Book:George B. Thomas, Jr./Calculus and Analytic Geometry/Fourth Edition
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George B. Thomas, Jr.: Calculus and Analytic Geometry (4th Edition)
Published $\text {1968}$, Addison Wesley
Subject Matter
Contents
- Preface to the Fourth Edition
- $1 \quad$ The rate of change of a function
- $1.1 \quad$ Introduction
- $1.2 \quad$ Coordinates
- $1.3 \quad$ Increments
- $1.4 \quad$ Slope of a straight line
- $1.5 \quad$ Equations of a straight line
- $1.6 \quad$ Functions and graphs
- $1.7 \quad$ Ways of combining functions
- $1.8 \quad$ Behavior of functions
- $1.9 \quad$ Slope of a curve
- $1.10 \quad$ Derivative of a function
- $1.11 \quad$ Velocity and rates
- $2 \quad$ Limits
- $2.1 \quad$ Definition of the limit of a function
- $2.2 \quad$ Theorems about limits
- $2.3 \quad$ More theorems about limits
- $2.4 \quad$ Infinity
- $2.5 \quad$ Limits applied to areas
- $3 \quad$ Derivatives of Algebraic Functions
- $3.1 \quad$ Polynomial functions and their derivatives
- $3.2 \quad$ Rational functions and their derivatives
- $3.3 \quad$ Inverse functions and their derivatives
- $3.4 \quad$ The increment of a function
- $3.5 \quad$ Composite functions
- $3.6 \quad$ Derivatives of composite functions: the chain rule
- $3.7 \quad$ The differentials $dx$ and $dy$
- $3.8 \quad$ Formulas for differentiation
- $3.9 \quad$ Continuity
- $4 \quad$ Applications
- $4.1 \quad$ Increasing or decreasing functions: the sign of $dy / dx$
- $4.2 \quad$ Related rates
- $4.3 \quad$ Significance of the sign
- $4.4 \quad$ Curve plotting
- $4.5 \quad$ Maxima and minima: theory
- $4.6 \quad$ Maxima and minima: problems
- $4.7 \quad$ Rolle's Theorem
- $4.8 \quad$ The Mean Value Theorem
- $4.9 \quad$ Extension of the Mean Value Theorem
- $5 \quad$ Integration
- $5.1 \quad$ Introduction
- $5.2 \quad$ The indefinite integral
- $5.3 \quad$ Applications of indefinite integration
- $5.4 \quad$ Brief review of trigonometry
- $5.5 \quad$ Differentiation and integration of sines and cosines
- $5.6 \quad$ Area under a curve
- $5.7 \quad$ Computation of areas as limits
- $5.8 \quad$ Areas by calculus
- $5.9 \quad$ The definite integral and the Fundamental Theorem of Integral Calculus
- $5.10 \quad$ The trapezoidal rule for approximating an integral
- $5.11 \quad$ Some comments on notation
- $5.12 \quad$ Summary
- $6 \quad$ Applications of the definite integral
- $6.1 \quad$ Introduction
- $6.2 \quad$ Area between two curves
- $6.3 \quad$ Distance
- $6.4 \quad$ Volumes
- $6.5 \quad$ Approximations
- $6.6 \quad$ Length of a plane curve
- $6.7 \quad$ Area of a surface of revolution
- $6.8 \quad$ Average value of a function
- $6.9 \quad$ Moments and center of mass
- $6.10 \quad$ The centroid
- $6.11 \quad$ The theorems of Pappus
- $6.12 \quad$ Hydrostatic pressure
- $6.13 \quad$ Work
- $7 \quad$ Transcendental functions
- $7.1 \quad$ The trigonometric functions
- $7.2 \quad$ The inverse trigonometric functions
- $7.3 \quad$ Derivatives of the inverse trigonometric functions
- $7.4 \quad$ The natural logarithm
- $7.5 \quad$ The derivative of $\ln x$
- $7.6 \quad$ Properties of natural logarithms
- $7.7 \quad$ Graph of $y = \ln x$
- $7.8 \quad$ The exponential function
- $7.9 \quad$ The functions $a^u$ and $\log_a u$
- $7.10 \quad$ Differential equations
- $8 \quad$ Hyperbolic functions
- $8.1 \quad$ Introduction
- $8.2 \quad$ Definitions and identities
- $8.3 \quad$ Derivatives and integrals
