Advanced Calculus

Contents......Page 8
Real Numbers......Page 12
Geometric Representation of Real Numbers......Page 13
Inequalities......Page 14
Logarithms......Page 15
Point Sets, Intervals......Page 16
Bounds......Page 17
Polar Form of Complex Numbers......Page 18
Mathematical Induction......Page 19
Theorems on Limits of Sequences......Page 36
Least Upper Bound and Greatest Lower Bound of a Sequence......Page 37
Infinite Series......Page 38
Functions......Page 54
Graph of a Function......Page 55
Inverse Functions, Principal Values......Page 56
Maxima and Minima......Page 57
Types of Functions......Page 58
Transcendental Functions......Page 59
Right- and Left-Hand Limits......Page 60
Special Limits......Page 61
Continuity in an Interval......Page 62
Piecewise Continuity......Page 63
Uniform Continuity......Page 64
The Concept and Definition of a Derivative......Page 82
Right- and Left-Hand Derivatives......Page 83
Differentials......Page 84
Rules for Differentiation......Page 86
Derivatives of Elementary Functions......Page 87
Mean Value Theorems......Page 88
L’Hospital’s Rules......Page 89
Applications......Page 90
Introduction of the Definite Integral......Page 108
Properties of Definite Integrals......Page 109
Mean Value Theorems for Integrals......Page 110
Connecting Integral and Differential Calculus......Page 111
Change of Variable of Integration......Page 112
Integrals of Elementary Functions......Page 113
Special Methods of Integration......Page 114
Numerical Methods for Evaluating Definite Integrals......Page 115
Arc Length......Page 116
Volumes of Revolution......Page 118
Functions of Two or More Variables......Page 136
Regions......Page 137
Iterated Limits......Page 138
Partial Derivatives......Page 139
Differentials......Page 140
Theorems on Differentials......Page 141
Implicit Functions......Page 142
Partial Derivatives Using Jacobians......Page 143
Transformations......Page 144
Mean Value Theorems......Page 145
Geometric Properties of Vectors......Page 172
Algebraic Properties of Vectors......Page 173
Components of a Vector......Page 174
Dot, Scalar, or Inner Product......Page 175
Cross or Vector Product......Page 176
Axiomatic Approach To Vector Analysis......Page 177
Vector Functions......Page 178
Geometric Interpretation of a Vector Derivative......Page 179
Gradient, Divergence, and Curl......Page 181
Vector Interpretation of Jacobians and Orthogonal Curvilinear Coordinates......Page 182
Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates......Page 183
Special Curvilinear Coordinates......Page 184
Applications to Geometry......Page 206
Integration Under the Integral Sign......Page 209
Maxima and Minima......Page 210
Applications to Errors......Page 211
Double Integrals......Page 232
Iterated Integrals......Page 233
Triple Integrals......Page 235
The Differential Element of Area in Polar Coordinates, Differential Elements of Area in Cylindrical and Spherical Coordinates......Page 236
Line Integrals......Page 254
Simple Closed Curves, Simply and Multiply Connected Regions......Page 256
Conditions for a Line Integral to Be Independent of the Path......Page 257
Surface Integrals......Page 258
The Divergence Theorem Stokes’s Theorem......Page 261
Stokes’s Theorem......Page 262
Definitions of Infinite Series and Their Convergence and Divergence......Page 290
Tests for Convergence and Divergence of Series of Constants......Page 291
Theorems on Absolutely Convergent Series......Page 293
Infinite Sequences and Series of Functions, Uniform Convergence......Page 294
Theorems on Uniformly Convergent Series......Page 295
Power Series......Page 296
Expansion of Functions in Power Series......Page 297
Taylor’s Theorem......Page 298
Some Important Power Series......Page 300
Taylor’s Theorem (For Two Variables)......Page 301
Improper Integrals of the First Kind (Unbounded Intervals)......Page 332
Convergence or Divergence of Improper Integrals of the First Kind......Page 333
Convergence Tests for Improper Integrals of the First Kind......Page 334
Improper Integrals of the Second Kind......Page 335
Special Improper Integrals of the Second Kind......Page 336
Convergence Tests for Improper Integrals of the Second Kind......Page 337
Special Tests for Uniform Convergence of Integrals......Page 338
Laplace Transforms......Page 339
Application......Page 340
Improper Multiple Integrals......Page 341
Fourier Series......Page 360
Dirichlet Conditions......Page 361
Differentiation and Integration of Fourier Series......Page 362
Boundary-Value Problems......Page 363
Orthogonal Functions......Page 366
The Fourier Integral......Page 388
Fourier Transforms......Page 389
Table of Values and Graph of the Gamma Function......Page 400
The Beta Function......Page 403
Dirichlet Integrals......Page 404
Limits and Continuity......Page 416
Integrals......Page 417
Cauchy’s Integral Formulas......Page 418
Laurent’s Series......Page 419
Branches and Branch Points......Page 420
Residue Theorem......Page 421
Evaluation of Definite Integrals......Page 422
C......Page 448
D......Page 449
E......Page 450
H......Page 451
I......Page 452
M......Page 453
R......Page 454
S......Page 455
Z......Page 456