Advanced calculus /

INDEX

A

Abel, N. H., 244

Abel's test for uniform convergence, 204

Abel's theorem, on interval of con-

vergence, 268

on power series, 276

Absolute convergence, of integrals,

337, 341, 344, 349

of series, 236, 240, 263, 268

Addition of series, 212, 240, 281

Adiabatic process, 208

Alternating series, 234

Angle, as a line integral, 171

solid, 171

Are length, 127

Area, as a double integral, 140

element of, 153, 201

as a line integral, 178

of surface, 161

Associative law for series, 212.

Attraction, law of, 172, 202, 369, 372

B

Bernoulli's inequality, 8

Bernoulli's numbers, 287, 289

Beta function, 376

Binomial series, 301

Binomial theorem, 301

Bolzano-Weierstrass theorem, 18

Bounded sets, 105

Bounds, 22

of a function, 23, 106

C

Cantor-Dedekind axiom, 3

Cauchy-Riemann equations, 206

Cauchy's integral test, 225

Cauchy's principal value, 347

Cauchy's product of series, 244

Cauchy's ratio test, 217n.

Cauchy's root test, 216

Cauchy's theorem, 53

Center of gravity, 137, 150, 165

Change of variables, in derivatives, 91

in integrals, 124, 147, 155, 199

Circular functions, 308

Closed curve, ares of, 178

simple, 179

Comparison test for series, 215

Complex number, a

Conditionally convergent series, 236

Conservative systems, 204

Continuity, 31, 35, 39, 58, 256, 275

equation of, 206

piece-wise, 34

uniform, 37, 38, 60

Convergence, 11

absolute, 236, 240, 263, 268, 337,

341, 344

criterion, 13, 14, 20, 210, 233

of integrals, 337, 341

testa for, 341, 349, 357

interval of, 248, 267, 260

radius of, 269

of series, 209, 247

tests for, 215, 225, 237, 262

uniform, 247, 263, 264, 355, 367,

408

Coordinate surfaces, 155

Coordinates, curvilinear, 148

cylindrical, 157

polar, 148, 152, 157

spherical, 157

Curve, simple, 179

Curvilinear coordinates, 148

Cut, 2

Cylindrical coordinates, 157

441
INDEX

Fresnel's integrals, 365

Function, 22, 58, 321

bounds of, 106

composite, 45, 47, 439

continuity of, 31, 35, 39, 59, 392

even, 391

homogeneous, 75

implicit, 68, 415-440

integrable, 109, 110, 335л.

jump in, 34

multiple-valued, 22

normal, 388

odd, 391

orthogonal, 388

periodie, 379n.

piece-wise continuous, 34

sectionally continuous, 34

single-valued, 22

uniformly continuous, 38

Functional dependence, 423, 433

167, 200, 421, 438

Functional determinant, 153, 157,

Fundamental theorem of integral calculus, 110, 118, 120

G

Gamma functions, 372

Gauss's test for series, 230

Gauss's theorem, 169n. Geometric series, 214

Gradient, 78

Gravitational potential (see Poten-tial)

Gravity, center of, 137, 159, 165

Gmen's theorem, 167, 170, 172, 181

H

Helix, 85, 86

Homogeneous functions, 75

Hyperbolic functions, 306

Hyperbolic paraboloid, 325

I

Implicit functions, 68, 415-440

ferentiation of, 71, 416

existence theorem on, 425

ger derivatives of, 80

Improper integrala, 336-377

multiple, 365

Indefinite integrals, 119

Indeterminate forma, 4

Induction, mathematical, 9, 431

differentiable, 43

Infinite integrals, 335, 347

443

testa for convergence of, 341, 349,

367

(See also Improper integrals)

Infinite series, 200-266

absolute convergence of, 236, 240,

253, 268

addition of, 212, 240, 281

alternating, 234

of arbitrary terms, 233

conditional convergence of, 236

convergence of, 200, 233, 236

differentiation of, 261

division of, 285

double, 246

expansion in, 208, 383

of functions, 247, 275

geometric, 214

integration of, 258, 309

multiplication of, 242, 281

of orthogonal functions, 380

of positive terms, 233

of power functions, 267-334

algebra, 280

applications, 291-334

calculations with, 285

expansion in, 291

integration, 300

reversion, 289

tests for convergence, 269

uniqueness theorem on, 279

remainder in, 234

sum of, 209, 395

tests for convergence, 215, 225

237, 262, 269

uniform convergence of, 247, 252

275

(See also Fourier series)

Infinitesimal, 648.

Infinity, 16

Integrable function, 109, 335m

Integral equation, 413
440

ADVANCED CALCULU

nal

Taylor's formula, 202, 293, 317 applications of, 208, 322

Taylor's series, 206

Tests, for integrals, 341, 349, 367 for series, 215, 225, 237, 262, 264

Total differential, 64 (See also Exact differential)

Transformation, of coordinates, 153 of integrals, 167, 170, 172, 181, 196 of inversion, 440

Trigonometric series (see Fourier series)

Triple integrals, 142 improper, 367 U

Undetermined multipliers, 328 Uniform continuity, 37, 60

Uniform convergence, of integrals, 355, 367 of series, 247, 262, 264, 408

Upper bound, of a function, 106 V

Velocity potential, 206 Vicinity, 8 Volume, element of, 156, 158, 159 of revolution, 128, 143, 155 W

Wallis's formula, 311 Weierstrass test, for integrals, 355 for series, 262 Work, as a line integral, 201

Y

Young's theorem, 80m.

Z

Zeta-function, 364, 365

PREFACE

There is a pronounced need of a book on advanced calculus that does not sacrifice rigor to such an extent as to become ineffectual as an instrument for developing a critical attitude toward analyti-cal processes, and yet which is sufficiently concrete to be useful to a student with one year of preparation in the calculus. I am under no delusion that this volume completely fills this need, and I shall feel generously repaid for my efforts if it should prove of some aid to those who are faced with the perplexing problem of instruction in analysis.

In preparing this book I have made every effort to keep in mind the difficulties of the reader who is encountering for the first time a serious body of mathematical doctrine. Some ideas that are innately difficult, but whose basic sources stem from geometry, are presented first from an intuitive point of view, so that the essentials can be grasped at once. I did not think it wise to include rigorous arithmetical proofs of such theorems as those on convergence of bounded monotone sequences (Sec. 6), the theorem of Bolzano-Weierstrass (Sec. 7), the theorem of Darboux (Sec. 35), and a few others. This is in accordance with the precept that the most effective means of thwarting interest in mathematics is by misdirecting rigor. A reader who is suffi-ciently sophisticated to feel the need of arithmetical proofs of these theorems will find them in the treatises to which I refer in the text. The material contained in this volume is so arranged as to minimize the need of irksome references to matters to be established later on. No difficult and essential proofs have been relegated to exercises to be worked out at the reader's leisure.

The subject of advanced calculus is not an easy one, and the working of the problems is essential to a mastery. There are numerous illustrative exercises and problems scattered through-out the text to aid the reader in gaining an insight into the beauty and the wide range of applications of analysis. A student with a good background in the calculus will be able to read this book without omissions.

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