Easily master the fundamental concepts of mathematical analysis with ADVANCED CALCULUS. Presented in a clear and simple way, this advanced caluclus text leads you to a precise understanding of the subject by providing you with the tools you need to succeed. A wide variety of exercises helps you gain a genuine understanding of the material and examples demonstrate the significance of what you learn. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge.
Author(s): Patrick M. Fitzpatrick
Edition: 2
Publisher: Brooks Cole
Year: 2005
Language: English
Pages: 609
ADVANCED CALCULUS......Page 2
CONTENTS......Page 6
PREFACE......Page 12
ABOUT THE AUTHOR......Page 19
PRELIMINARIES......Page 20
1.1 THE COMPLETENESS AXIOM AND SOME OF ITS CONSEQUENCES......Page 24
1.2 THE DISTRIBUTION OF THE INTEGERS AND THE RATIONAL NUMBERS......Page 31
1.3 INEQUALITIES AND IDENTITIES......Page 35
2.1 THE CONVERGENCE OF SEQUENCES......Page 42
2.2 SEQUENCES AND SETS......Page 54
2.3 THE MONOTONE CONVERGENCE THEOREM......Page 57
2.4 THE SEQUENTIAL COMPACTNESS THEOREM......Page 62
2.5 COVERING PROPERTIES OF SETS*......Page 66
3.1 CONTINUITY......Page 72
3.2 THE EXTREME VALUE THEOREM......Page 77
3.3 THE INTERMEDIATE VALUE THEOREM......Page 81
3.4 UNIFORM CONTINUITY......Page 85
3.5 THE epsilon-delta CRITERION FOR CONTINUITY......Page 89
3.6 IMAGES AND INVERSES; MONOTONE FUNCTIONS......Page 93
3.7 LIMITS......Page 100
4.1 THE ALGEBRA OF DERIVATIVES......Page 106
4.2 DIFFERENTIATING INVERSES AND COMPOSITIONS......Page 115
4.3 THE MEAN VALUE THEOREM AND ITS GEOMETRIC CONSEQUENCES......Page 120
4.4 THE CAUCHY MEAN VALUE THEOREM AND ITS ANALYTIC CONSEQUENCES......Page 130
4.5 THE NOTATION OF LEIBNITZ......Page 132
5.1 SOLUTIONS OF DIFFERENTIAL EQUATIONS......Page 135
5.2 THE NATURAL LOGARITHM AND EXPONENTIAL FUNCTIONS......Page 137
5.3 THE TRIGONOMETRIC FUNCTIONS......Page 144
5.4 THE INVERSE TRIGONOMETRIC FUNCTIONS......Page 151
6.1 DARBOUX SUMS; UPPER AND LOWER INTEGRALS......Page 154
6.2 THE ARCHIMEDES-RIEMANN THEOREM......Page 161
6.3 ADDITIVITY, MONOTONICITY, AND LINEARITY......Page 169
6.4 CONTINUITY AND INTEGRABILITY......Page 174
6.5 THE FIRST FUNDAMENTAL THEOREM: INTEGRATING DERIVATIVES......Page 179
6.6 THE SECOND FUNDAMENTAL THEOREM: DIFFERENTIATING INTEGRALS......Page 184
7.1 SOLUTIONS OF DIFFERENTIAL EQUATIONS......Page 194
7.2 INTEGRATION BY PARTS AND BY SUBSTITUTION......Page 197
7.3 THE CONVERGENCE OF DARBOUX AND RIEMANN SUMS......Page 202
7.4 THE APPROXIMATION OF INTEGRALS......Page 209
8.1 TAYLOR POLYNOMIALS......Page 218
8.2 THE LAGRANGE REMAINDER THEOREM......Page 222
8.3 THE CONVERGENCE OF TAYLOR POLYNOMIALS......Page 228
8.4 A POWER SERIES FOR THE LOGARITHM......Page 231
8.5 THE CAUCHY INTEGRAL REMAINDER THEOREM......Page 234
8.6 A NONANALYTIC, INFINITELY DIFFERENTIABLE FUNCTION......Page 240