- $8.4 \quad$ Geometric meaning of the hyperbolic radian
- $8.5 \quad$ The inverse hyperbolic functions
- $8.6 \quad$ The hanging cable
- $9 \quad$ Methods of integration
- $9.1 \quad$ Basic formulas
- $9.2 \quad$ Powers of trigonometric functions
- $9.3 \quad$ Even powers of sines and cosines
- $9.4 \quad$ Integrals with terms $\sqrt {a^2 - u^2}$, $\sqrt {a^2 + u^2}$, $\sqrt {u^2 - a^2}$, $a^2 + u^2$, $a^2 - u^2$
- $9.5 \quad$ Integrals with $a x^2 + b x + c$
- $9.6 \quad$ Integration by the method of partial fractions
- $9.7 \quad$ Integration by parts
- $9.8 \quad$ Integration of rational functions of $\sin x$ and $\cos x$, and other trigonometric integrals
- $9.9 \quad$ Further substitutions
- $9.10 \quad$ Improper integrals
- $9.11 \quad$ Numerical methods for approximating definite integrals
- $10 \quad$ Plane analytic geometry
- $10.1 \quad$ Curves and equations
- $10.2 \quad$ Tangents and normals
- $10.3 \quad$ Newton's method for approximating roots of equations
- $10.4 \quad$ Distance between two points: equations of loci
- $10.5 \quad$ The circle
- $10.6 \quad$ The parabola
- $10.7 \quad$ The ellipse
- $10.8 \quad$ The hyperbola
- $10.9 \quad$ Second-degree curves
- $10.10 \quad$ Invariants and the discriminant
- $10.11 \quad$ Sections of a cone
- $11 \quad$ Polar coordinates
- $11.1 \quad$ The polar coordinate system
- $11.2 \quad$ Graphs of polar equations
- $11.3 \quad$ Polar equations of the conic sections and other curves
- $11.4 \quad$ The angle $\bspsi$ between the radius vector and the tangent line
- $11.5 \quad$ Plane areas in polar coordinates
- $12 \quad$ Vectors and parametric equations
- $12.1 \quad$ Parametric equations in kinematics
- $12.2 \quad$ Parametric equations in analytic geometry
- $12.3 \quad$ Vector components and the unit vectors $\mathbf i$ and $\mathbf j$
- $12.4 \quad$ Space coordinates
- $12.5 \quad$ Vectors in space
- $12.6 \quad$ The scalar product of two vectors
- $12.7 \quad$ The vector product of two vectors
- $12.8 \quad$ Equations of lines and planes
- $12.9 \quad$ Products of three or more vectors
- $12.10 \quad$ Loci in space: cylinders
- $12.11 \quad$ Quadric surfaces
- $13 \quad$ Linear algebra: vectors in $n$-space
- $13.1 \quad$ Vectors in Euclidean $n$-space
- $13.2 \quad$ Matrices and simultaneous linear equations: notation
- $13.3 \quad$ Matrices and simultaneous linear equations: computational techniques
- $13.4 \quad$ Linear independence and linear dependence of vectors
- $13.5 \quad$ Matrices and linear transformations
- $14 \quad$ Vector functions and their derivatives
- $14.1 \quad$ Introduction
- $14.2 \quad$ Velocity and acceleration
- $14.3 \quad$ Tangential vectors
- $14.4 \quad$ Curvature and normal vectors
- $14.5 \quad$ Differentiation of products of vectors
- $14.6 \quad$ Polar and cylindrical coordinates
- $15 \quad$ Partial differentiation
- $15.1 \quad$ Functions of two or more variables
- $15.2 \quad$ The direction derivative: special cases
- $15.3 \quad$ Tangent plane and normal line
- $15.4 \quad$ Approximate value of $\Delta w$
- $15.5 \quad$ The directional derivative: general case
- $15.6 \quad$ The gradient
- $15.7 \quad$ The chain rule for partial derivatives
- $15.8 \quad$ The total differential
- $15.9 \quad$ Maxima and minima of functions of two independent variables
- $15.10 \quad$ The method of least squares
- $15.11 \quad$ Maxima and minima of functions of several independent variables
