The Cross Product ..... 808
Discovery Project = The Geometry of a Tetrahedron ..... 816
12.5 Equations of Lines and Planes ..... 816
Laboratory Project = Putting 3D in Perspective ..... 826
12.6 Cylinders and Quadric Surfaces ..... 827
Review ..... 834
Problems Plus ..... 837
13 Vector Functions ..... 839
13.1 Vector Functions and Space Curves ..... 840
13.2 Derivatives and Integrals of Vector Functions ..... 847
13.3 Arc Length and Curvature ..... 853
13.4 Motion in Space: Velocity and Acceleration ..... 862
Applied Project = Kepler's Laws ..... 872
Review ..... 873
Problems Plus ..... 876
14 Partial Derivatives ..... 877
14.1 Functions of Several Variables ..... 878
14.2 Limits and Continuity ..... 892
14.3 Partial Derivatives ..... 900
14.4 Tangent Planes and Linear Approximations ..... 915
14.5 The Chain Rule ..... 924
14.6 Directional Derivatives and the Gradient Vector ..... 933
14.7 Maximum and Minimum Values ..... 946
Applied Project = Designing a Dumpster ..... 956
Discovery Project = Quadratic Approximations and Critical Points ..... 956
14.8 Lagrange Multipliers ..... 957
Applied Project $\cdot$ Rocket Science ..... 964
Applied Project - Hydro-Turbine Optimization ..... 966
Review ..... 967
Problems Plus ..... 971
15 Multiple Integrals ..... 973
15.1 Double Integrals over Rectangles ..... 974
15.2 Iterated Integrals ..... 982
15.3 Double Integrals over General Regions ..... 988
15.4 Double Integrals in Polar Coordinates ..... 997
15.5 Applications of Double Integrals ..... 1003
15.6 Surface Area ..... 1013
15.7 Triple Integrals ..... 1017
Discovery Project - Volumes of Hyperspheres ..... 1027
15.8 Triple Integrals in Cylindrical Coordinates ..... 1027
Discovery Project - The Intersection of Three Cylinders ..... 1032
15.9 Triple Integrals in Spherical Coordinates ..... 1033
Applied Project $\cdot$ Roller Derby ..... 1039
15.10 Change of Variables in Multiple Integrals ..... 1040
Review ..... 1049
Problems Plus ..... 1053
16 Vector Calculus ..... 1055
16.1 Vector Fields ..... 1056
16.2 Line Integrals ..... 1063
16.3 The Fundamental Theorem for Line Integrals ..... 1075
16.4 Green's Theorem ..... 1084
16.5 Curl and Divergence ..... 1091
16.6 Parametric Surfaces and Their Areas ..... 1099
16.7 Surface Integrals ..... 1110
16.8 Stokes' Theorem ..... 1122
Writing Project - Three Men and Two Theorems ..... 1128
16.9 The Divergence Theorem ..... 1128
16.10 Summary ..... 1135
Review ..... 1136
Problems Plus ..... 1139
17 Second-Order Differential Equations ..... 1141
17.1 Second-Order Linear Equations ..... 1142
17.2 Nonhomogeneous Linear Equations ..... 1148
17.3 Applications of Second-Order Differential Equations ..... 1156
17.4 Series Solutions ..... 1164
Review ..... 1169
Appendixes ..... A1
A Numbers, Inequalities, and Absolute Values ..... A2
B Coordinate Geometry and Lines ..... A10
C Graphs of Second-Degree Equations ..... A16
D Trigonometry ..... A24
E Sigma Notation ..... A34
F Proofs of Theorems ..... A39
G The Logarithm Defined as an Integral ..... A50
H Complex Numbers ..... A57
I Answers to Odd-Numbered Exercises ..... A65
Index ..... A135
# Preface
#### Abstract
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
GEORGE POLYA
The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus-both for its practical power and its surprising beauty. In this edition, as in the first six editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement.
The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation:
Focus on conceptual understanding.
I have tried to implement this goal through the Rule of Three: "Topics should be presented geometrically, numerically, and algebraically." Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. The Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well.
In writing the seventh edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum.
## Alternative Versions
I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions.
