Advanced Calculus, 3rd Edition,9780471025665

Advanced Calculus, 3rd Edition

by ;
Edition: 3rd
ISBN13: 9780471025665
ISBN10: 0471025666
Format: Paperback
Pub. Date: 1991-01-16
Publisher(s): Wiley
List Price: $319.34

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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

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