导语

  

内容提要

  

    韦尔、哈斯、吉尔当诺主的《托马斯微积分(影印版上第11版)(英文版)》1951年出版第1版，是一本深受美国广大教师和学生欢迎的教材，不少学校和教师采用它作为微积分课程的教材，在相当一段时间里，它是麻省理工学院微积分课程所用的教材之一。
    本书具有以下几个突出特色：取材于科学和工程领域中的重要应用实例以及配置丰富的习题；对每个重要专题均用语言的、代数的、数值的、图像的方式予以陈述i重视数值计算和程序应用；切实融入数学建模和数学实验的思想和方法；每个新专题都通过清楚的、易于理解的例子启发式地引入，可读性强；配有丰富的教学资源，可用于教师教学和学生学习。

Preface
Pretiminaries
1.1 Real Numbers and the Real Line
1.2 Lines, Circles, and Parabolas
1.3 Functions and Their Graphs
1.4 Identifying Functions; Mathematical Models
1.5 Combining Functions; Shifting and Scaling Graphs
1.6 Trigonometric Functions
1.7 Graphing with Calculators and Computers
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Limits and Continuity
2.1 Rates of Change and Limits
2.2 Calculating Limits Using the Limit Laws
2.3 The Precise Definition of a Limit
2.4 One-Sided Limits and Limits at Infinity
2.5 Infinite Limits and Vertical Asymptotes
2.6 Continuity
2.7 Tangents and Derivatives
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Differentiation
3.1 The Derivative as a Function
3.2 Differentiation Rules
3.3 The Derivative as a Rate of Change
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule and Parametric Equations
3.6 Implicit Differentiation
3.7 Related Rates
3.8 Linearization and Differentials
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
AppticaUons of Derivatives
4.1 Extreme Values of Functions
4.2 The Mean Value Theorem
4.3 Monotonic Functions and the First Derivative Test
4.4 Concavity and Curve Sketching
4.5 Applied Optimization Problems
4.6 Indeterminate Forms and IgH6pital's Rule
4.7 Newton's Method
4.8 Antiderivatives
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Integration
5.1 Estimating with Finite Sums
5.2 Sigma Notation and Limits of Finite Sums
5.3 The Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Indefinite Integrals and the Substitution Rule
5.6 Substitution and Area Between Curves
QUESTIONS TO GUIDE YoUR REvIEw
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Apptications of Definite Integrats
6.1 Volumes by Slicing and Rotation About an Axis
6.2 Volumes by Cylindrical Shells
6.3 Lengths of Plane Curves
6.4 Moments and Centers of Mass
6.5 Areas of Surfaces of Revolution and the Theorems of Pappus
6.6 Work
6.7 Fluid Pressures and Forces
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Transcendentat Functions
7.1 Inverse Functions and Their Derivatives
7.2 Natural Logarithms
7.3 The Exponential Function
7.4 ax and logax
7.5 Exponential Growth and Decay
7.6 Relative Rates of Growth
7.7 Inverse Trigonometric Functions
7.8 Hyperbolic Functions
QUESTIONS TO GLADE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Techniques of Integration 5
8.1 Basic Integration Formulas
8.2 Integration by Parts
8.3 Integration of Rational Functions by Partial Fractions
8.4 Trigonometric Integrals
8.5 Trigonometric Substitutions
8.6 Integral Tables and Comouter Algebra Systems
8.7 Numerical Integration
8.8 Improper Integrals
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Further Applications of Integration
9.1 Slope Fields and Separable Differential Equations
9.2 First-Order Linear Differential Equations
9.3 Euler's Method
9.4 Graphical Solutions of Autonomous Differential Equations
9.5 Applications of First-Order Differential Equations
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Conic Sections and Polar Coordinates
10.1 Conic Sections and Quadratic Equations
10.2 Classifying Conic Sections by Eccentricity
10.3 Quadratic Equations and Rotations
10.4 Conics and Parametric Equations; The Cycloid
10.5 Polar Coordinates
10.6 Graphing in Polar Coordinates
10.7 Areas and Lengths in Polar Coordinates
10.8 Conic Sections in Polar Coordinates
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Infinite Sequences and Series
11.1 Sequences
11.2 Infinite Series
11.3 The Integral Test
11.4 Comparison Tests
11.5 The Ratio and Root Tests _
11.6 Alternating Series, Absolute and Conditional Convergence
11.7 Power Series
11.8 Taylor and Maclaurin Series
11.9 Convergence of Taylor Series; Error Estimates
11.10 Applications of Power Series
11.11 Fourier Series
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Vectors and the Geometry of Space
12.1 Three-Dimensional Coordinate Systems
12.2 Vectors
12.3 The Dot Product
12.4 The Cross Product
12.5 Lines and Planes in Space
12.6 Cylinders and Quadric Surfaces
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Vector-Va[ued Functions and Motion in Space
13.1 Vector Functions 906
13.2 Modeling Projectile Motion 920
13.3 Arc Length and the Unit Tangent Vector T 931
13.4 Curvature and the Unit Normal Vector N 936
13.5 Torsion and the Unit Binormal Vector B 943
13.6 Planetary Motion and Satellites 950
QUESTIONS TO GUIDE YOUR REVIEW 959
PRACTICE EXERCISES 960
ADDITIONAL AND ADVANCED EXERCISES 962
Partiat Derivatives
14. l Functions of Several Variables
14.2 Limits and Continuity in Higher Dimensions
14.3 Partial Derivatives
14.4 The Chain Rule
14.5 Directional Derivatives and Gradient Vectors
14.6 Tangent Planes and Differentials
14.7 Extreme Values and Saddle Points
14.8 Lagrange Multipliers
14.9 Partial Derivatives with Constrained Variables
14.10 Taylor's Formula for Two Variables
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
MutUpte Integrats
15.1 Double Integrals
15.2 Areas, Moments, and Centers of Mass
15.3 Double Integrals in Polar Form
15.4 Triple Integrals in Rectangular Coordinates
15.5 Masses and Moments in Three Dimensions
15.6 Triple Integrals in Cylindrical and Spherical Coordinates
15.7 Substitutions in Multiple Integrals
QUESTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Integration in Vector Fietds
16.1 Line Integrals
16.2 Vector Fields, Work, Circulation, and Flux
16.3 Path Independence, Potential Functions, and Conservative Fields
16.4 Green's Theorem in the Plane
16.5 Surface Area and Surface Integrals
16.6 Parametrized Surfaces
16.7 Stokes' Theorem
16.8 The Divergence Theorem and a Unified Theory
QUESTIONS TO GUIDE YOUR RnVIEW
PRACTICE EXERCISES
ADDITIONAL AND ADVANCED EXERCISES
Appendices
A.1 Mathematical Induction
A.2 Proofs of Limit Theorems
A.3 Commonly Occurring Limits
A.4 Theory of the Real Numbers
A.5 Complex Numbers
A.6 The Distributive Law for Vector Cross Products
A.7 The Mixed Derivative Theorem and the Increment Theorem
A.8 The Area ofa Parallelogram's Projection on a Plane
A.9 Basic Algebra, Geometry, and Trigonometry Formulas
Answers
Index
A Brief TaMe of Integrals
Credits