Advanced Calculus, 3rd Edition

by Taylor, Angus E.; Mann, W. Robert

• ISBN13: 9780471025665
• ISBN10: 0471025666
• Edition: 3rd
• Format: Paperback
• Copyright: 1991-01-16
• Publisher: Wiley

Summary

Outlines theory and techniques of calculus, emphasizing strong understanding of concepts, and the basic principles of analysis. Reviews elementary and intermediate calculus and features discussions of elementary-point set theory, and properties of continuous functions.

Author Biography

Angus Ellis Taylor (October 13, 1911 – April 6, 1999) was a mathematician and professor at various universities in the University of California system. He earned his undergraduate degree at Harvard summa cum laude in 1933 and his PhD at Caltech in 1936 under Aristotle Michal with a dissertation on analytic functions. By 1944 he had risen to full professor at UCLA, whose mathematics department he later chaired (1958–1964). Taylor was also an astute administrator and eventually rose through the UC system to become provost and then chancellor of UC Santa Cruz. He authored a number of mathematical texts, one of which, Advanced Calculus (1955, Ginn and Co.), became a standard for a generation of mathematics students.

Table of Contents

Fundamentals of Elementary Calculus
Introduction	1	(1)
Functions	2	(10)
Derivatives	12	(8)
Maxima and Minima	20	(6)
The Law of the Mean (The Mean-Value Theorem for Derivatives)	26	(6)
Differentials	32	(3)
The Inverse of Differentiation	35	(3)
Definite Integrals	38	(7)
The Mean-Value Theorem for Integrals	45	(1)
Variable Limits of Integration	46	(3)
The Integral of a Derivative	49	(4)
Limits	53	(1)
Limits of Functions of a Continuous Variable	54	(4)
Limits of Sequences	58	(9)
The Limit Defining a Definite Integral	67	(1)
The Theorem on Limits of Sums, Products, and Quotients	67	(5)
The Real Number System
Numbers	72	(1)
The Field of Real Numbers	72	(2)
Inequalities. Absolute Value	74	(1)
The Principle of Mathematical Induction	75	(2)
The Axiom of Continuity	77	(1)
Rational and Irrational Numbers	78	(1)
The Axis of Reals	79	(1)
Least Upper Bounds	80	(2)
Nested Intervals	82	(3)
Continuous Functions
Continuity	85	(1)
Bounded Functions	86	(2)
The Attainment of Extreme Values	88	(2)
The Intermediate-Value Theorem	90	(5)
Extensions of the Law of the Mean
Introduction	95	(1)
Cauchy's Generalized Law of the Mean	95	(2)
Taylor's Formula with Integral Remainder	97	(2)
Other Forms of the Remainder	99	(6)
An Extension of the Mean-Value Theorem for Integrals	105	(1)
L'Hospital's Rule	106	(10)
Functions of Several Variables
Functions and Their Regions of Definition	116	(1)
Point Sets	117	(5)
Limits	122	(3)
Continuity	125	(2)
Modes of Representing a Function	127	(3)
The Elements of Partial Differentiation
Partial Derivatives	130	(2)
Implicit Functions	132	(3)
Geometrical Significance of Partial Derivatives	135	(3)
Maxima and Minima	138	(6)
Differentials	144	(10)
Composite Functions and the Chain Rule	154	(8)
An Application in Fluid Mechanics	162	(2)
Second Derivatives by the Chain Rule	164	(4)
Homogeneous Functions. Euler's Theorem	168	(4)
Derivatives of Implicit Functions	172	(5)
Extremal Problems with Constraints	177	(5)
Lagrange's Method	182	(7)
Quadratic Forms	189	(7)
General Theorems of Partial Differentiation
Preliminary Remarks	196	(1)
Sufficient Conditions for Differentiability	197	(2)
Changing the Order of Differentiation	199	(3)
Differentials of Composite Functions	202	(2)
The Law of the Mean	204	(3)
Taylor's Formula and Series	207	(4)
Sufficient Conditions for a Relative Extreme	211	(11)
Implicit-Function Theorems
The Nature of the Problem of Implicit Functions	222	(2)
The Fundamental Theorem	224	(3)
Generalization of the Fundamental Theorem	227	(3)
Simultaneous Equations	230	(7)
The Inverse Function Theorem With Applications
Introduction	237	(4)
The Inverse Function Theorem in Two Dimensions	241	(6)
Mappings	247	(5)
Successive Mappings	252	(3)
Transformations of Co-ordinates	255	(3)
Curvilinear Co-ordinates	258	(5)
Identical Vanishing of the Jacobian. Functional Dependence	263	(5)
Vectors and Vector Fields
Purpose of the Chapter	268	(1)
Vectors in Euclidean Space	268	(5)
Orthogonal Unit Vectors in R3	273	(1)
The Vector Space Rn	274	(6)
Cross Products in R3	280	(3)
Rigid Motions of the Axes	283	(3)
Invariants	286	(5)
Scalar Point Functions	291	(2)
Vector Point Functions	293	(2)
The Gradient of a Scalar Field	295	(5)
The Divergence of a Vector Field	300	(5)
The Curl of a Vector Field	305	(4)
Linear Transformations
Introduction	309	(3)
Linear Transformations	312	(1)
The Vector Space L(Rn, Rm)	313	(1)
Matrices and Linear Transformations	313	(3)
Some Special Cases	316	(2)
Norms	318	(1)
Metrics	319	(1)
Open Sets and Continuity	320	(4)
A Norm on L(Rn, Rm)	324	(3)
L(Rn)	327	(3)
The Set of Invertible Operators	330	(5)