8.7 THE WEIERSTRASS APPROXIMATION THEOREM......Page 242
9.1 SEQUENCES AND SERIES OF NUMBERS......Page 247
9.2 POINTWISE CONVERGENCE Of SEQUENCES OF FUNCTIONS......Page 260
9.3 UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS......Page 264
9.4 THE UNIFORM LIMIT OF FUNCTIONS......Page 268
9.5 POWER SERIES......Page 274
9.6 A CONTINUOUS NOWHERE DIFFERENTIABLE FUNCTION......Page 283
10.1 THE LINEAR STRUCTURE OF IR^n AND THE SCALAR PRODUCT......Page 288
10.2 CONVERGENCE OF SEQUENCES IN IR^n......Page 296
10.3 OPEN SETS AND CLOSED SETS IN IR^n......Page 301
11.1 CONTINUOUS FUNCTIONS AND MAPPINGS......Page 309
11.2 SEQUENTIAL COMPACTNESS, EXTREME VALUES, AND UNIFORM CONTINUITY......Page 317
11.3 PATHWISE CONNECTEDNESS AND THE INTERMEDIATE VALUE THEOREM......Page 323
11.4 CONNECTEDNESS AND THE INTERMEDIATE VALUE PROPERTY......Page 329
12.1 OPEN SETS, CLOSED SETS, AND SEQUENTIAL CONVERGENCE......Page 333
12.2 COMPLETENESS AND THE CONTRACTION MAPPING PRINCIPLE......Page 341
12.3 THE EXISTENCE THEOREM FOR NONLINEAR DIFFERENTIAL EQUATIONS......Page 347
12.4 CONTINUOUS MAPPINGS BETWEEN METRIC SPACES......Page 356
12.5 SEQUENTIAL COMPACTNESS AND CONNECTEDNESS......Page 361
13.1 LIMITS......Page 367
13.2 PARTIAL DERIVATIVES......Page 372
13.3 THE MEAN VALUE THEOREM AND DIRECTIONAL DERIVATIVES......Page 383
14.1 FIRST-ORDER APPROXIMATION, TANGENT PLANES, AND AFFINE FUNCTIONS......Page 391
14.2 QUADRATIC FUNCTIONS, HESSIAN MATRICES, AND SECOND DERIVATIVES......Page 399
14.3 SECOND-ORDER APPROXIMATION AND THE SECOND-DERIVATIVE TEST......Page 406
15.1 LINEAR MAPPINGS AND MATRICES......Page 413
15.2 THE DERIVATIVE MATRIX AND THE DIFFERENTIAL......Page 426
15.3 THE CHAIN RULE......Page 433
16.1 FUNCTIONS OF A SINGLE VARIABLE AND MAPS IN THE PLANE......Page 440
16.2 STABILITY OF NONLINEAR MAPPINGS......Page 448
16.3 A MINIMIZATION PRINCIPLE AND THE GENERAL INVERSE FUNCTION THEOREM......Page 452
17.1 A SCALAR EQUATION IN TWO UNKNOWNS: DINI'S THEOREM......Page 459
17.2 THE GENERAL IMPLICIT FUNCTION THEOREM......Page 468
17.3 EQUATIONS OF SURFACES AND PATHS IN IR^3......Page 473
17.4 CONSTRAINED EXTREMA PROBLEMS AND LAGRANGE MULTIPLIERS......Page 479
18.1 INTEGRATION OF FUNCTIONS ON GENERALIZED RECTANGLES......Page 489
18.2 CONTINUITY AND INTEGRABILITY......Page 501
18.3 INTEGRATION OF FUNCTIONS ON JORDAN DOMAINS......Page 508
19.1 FUBINI'S THEOREM......Page 517
19.2 THE CHANGE OF VARIABLES THEOREM: STATEMENTS AND EXAMPLES......Page 524
19.3 PROOF OF THE CHANGE OF VARIABLES THEOREM......Page 529
20.1 ARCLENGTH AND LINE INTEGRALS......Page 539
20.2 SURFACE AREA AND SURFACE INTEGRALS......Page 552
20.3 THE INTEGRAL FORMULAS OF GREEN AND STOKES......Page 562
A.1 THE FIELD AXIOMS AND THEIR CONSEQUENCES......Page 578
A.2 THE POSITIVITY AXIOMS AND THEIR CONSEQUENCES......Page 582
APPENDIX B: LINEAR ALGEBRA......Page 584
INDEX......Page 600
