Schaum's Outline of Advanced Calculus

✍ Scribed by Wrede, Robert C;Spiegel, Murray R

• Publisher: McGraw-Hill Education; McGraw-Hill Trade
• Year: 2001
• Tongue: English
• Leaves: 443
• Edition: 2
• Category: Library

✦ Synopsis

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Coverage of the most up-to-date developments in your course field
In-depth review of practices and applications
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✦ Table of Contents

Copyright......Page 3
PREFACE......Page 5
CONTENTS......Page 7
REAL NUMBERS......Page 11
OPERATIONS WITH REAL NUMBERS......Page 12
EXPONENTS AND ROOTS......Page 13
AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM......Page 14
LIMIT POINTS......Page 15
THE COMPLEX NUMBER SYSTEM......Page 16
POLAR FORM OF COMPLEX NUMBERS......Page 17
MATHEMATICAL INDUCTION......Page 18
THEOREMS ON LIMITS OF SEQUENCES......Page 33
LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE......Page 34
INFINITE SERIES......Page 35
FUNCTIONS......Page 49
BOUNDED FUNCTIONS......Page 50
INVERSE FUNCTIONS. PRINCIPAL VALUES......Page 51
MAXIMA AND MINIMA......Page 52
TYPES OF FUNCTIONS......Page 53
TRANSCENDENTAL FUNCTIONS......Page 54
THEOREMS ON LIMITS......Page 55
CONTINUITY......Page 56
THEOREMS ON CONTINUITY......Page 57
UNIFORM CONTINUITY......Page 58
THE CONCEPT AND DEFINITION OF A DERIVATIVE......Page 75
DIFFERENTIALS......Page 77
IMPLICIT DIFFERENTIATION......Page 79
RULES FOR DIFFERENTIATION......Page 80
HIGHER ORDER DERIVATIVES......Page 81
L’HOSPITAL’S RULES......Page 82
APPLICATIONS......Page 83
INTRODUCTION OF THE DEFINITE INTEGRAL......Page 100
PROPERTIES OF DEFINITE INTEGRALS......Page 101
MEAN VALUE THEOREMS FOR INTEGRALS......Page 102
THE FUNDAMENTAL THEOREM OF THE CALCULUS......Page 104
INTEGRALS OF ELEMENTARY FUNCTIONS......Page 105
IMPROPER INTEGRALS......Page 107
APPLICATIONS......Page 108
ARC LENGTH......Page 109
VOLUMES OF REVOLUTION......Page 110
THREE-DIMENSIONAL RECTANGULAR COORDINATE SYSTEMS......Page 126
REGIONS......Page 127
LIMITS......Page 128
PARTIAL DERIVATIVES......Page 129
DIFFERENTIALS......Page 130
THEOREMS ON DIFFERENTIALS......Page 131
EULER’S THEOREM ON HOMOGENEOUS FUNCTIONS......Page 132
PARTIAL DERIVATIVES USING JACOBIANS......Page 133
TRANSFORMATIONS......Page 134
MEAN VALUE THEOREM......Page 135
GEOMETRIC PROPERTIES......Page 160
ALGEBRAIC PROPERTIES OF VECTORS......Page 161
RECTANGULAR (ORTHOGONAL) UNIT VECTORS......Page 162
DOT OR SCALAR PRODUCT......Page 163
CROSS OR VECTOR PRODUCT......Page 164
AXIOMATIC APPROACH TO VECTOR ANALYSIS......Page 165
LIMITS, CONTINUITY, AND DERIVATIVES OF VECTOR FUNCTIONS......Page 166
GEOMETRIC INTERPRETATION OF A VECTOR DERIVATIVE......Page 167
GRADIENT, DIVERGENCE, AND CURL......Page 168
FORMULAS INVOLVING v......Page 169
VECTOR INTERPRETATION OF JACOBIANS, ORTHOGONAL CURVILINEAR COORDINATES......Page 170
SPECIAL CURVILINEAR COORDINATES......Page 171
APPLICATIONS TO GEOMETRY......Page 193
INTEGRATION UNDER THE INTEGRAL SIGN......Page 196
MAXIMA AND MINIMA......Page 197
METHOD OF LAGRANGE MULTIPLIERS FOR MAXIMA AND MINIMA......Page 198
APPLICATIONS TO ERRORS......Page 199
DOUBLE INTEGRALS......Page 217
ITERATED INTEGRALS......Page 218
TRIPLE INTEGRALS......Page 220
THE DIFFERENTIAL ELEMENT OF AREA IN POLAR COORDINATES, DIFFERENTIAL ELEMENTS OF AREA IN CYLINDRAL AND SPHERICAL COORDINATES......Page 221
LINE INTEGRALS......Page 239
PROPERTIES OF LINE INTEGRALS EXPRESSED FOR PLANE CURVES......Page 241
CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH......Page 242
SURFACE INTEGRALS......Page 243
THE DIVERGENCE THEOREM......Page 246
STOKES’ THEOREM......Page 247
DEFINITIONS OF INFINITE SERIES AND THEIR CONVERGENCE AND DIVERGENCE......Page 275
TESTS FOR CONVERGENCE AND DIVERGENCE OF SERIES OF CONSTANTS......Page 276
INFINITE SEQUENCES AND SERIES OF FUNCTIONS, UNIFORM CONVERGENCE......Page 279
SPECIAL TESTS FOR UNIFORM CONVERGENCE OF SERIES......Page 280
THEOREMS ON UNIFORMLY CONVERGENT SERIES......Page 281
OPERATIONS WITH POWER SERIES......Page 282
TAYLOR’S THEOREM......Page 283
SOME IMPORTANT POWER SERIES......Page 285
TAYLOR’S THEOREM (FOR TWO VARIABLES)......Page 286
IMPROPER INTEGRALS OF THE FIRST KIND (Unbounded Intervals)......Page 316
CONVERGENCE OR DIVERGENCE OF IMPROPER INTEGRALS OF THE FIRST KIND......Page 317
CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE FIRST KIND......Page 318
CAUCHY PRINCIPAL VALUE......Page 320
CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE SECOND KIND......Page 321
SPECIAL TESTS FOR UNIFORM CONVERGENCE OF INTEGRALS......Page 323
LAPLACE TRANSFORMS......Page 324
APPLICATION......Page 325
IMPROPER MULTIPLE INTEGRALS......Page 326
PERIODIC FUNCTIONS......Page 346
DIRICHLET CONDITIONS......Page 347
PARSEVAL’S IDENTITY......Page 348
BOUNDARY-VALUE PROBLEMS......Page 349
ORTHOGONAL FUNCTIONS......Page 352
THE FOURIER INTEGRAL......Page 373
FOURIER TRANSFORMS......Page 374
TABLES OF VALUES AND GRAPH OF THE GAMMA FUNCTION......Page 385
THE BETA FUNCTION......Page 388
DIRICHLET INTEGRALS......Page 389
FUNCTIONS......Page 402
CAUCHY-RIEMANN EQUATIONS......Page 403
CAUCHY’S INTEGRAL FORMULAS......Page 404
LAURENT’S SERIES......Page 405
BRANCHES AND BRANCH POINTS......Page 406
RESIDUE THEOREM......Page 407
EVALUATION OF DEFINITE INTEGRALS......Page 408
INDEX......Page 435

✦ Subjects

Science;Mathematics;Nonfiction;Reference;Calculus