Elementary Classical Analysis

by Marsden, Jerrold E.; Hoffman, Michael J.

• ISBN13: 9780716721055
• ISBN10: 0716721058
• Edition: 2nd
• Format: Hardcover
• Copyright: 1993-03-15
• Publisher: W. H. Freeman

Summary

Designed for courses in advanced calculus and introductory real analysis,Elementary Classical Analysisstrikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.

Table of Contents

PREFACE	ix	(2)
PREFACE TO THE FIRST EDITION	xi
INTRODUCTION: SETS AND FUNCTIONS	1	(24)
Supplement on the Axioms of Set Theory	7	(11)
Worked Examples	18	(2)
Exercises	20	(5)
1 THE REAL LINE AND EUCLIDEAN SPACE	25	(78)
1.1 Ordered Fields and the Number System	25	(10)
1.2 Completeness and the Real Number System	35	(10)
1.3 Least Upper Bounds	45	(4)
1.4 Cauchy Sequences	49	(3)
1.5 Cluster Points; lim inf and lim sup	52	(5)
1.6 Euclidean Space	57	(7)
1.7 Norms, Inner Products, and Metrics	64	(6)
1.8 The Complex Numbers	70	(9)
Theorem Proofs	79	(16)
Worked Examples	95	(2)
Exercises	97	(6)
2 THE TOPOLOGY OF EUCLIDEAN SPACE	103	(48)
2.1 Open Sets	104	(4)
2.2 Interior of a Set	108	(2)
2.3 Closed Sets	110	(3)
2.4 Accumulation Points	113	(3)
2.5 Closure of a Set	116	(2)
2.6 Boundary of a Set	118	(2)
2.7 Sequences	120	(3)
2.8 Completeness	123	(2)
2.9 Series of Real Numbers and Vectors	125	(5)
Theorem Proofs	130	(10)
Worked Examples	140	(3)
Exercises	143	(8)
3 COMPACT AND CONNECTED SETS	151	(26)
3.1 Compactness	151	(4)
3.2 The Heine-Borel Theorem	155	(2)
3.3 Nested Set Property	157	(3)
3.4 Path-Connected Sets	160	(3)
3.5 Connected Sets	163	(2)
Theorem Proofs	165	(5)
Worked Examples	170	(2)
Exercises	172	(5)
4 CONTINUOUS MAPPINGS	177	(60)
4.1 Continuity	177	(5)
4.2 Images of Compact and Connected Sets	182	(2)
4.3 Operations on Continuous Mappings	184	(4)
4.4 The Boundedness of Continuous Functions on Compact Sets	188	(3)
4.5 The Intermediate Value Theorem	191	(3)
4.6 Uniform Continuity	194	(2)
4.7 Differentiation of Functions of One Variable	196	(8)
4.8 Integration of Functions of One Variable	204	(7)
Theorem Proofs	211	(16)
Worked Examples	227	(4)
Exercises	231	(6)
5 UNIFORM CONVERGENCE	237	(90)
5.1 Pointwise and Uniform Convergence	237	(7)
5.2 The Weierstrass M Test	244	(3)
5.3 Integration and Differentiation of Series	247	(7)
5.4 The Elementary Functions	254	(14)
5.5 The Space of Continuous Functions	268	(4)
5.6 The Arzela-Ascoli Theorem	272	(3)
5.7 The Contraction Mapping Principle and Its Applications	275	(8)
5.8 The Stone-Weierstrass Theorem	283	(4)
5.9 The Dirichlet and Abel Tests	287	(2)
5.10 Power Series and Cesaro and Abel Summability	289	(5)
Theorem Proofs	294	(19)
Worked Examples	313	(3)
Exercises	316	(11)
6 DIFFERENTIABLE MAPPINGS	327	(64)
6.1 Definition of the Derivative	327	(4)
6.2 Matrix Representation	331	(3)
6.3 Continuity of Differentiable Mappings; Differentiable Paths	334	(6)
6.4 Conditions for Differentiability	340	(5)
6.5 The Chain Rule	345	(4)
6.6 Product Rule and Gradients	349	(4)
6.7 The Mean Value Theorem	353	(2)
6.8 Taylor's Theorem and Higher Derivatives	355	(7)
6.9 Maxima and Minima	362	(5)
Theorem Proofs	367	(13)
Worked Examples	380	(3)
Exercises	383	(8)
7 THE INVERSE AND IMPLICIT FUNCTION THEOREMS AND RELATED TOPICS	391	(54)
7.1 Inverse Function Theorem	392	(5)
7.2 Implicit Function Theorem	397	(4)
7.3 The Domain-Straightening Theorem	401	(2)
7.4 Further Consequences of the Implicit Function Theorem	403	(4)
7.5 An Existence Theorem for Ordinary Differential Equations	407	(4)
7.6 The Morse Lemma	411	(3)
7.7 Constrained Extrema and Lagrange Multipliers	414	(6)
Theorem Proofs	420	(15)
Worked Examples	435	(3)
Exercises	438	(7)
8 INTEGRATION	445	(52)
8.1 Integrable Functions	445	(6)
8.2 Volume and Sets of Measure Zero	451	(3)
8.3 Lebesgue's Theorem	454	(3)
8.4 Properties of the Integral	457	(2)
8.5 Improper Integrals	459	(7)
8.6 Some Convergence Theorems	466	(3)
8.7 Introduction to Distributions	469	(3)
Theorem Proofs	472	(10)
Worked Examples	487	(3)
Exercises	490	(7)
9 FUBINI'S THEOREM AND THE CHANGE OF VARIABLES FORMULA	497	(46)
9.1 Introduction	497	(3)
9.2 Fubini's Theorem	500	(5)
9.3 Change of Variables Theorem	505	(3)
9.4 Polar Coordinates	508	(2)
9.5 Spherical and Cylindrical Coordinates	510	(3)
9.6 A Note on the Lebesgue Integral	513	(1)
9.7 Interchange of Limiting Operations	514	(7)
Theorem Proofs	521	(10)
Worked Examples	531	(4)
Exercises	535	(8)
10 FOURIER ANALYSIS	543	(120)
10.1 Inner Product Spaces	545	(6)
10.2 Orthogonal Families of Functions	551	(9)
10.3 Completeness and Convergence Theorems	560	(10)
10.4 Functions of Bounded Variation and Fejer Theory (Optional)	570	(3)
10.5 Computation of Fourier Series	573	(14)
10.6 Further Convergence Theorems	587	(6)
10.7 Applications	593	(12)
10.8 Fourier Integrals	605	(5)
10.9 Quantum Mechanical Formalism	610	(8)
Theorem Proofs	618	(26)
Worked Examples	644	(6)
Exercises	650	(13)
APPENDIX A: MISCELLANEOUS EXERCISES	663	(14)
APPENDIX B: REFERENCES AND SUGGESTIONS FOR FURTHER STUDY	677	(6)
APPENDIX C: ANSWERS AND SUGGESTIONS FOR SELECTED ODD-NUMBERED EXERCISES	683	(46)
INDEX	729