Advanced Calculus Second Edition

by Widder, David V.

• ISBN13: 9780486661032
• ISBN10: 0486661032
• Edition: 2nd
• Format: Paperback
• Copyright: 1989-08-01
• Publisher: Dover Publications

Summary

Classic text leads from elementary calculus into more theoretic problems. Precise approach with definitions, theorems, proofs, examples and exercises. Topics include partial differentiation, vectors, differential geometry, Stieltjes integral, infinite series, gamma function, Fourier series, Laplace transform, much more. Numerous graded exercises with selected answers. 1961 edition.

Table of Contents

Partial Differentiation	p. 1
Introduction	p. 1
Functions of One Variable	p. 6
Functions of Several Variables	p. 11
Homogeneous Functions. Higher Derivatives	p. 19
Implicit Functions	p. 23
Simultaneous Equations. Jacobians	p. 27
Dependent and Independent Variables	p. 33
Differentials. Directional Derivatives	p. 36
Taylor's Theorem	p. 43
Jacobians	p. 48
Equality of Cross Derivatives	p. 52
Implicit Functions	p. 55
Vectors	p. 62
Introduction	p. 62
Solid Analytic Geometry	p. 66
Space Curves	p. 70
Surfaces	p. 75
A Symbolic Vector	p. 79
Invariants	p. 84
Differential Geometry	p. 88
Arc Length of a Space Curve	p. 88
Osculating Plane	p. 92
Curvature and Torsion	p. 97
Frenet-Serret Formulas	p. 101
Surface Theory	p. 105
Fundamental Differential Forms	p. 109
Mercator Maps	p. 113
Applications of Partial Differentiation	p. 118
Maxima and minima	p. 118
Functions of Two Variables	p. 122
Sufficient Conditions	p. 126
Functions of Three Variables	p. 132
Lagrange's Multipliers	p. 135
Families of Plane Curves	p. 141
Families of Surfaces	p. 145
Stieltjes Integral	p. 149
Introduction	p. 149
Properties of the Integral	p. 155
Integration by Parts	p. 159
Laws of the Mean	p. 161
Physical Applications	p. 165
Continuous Functions	p. 170
Existence of Stieltjes Integrals	p. 176
Multiple Integrals	p. 181
Introduction	p. 181
Properties of Double Integrals	p. 185
Evaluation of Double Integrals	p. 188
Polar Coordinates	p. 191
Change in Order of Integration	p. 195
Applications	p. 198
Further Applications	p. 202
Triple Integrals	p. 206
Other Coordinates	p. 210
Existence of Double Integrals	p. 213
Line and Surface Integrals	p. 217
Introduction	p. 217
Green's Theorem	p. 223
Application	p. 227
Surface Integrals	p. 232
Change of Variable in Multiple Integrals	p. 239
Line Integrals in Space	p. 246
Limits and Indeterminate Forms	p. 257
The Indeterminate Form 0/0	p. 257
The Indeterminate Form
Other Indeterminate Forms	p. 268
Other Methods. Orders of Infinity	p. 271
Superior and Inferior Limits	p. 277
Infinite Series	p. 284
Convergence of Series. Comparison Tests	p. 284
Convergence Tests	p. 288
Absolute Convergence. Altering Series	p. 292
Limit Tests	p. 295
Uniform Convergence	p. 298
Applications	p. 303
Divergent Series	p. 307
Miscellaneous Methods	p. 313
Power Series	p. 318
Convergence of Improper Integrals	p. 323
Introduction	p. 323
Type I. Limit Tests	p. 330
Type I. Conditional Convergence	p. 333
Type III	p. 337
Combination of Types	p. 342
Uniform Convergence	p. 347
Properties of Proper Integrals	p. 350
Application of Uniform Convergence	p. 356
Divergent Integrals	p. 361
Integral Inequalities	p. 364
The Gamma Function. Evaluation of Definite Integrals	p. 367
Introduction	p. 367
The Beta Function	p. 373
Evaluation of Definite Integrals	p. 378
Stirling's Formula	p. 383
Fourier Series	p. 393
Introduction	p. 393
Several Classes of Functions	p. 398
Convergence of a Fourier Series to Its Defining Function	p. 403
Extensions and Applications	p. 408
Vibrating String	p. 413
Summability of Fourier Series	p. 421
Applications	p. 425
Fourier Integral	p. 431
The Laplace Transform	p. 436
Introduction	p. 436
Region of Convergence	p. 440
Absolute and Uniform Convergence	p. 445
Operational Properties of the Transform	p. 448
Resultant	p. 452
Tables of Transforms	p. 455
Uniqueness	p. 459
Inversion	p. 461
Representation	p. 465
Related Transforms	p. 468
Applications of the Laplace Transform	p. 474
Introduction	p. 474
Linear Differential Equation	p. 478
The General Homogeneous Case	p. 482
The Nonhomogeneous Case	p. 487
Difference Equations	p. 490
Partial Differential Equations	p. 494
Selected Answers	p. 499
Index of Symbols	p. 511
Index	p. 513
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