Calculus: A modern, rigorous approach

Contents......Page 3
Elementary Mathematical Logic......Page 5
Sets......Page 14
Maps......Page 23
Limits of Sequences of Real Numbers......Page 36
Continuous Functions......Page 55
Differentiation......Page 79
Applications of Differentiation......Page 94
Riemann Integration......Page 140
Change of Variables......Page 163
Integration by Parts......Page 173
Partial Fractions......Page 186
Approximate Calculation of Integrals......Page 196
Improper Integrals......Page 202
Series of Real Numbers......Page 223
Series of Functions......Page 248
Metric Spaces......Page 292
Vector Spaces......Page 298
Conic Sections......Page 323
Polar Coordinates......Page 330
Quadric Surfaces......Page 338
Cylindrical and Spherical Coordinates......Page 346
Limits in Rn......Page 351
Paths in Rn......Page 356
Vector-valued Functions of Several Variables......Page 375
Derivatives of Vector-valued Functions of Several Variables......Page 394
Applications of Differentiation......Page 423
The Riemann Integral in n-dimensions......Page 446
Applications of Multiple Integrals......Page 460
Vector Calculus......Page 471
Generalizations of the Fundamental Theorem of Calculus......Page 483
Green\'s Theorem......Page 486
Stokes\' Theorem......Page 502
Gauss\' Theorem......Page 512
Construction of the Real Number System......Page 529
Lebesgue\'s Criterion of Riemann-integrability......Page 542
Properties of the Determinant......Page 546
The Inverse Mapping Theorem......Page 563
References......Page 566
Index of Notation......Page 569
Index of Terminology......Page 571