- Calculus: Early Transcendentals, Seventh Edition, Hybrid Version, is similar to the present textbook in content and coverage except that all end-of-section exercises are available only in Enhanced WebAssign. The printed text includes all end-of-chapter review material.
- Calculus, Seventh Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the second semester.
- Calculus, Seventh Edition, Hybrid Version, is similar to Calculus, Seventh Edition, in content and coverage except that all end-of-section exercises are available only in Enhanced WebAssign. The printed text includes all end-of-chapter review material.
- Essential Calculus is a much briefer book ( 800 pages), though it contains almost all of the topics in Calculus, Seventh Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website.
- Essential Calculus: Early Transcendentals resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3.
- Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters.
- Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking Engineering and Physics courses concurrently with calculus.
- Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences.
# What's New in the Seventh Edition?
The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I've incorporated into this edition:
- Some material has been rewritten for greater clarity or for better motivation. See, for instance, the introduction to maximum and minimum values on page 274, the introduction to series on page 703, and the motivation for the cross product on page 808.
- New examples have been added (see Example 4 on page 1021 for instance). And the solutions to some of the existing examples have been amplified. A case in point: I added details to the solution of Example 2.3.11 because when I taught Section 2.3 from the sixth edition I realized that students need more guidance when setting up inequalities for the Squeeze Theorem.
- The art program has been revamped: New figures have been incorporated and a substantial percentage of the existing figures have been redrawn.
- The data in examples and exercises have been updated to be more timely.
- Three new projects have been added: The Gini Index (page 429) explores how to measure income distribution among inhabitants of a given country and is a nice application of areas between curves. (I thank Klaus Volpert for suggesting this project.) Families of Implicit Curves (page 217) investigates the changing shapes of implicitly defined curves as parameters in a family are varied. Families of Polar Curves (page 664) exhibits the fascinating shapes of polar curves and how they evolve within a family.
- The section on the surface area of the graph of a function of two variables has been restored as Section 15.6 for the convenience of instructors who like to teach it after double integrals, though the full treatment of surface area remains in Chapter 16.
- I continue to seek out examples of how calculus applies to so many aspects of the real world. On page 909 you will see beautiful images of the earth's magnetic field strength and its second vertical derivative as calculated from Laplace's equation. I thank Roger Watson for bringing to my attention how this is used in geophysics and mineral exploration.
- More than $25 \%$ of the exercises in each chapter are new. Here are some of my favorites: $1.6 .58,2.6 .51,2.8 .13-14,3.3 .56,3.4 .67,3.5 .69-72,3.7 .22,4.3 .86$, $5.2 .51-53,6.4 .30,11.2 .49-50,11.10 .71-72,12.1 .44,12.4 .43-44$, and Problems 4, 5 , and 8 on pages $837-38$.
# Technology Enhancements
- The media and technology to support the text have been enhanced to give professors greater control over their course, to provide extra help to deal with the varying levels of student preparedness for the calculus course, and to improve support for conceptual understanding. New Enhanced WebAssign features including a customizable Cengage YouBook, Just in Time review, Show Your Work, Answer Evaluator, Personalized Study Plan, Master Its, solution videos, lecture video clips (with associated questions), and Visualizing Calculus (TEC animations with associated questions) have been developed to facilitate improved student learning and flexible classroom teaching.
- Tools for Enriching Calculus (TEC) has been completely redesigned and is accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.
## Features
CONCEPTUAL EXERCISES The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35-40, $2.8 .43-46,9.1 .11-13,10.1 .24-27,11.10 .2,13.2 .1-2,13.3 .33-39,14.1 .1-2,14.1 .32-42$, $14.3 .3-10,14.6 .1-2,14.7 .3-4,15.1 .5-10,16.1 .11-18,16.2 .17-18$, and $16.3 .1-2$ ).
Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.10, 2.8.58, 4.3.63-64, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.39$40,3.7 .27$, and 9.4 .2 ).
GRADED EXERCISE SETS Each exercise set is carefully graded, progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems involving applications and proofs.