Differential Calculus of Functions From Rn to Rm
Introduction	335	(1)
The Differential and the Derivative	336	(4)
The Component Functions and Differentiability	340	(3)
Directional Derivatives and the Method of Steepest Descent	343	(4)
Newton's Method	347	(3)
A Form of the Law of the Mean for Vector Functions	350	(2)
The Hessian and Extreme Values	352	(2)
Continuously Differentiable Functions	354	(1)
The Fundamental Inversion Theorem	355	(6)
The Implicit Function Theorem	361	(5)
Differentiation of Scalar Products of Vector Valued Functions of a Vector Variable	366	(10)
Double and Triple Integrals
Preliminary Remarks	376	(1)
Motivations	376	(3)
Definition of a Double Integral	379	(2)
Some Properties of the Double Integral	381	(1)
Inequalities. The Mean-Value Theorem	382	(1)
A Fundamental Theorem	383	(1)
Iterated Integrals. Centroids	384	(6)
Use of Polar Co-ordinates	390	(5)
Applications of Double Integrals	395	(6)
Potentials and Force Fields	401	(3)
Triple Integrals	404	(5)
Applications of Triple Integrals	409	(3)
Cylindrical Co-ordinates	412	(1)
Spherical Co-ordinates	413	(4)
Curves and Surfaces
Introduction	417	(1)
Representations of Curves	417	(1)
Arc Length	418	(3)
The Tangent Vector	421	(2)
Principal normal. Curvature	423	(2)
Binormal. Torsion	425	(3)
Surfaces	428	(5)
Curves on a Surface	433	(4)
Surface Area	437	(8)
Line and Surface Integrals
Introduction	445	(1)
Point Functions on Curves and Surfaces	445	(1)
Line Integrals	446	(5)
Vector Functions and Line Integrals. Work	451	(4)
Partial Derivatives at the Boundary of a Region	455	(2)
Green's Theorem in the Plane	457	(6)
Comments on the Proof of Green's Theorem	463	(2)
Transformations of Double Integrals	465	(4)
Exact Differentials	469	(5)
Line Integrals Independent of the Path	474	(4)
Further Discussion of Surface Area	478	(2)
Surface Integrals	480	(4)
The Divergence Theorem	484	(8)
Green's Identities	492	(2)
Transformation of Triple Integrals	494	(5)
Stokes's Theorem	499	(6)
Exact Differentials in Three Variables	505	(7)
Point-Set Theory
Preliminary Remarks	512	(1)
Finite and Infinite Sets	512	(2)
Point Sets on a Line	514	(3)
The Bolzano-Weierstrass Theorem	517	(1)
Convergent Sequences on a Line	518	(2)
Point Sets in Higher Dimensions	520	(1)
Convergent Sequences in Higher Dimensions	521	(1)
Cauchy's Convergence Condition	522	(1)
The Heine-Borel Theorem	523	(4)
Fundamental Theorems on Continuous Functions
Purpose of the Chapter	527	(1)
Continuity and Sequential Limits	527	(2)
The Boundedness Theorem	529	(1)
The Extreme-Value Theorem	529	(1)
Uniform Continuity	529	(3)
Continuity of Sums, Products, and Quotients	532	(1)
Persistence of Sign	532	(1)
The Intermediate-Value Theorem	533	(2)
The Theory of Integration
The Nature of the Chapter	535	(1)
The Definition of Integrability	535	(4)
The Integrability of Continuous Functions	539	(1)
Integrable Functions with Discontinuities	540	(2)
The Integral as a Limit of Sums	542	(3)
Duhamel's Principle	545	(3)
Further Discussion of Integrals	548	(1)
The Integral as a Function of the Upper Limit	548	(2)
The Integral of a Derivative	550	(1)
Integrals Depending on a Parameter	551	(3)
Riemann Double Integrals	554	(3)
Double Integrals and Iterated Integrals	557	(2)
Triple Integrals	559	(1)
Improper Integrals	559	(1)
Stieltjes Integrals	560	(6)
Infinite Series
Definitions and Notation	566	(3)
Taylor's Series	569	(3)
A Series for the Inverse Tangent	572	(1)
Series of Nonnegative Terms	573	(4)
The Integral Test	577	(2)
Ratio Tests	579	(2)
Absolute and Conditional Convergence	581	(4)
Rearrangement of Terms	585	(2)
Alternating Series	587	(3)
Tests for Absolute Convergence	590	(7)
The Binomial Series	597	(3)
Multiplication of Series	600	(4)
Dirichlet's Test	604	(6)
Uniform Convergence
Functions Defined by Convergent Sequences	610	(3)
The Concept of Uniform Convergence	613	(5)
A Comparison Test for Uniform Convergence	618	(2)
Continuity of the Limit Function	620	(1)
Integration of Sequences and Series	621	(3)
Differentiation of Sequences and Series	624	(3)
Power Series
General Remarks	627	(1)
The Interval of Convergence	627	(5)
Differentiation of Power Series	632	(7)
Division of Power Series	639	(4)
Abel's Theorem	643	(4)
Inferior and Superior Limits	647	(3)
Real Analytic Functions	650	(4)
Improper Integrals
Preliminary Remarks	654	(2)
Positive Integrands. Integrals of the First Kind	656	(5)
Integrals of the Second Kind	661	(3)
Integrals of Mixed Type	664	(2)
The Gamma Function	666	(4)
Absolute Convergence	670	(3)
Improper Multiple Integrals. Finite Regions	673	(5)
Improper Multiple Integrals. Infinite Regions	678	(4)
Functions Defined by Improper Integrals	682	(8)
Laplace Transforms	690	(3)
Repeated Improper Integrals	693	(2)
The Beta Function	695	(4)
Stirling's Formula	699	(10)
Answers to Selected Exercises	709	(18)
Index	727