- $15.12 \quad$ Higher-order derivatives
- $15.13 \quad$ Exact differentials
- $15.14 \quad$ Derivatives of integrals
- $16 \quad$ Multiple integrals
- $16.1 \quad$ Double integrals
- $16.2 \quad$ Area by double integration
- $16.3 \quad$ Physical applications
- $16.4 \quad$ Polar coordinates
- $16.5 \quad$ Triple integrals: volume
- $16.6 \quad$ Cylindrical coordinates
- $16.7 \quad$ Physical applications of triple integration
- $16.8 \quad$ Spherical coordinates
- $16.9 \quad$ Surface area
- $17 \quad$ Vector analysis
- $17.1 \quad$ Introduction: vector fields
- $17.2 \quad$ Surface integrals
- $17.3 \quad$ Line integrals
- $17.4 \quad$ Two-dimensional fields: line integrals in the plane and their relation to surface integrals on cylinders
- $17.5 \quad$ Green's theorem
- $17.6 \quad$ Divergence theorem
- $17.7 \quad$ Stokes' theorem
- $18 \quad$ Infinite series
- $18.1 \quad$ Introduction and definitions
- $18.2 \quad$ Tests for convergence of a series of constants
- $18.3 \quad$ Power series expansions of functions
- $18.4 \quad$ Taylor's theorem with remainder
- $18.5 \quad$ Application to max-min theory for functions of two independent variables
- $18.6 \quad$ Computations
- $18.7 \quad$ Indeterminate forms
- $18.8 \quad$ Fourier series
- $18.9 \quad$ Convergence of power series: absolute convergence
- $18.10 \quad$ Alternating series: conditional convergence
- $19 \quad$ Complex numbers and functions
- $19.1 \quad$ Invented number systems
- $19.2 \quad$ The Argand diagram
- $19.3 \quad$ The complex variable
- $19.4 \quad$ Derivatives
- $19.5 \quad$ Cauchy-Riemann differential equations
- $19.6 \quad$ Complex series
- $19.7 \quad$ Certain elementary functions
- $19.8 \quad$ Logarithms
- $20 \quad$ Differential equations
- $20.1 \quad$ Introduction
- $20.2 \quad$ Solutions
- $20.3 \quad$ First-order equations with variables separable
- $20.4 \quad$ First-order homogeneous equations
- $20.5 \quad$ First-order linear equations
- $20.6 \quad$ First-order equations with exact differentials
- $20.7 \quad$ Special types of second-order equations
- $20.8 \quad$ Linear equations with constant coefficients
- $20.9 \quad$ Homogeneous linear second-order differential equations with constant coefficients
- $20.10 \quad$ Nonhomogeneous linear second-order differental equations with constant coefficients
- $20.11 \quad$ Higher-order linear differential equations with constant coefficients
- $20.12 \quad$ Vibrations
- $20.13 \quad$ Poisson probability distribution
- Appendix $\text I \quad$ Determinants and linear equations
- $\text A.1 \quad$ Indroduction
- $\text A.2 \quad$ Determinants and linear equations
- $\text A.3 \quad$ Determinants of order three
- $\text A.4 \quad$ Determinants of order $n$
- $\text A.5 \quad$ Properties of determinants of order $n$
- $\text A.6 \quad$ Expansion by cofactors
- $\text A.7 \quad$ Solution of simultaneous linear equations
- $\text A.8 \quad$ Homogeneous linear equations
- Appendix $\text {II} \quad$ Formulas from elementary mathematics
- Appendix $\text {III} \quad$ Tables of functions
- Answers to exercises
- Index
Further Editions
- 1951: George B. Thomas, Jr.: Calculus and Analytic Geometry
- 1996: George B. Thomas, Jr. and Ross L. Finney: Calculus and Analytic Geometry (9th ed.)
Source work progress
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous): Front endpapers: A Brief Table of Integrals: $69$.
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