Advanced calculus /
S.Loaeza
Advanced calculus / - 13 - EUA Mc Grall Hill 1990 - 365 Contiene gráficos 14.5cm de ancho X 21cm de largo - Serie .
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.
9786078273188
LCC
Advanced calculus / - 13 - EUA Mc Grall Hill 1990 - 365 Contiene gráficos 14.5cm de ancho X 21cm de largo - Serie .
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.
9786078273188
LCC
