Advanced Calculus

by Folland, Gerald B.

• ISBN13: 9780130652652
• ISBN10: 0130652652
• Edition: 1st
• Format: Paperback
• Copyright: 2001-12-21
• Publisher: Pearson

Summary

This book presents a unified view of calculus in which theory and practice reinforces each other. It is about the theory and applications of derivatives (mostly partial), integrals, (mostly multiple or improper), and infinite series (mostly of functions rather than of numbers), at a deeper level than is found in the standard calculus books. Chapter topics cover: Setting the Stage, Differential Calculus, The Implicit Function Theorem and Its Applications, Integral Calculus, Line and Surface Integrals--Vector Analysis, Infinite Series, Functions Defined by Series and Integrals, and Fourier Series. For individuals with a sound knowledge of the mechanics of one-variable calculus and an acquaintance with linear algebra.

Table of Contents

Preface	ix
Setting the Stage	1	(42)
Euclidean Spaces and Vectors	4	(5)
Subsets of Euclidean Space	9	(3)
Limits and Continuity	12	(7)
Sequences	19	(5)
Completeness	24	(6)
Compactness	30	(3)
Connectedness	33	(6)
Uniform Continuity	39	(4)
Differential Calculus	43	(70)
Differentiability in One Variable	43	(10)
Differentiability in Several Variables	53	(9)
The Chain Rule	62	(8)
The Mean Value Theorem	70	(3)
Functional Relations and Implicit Functions: A First Look	73	(4)
Higher-Order Partial Derivatives	77	(8)
Taylor's Theorem	85	(10)
Critical Points	95	(5)
Extreme Value Problems	100	(6)
Vector-Valued Functions and Their Derivatives	106	(7)
The Implicit Function Theorem and Its Applications	113	(34)
The Implicit Function Theorem	113	(7)
Curves in the Plane	120	(6)
Surfaces and Curves in Space	126	(7)
Transformations and Coordinate Systems	133	(7)
Functional Dependence	140	(7)
Integral Calculus	147	(64)
Integration on the Line	147	(11)
Integration in Higher Dimensions	158	(10)
Multiple Integrals and Iterated Integrals	168	(9)
Change of Variables for Multiple Integrals	177	(11)
Functions Defined by Integrals	188	(5)
Improper Integrals	193	(9)
Improper Multiple Integrals	202	(5)
Lebesgue Measure and the Lebesgue Integral	207	(4)
Line and Surface Integrals; Vector Analysis	211	(68)
Arc Length and Line Integrals	212	(10)
Green's Theorem	222	(6)
Surface Area and Surface Integrals	228	(8)
Vector Derivatives	236	(3)
The Divergence Theorem	239	(4)
Some Application to Physics	243	(9)
Stoke's Theorem	252	(6)
Integrating Vector Derivatives	258	(9)
Higher Dimensions and Differential Forms	267	(12)
Infinite Series	279	(32)
Definitions and Examples	279	(5)
Series with Nonnegative Terms	284	(11)
Absolute and Conditional Convergence	295	(5)
More Convergence Tests	300	(6)
Double Series; Products of Series	306	(5)
Functions Defined by Series and Integrals	311	(44)
Sequences and Series of Functions	311	(9)
Integrals and Derivatives of Sequences and Series	320	(3)
Power Series	323	(10)
The Complex Exponential and Trig Functions	333	(3)
Functions Defined by Improper Integrals	336	(6)
The Gamma Function	342	(8)
Stirling's Formula	350	(5)
Fourier Series	355	(86)
Periodic Functions and Fourier Series	355	(7)
Convergence of Fourier Series	362	(10)
Derivatives, Integrals, and Uniform Convergence	372	(5)
Fourier Series on Intervals	377	(4)
Applications to Differential Equations	381	(11)
The Infinite-Dimensional Geometry of Fourier Series	392	(9)
The Isoperimetric Inequality	401	(4)
Appendices
A Summary of Linear Algebra	405	(14)
A.1 Vectors	405	(1)
A.2 Linear Maps and Matrices	406	(3)
A.3 Row Operations and Echelon Forms	409	(2)
A.4 Determinants	411	(2)
A.5 Linear Independence	413	(1)
A.6 Subspaces; Dimension; Rank	414	(2)
A.7 Invertibility	416	(1)
A.8 Eigenvectors and Eigenvalues	417	(2)
B Some Technical Proofs	419	(22)
B.1 The Heine-Borel Theorem	419	(1)
B.2 The Implicit Function Theorem	420	(2)
B.3 Approximation by Riemann Sums	422	(2)
B.4 Double Integrals and Iterated Integrals	424	(1)
B.5 Change of Variables for Multiple Integrals	425	(7)
B.6 Improper Multiple Integrals	432	(1)
B.7 Green's Theorem and the Divergence Theorem	433	(8)
Answers to Selected Exercises	441	(12)
Bibliography	453	(2)
Index	455