REAL-WORLD DATA My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.36 (percentage of the population under age 18), Exercise 5.1.16 (velocity of the space
shuttle Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the wind-chill index as a function of air temperature and wind speed (Example 2 in Section 14.1). Partial derivatives are introduced in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 3 in Section 14.4). Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20-21, 2006 (Example 4 in Section 15.1). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.
PROJECTS One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity. Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare present-day methods with those of the founders of calculus-Fermat's method for finding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.7), and intersections of three cylinders (after Section 15.8). Additional projects can be found in the Instructor's Guide (see, for instance, Group Exercise 5.1: Position from Samples).
PROBLEM SOLVING Students usually have difficulties with problems for which there is no single well-defined procedure for obtaining the answer. I think nobody has improved very much on George Polya's four-stage problem-solving strategy and, accordingly, I have included a version of his problem-solving principles following Chapter 1. They are applied, both explicitly and implicitly, throughout the book. After the other chapters I have placed sections called Problems Plus, which feature examples of how to tackle challenging calculus problems. In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: "A mathematical problem should be difficult in order to entice us, yet not inaccessible lest it mock our efforts." When I put these challenging problems on assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problem-solving principles are relevant.
TECHNOLOGY The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required. The icon 20 indicates an exercise that definitely requires the use of such technology, but that is not to say that it can't be used on the other exercises as well. The symbol [CAS] is reserved for problems in which the full resources of a computer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required. But technology doesn't make pencil and paper
obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate.
TOOLS FOR TEC is a companion to the text and is intended to enrich and complement its contents. (It ENRICHING ${ }^{\text {TM }}$ CALCULUS is now accessible in Enhanced WebAssign, CourseMate, and PowerLecture. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules.
HOMEWORK HINTS Homework Hints presented in the form of questions try to imitate an effective teaching assistant by functioning as a silent tutor. Hints for representative exercises (usually oddnumbered) are included in every section of the text, indicated by printing the exercise number in red. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress, and are available to students at stewartcalculus.com and in CourseMate and Enhanced WebAssign.
ENHANCED WebAssign Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the seventh edition we have been working with the calculus community and WebAssign to develop a more robust online homework system. Up to $70 \%$ of the exercises in each section are assignable as online homework, including free response, multiple choice, and multi-part formats.
The system also includes Active Examples, in which students are guided in step-by-step tutorials through text examples, with links to the textbook and to video solutions. New enhancements to the system include a customizable eBook, a Show Your Work feature, Just in Time review of precalculus prerequisites, an improved Assignment Editor, and an Answer Evaluator that accepts more mathematically equivalent answers and allows for homework grading in much the same way that an instructor grades.
www.stewartcalculus.com This site includes the following.
- Homework Hints
- Algebra Review
- Lies My Calculator and Computer Told Me
- History of Mathematics, with links to the better historical websites
- Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes
- Archived Problems (Drill exercises that appeared in previous editions, together with their solutions)
- Challenge Problems (some from the Problems Plus sections from prior editions)
- Links, for particular topics, to outside web resources
- Selected Tools for Enriching Calculus (TEC) Modules and Visuals
# Content
Diagnostic Tests The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.
A Preview of Calculus This is an overview of the subject and includes a list of questions to motivate the study of calculus.
1 Functions and Models From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view.
2 Limits and Derivatives The material on limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 2.4 , on the precise $\varepsilon-\delta$ definition of a limit, is an optional section. Sections 2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the examples and exercises explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.8.
3 Differentiation Rules All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponential growth and decay are covered in this chapter.
4 Applications of Differentiation The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your head $42^{\circ}$ to see the top of a rainbow.
5 Integrals The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed. (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.
6 Applications of Integration Here I present the applications of integration-area, volume, work, average value-that can reasonably be done without specialized techniques of integration. General methods are emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral.
7 Techniques of Integration All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I present a strategy for integration. The use of computer algebra systems is discussed in Section 7.6.
8 Further Applications of Integration Here are the applications of integration-arc length and surface area-for which it is useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also included a section on probability. There are more applications here than can realistically be covered in a given course. Instructors should select applications suitable for their students and for which they themselves have enthusiasm. | Calculus Early Transcendentals, 7th Edition (James Stewart) (Z-Library) (converted to markdown via Mistral OCR) |
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