S.Loaeza <br/><br/>
Advanced calculus / 

 - 13 
 - EUA   Mc Grall Hill   1990 
 - 365  Contiene gráficos    14.5cm de ancho X 21cm de largo  
 - Serie  .
<br/><br/>INDEX<br/><br/>A<br/><br/>Abel, N. H., 244<br/><br/>Abel's test for uniform convergence, 204<br/><br/>Abel's theorem, on interval of con-<br/><br/>vergence, 268<br/><br/>on power series, 276<br/><br/>Absolute convergence, of integrals,<br/><br/>337, 341, 344, 349<br/><br/>of series, 236, 240, 263, 268<br/><br/>Addition of series, 212, 240, 281<br/><br/>Adiabatic process, 208<br/><br/>Alternating series, 234<br/><br/>Angle, as a line integral, 171<br/><br/>solid, 171<br/><br/>Are length, 127<br/><br/>Area, as a double integral, 140<br/><br/>element of, 153, 201<br/><br/>as a line integral, 178<br/><br/>of surface, 161<br/><br/>Associative law for series, 212.<br/><br/>Attraction, law of, 172, 202, 369, 372<br/><br/>B<br/><br/>Bernoulli's inequality, 8<br/><br/>Bernoulli's numbers, 287, 289<br/><br/>Beta function, 376<br/><br/>Binomial series, 301<br/><br/>Binomial theorem, 301<br/><br/>Bolzano-Weierstrass theorem, 18<br/><br/>Bounded sets, 105<br/><br/>Bounds, 22<br/><br/>of a function, 23, 106<br/><br/>C<br/><br/>Cantor-Dedekind axiom, 3<br/><br/>Cauchy-Riemann equations, 206<br/><br/>Cauchy's integral test, 225<br/><br/>Cauchy's principal value, 347<br/><br/>Cauchy's product of series, 244<br/><br/>Cauchy's ratio test, 217n.<br/><br/>Cauchy's root test, 216<br/><br/>Cauchy's theorem, 53<br/><br/>Center of gravity, 137, 150, 165<br/><br/>Change of variables, in derivatives, 91<br/><br/>in integrals, 124, 147, 155, 199<br/><br/>Circular functions, 308<br/><br/>Closed curve, ares of, 178<br/><br/>simple, 179<br/><br/>Comparison test for series, 215<br/><br/>Complex number, a<br/><br/>Conditionally convergent series, 236<br/><br/>Conservative systems, 204<br/><br/>Continuity, 31, 35, 39, 58, 256, 275<br/><br/>equation of, 206<br/><br/>piece-wise, 34<br/><br/>uniform, 37, 38, 60<br/><br/>Convergence, 11<br/><br/>absolute, 236, 240, 263, 268, 337,<br/><br/>341, 344<br/><br/>criterion, 13, 14, 20, 210, 233<br/><br/>of integrals, 337, 341<br/><br/>testa for, 341, 349, 357<br/><br/>interval of, 248, 267, 260<br/><br/>radius of, 269<br/><br/>of series, 209, 247<br/><br/>tests for, 215, 225, 237, 262<br/><br/>uniform, 247, 263, 264, 355, 367,<br/><br/>408<br/><br/>Coordinate surfaces, 155<br/><br/>Coordinates, curvilinear, 148<br/><br/>cylindrical, 157<br/><br/>polar, 148, 152, 157<br/><br/>spherical, 157<br/><br/>Curve, simple, 179<br/><br/>Curvilinear coordinates, 148<br/><br/>Cut, 2<br/><br/>Cylindrical coordinates, 157<br/><br/>441<br/>INDEX<br/><br/>Fresnel's integrals, 365<br/><br/>Function, 22, 58, 321<br/><br/>bounds of, 106<br/><br/>composite, 45, 47, 439<br/><br/>continuity of, 31, 35, 39, 59, 392<br/><br/>even, 391<br/><br/>homogeneous, 75<br/><br/>implicit, 68, 415-440<br/><br/>integrable, 109, 110, 335л.<br/><br/>jump in, 34<br/><br/>multiple-valued, 22<br/><br/>normal, 388<br/><br/>odd, 391<br/><br/>orthogonal, 388<br/><br/>periodie, 379n.<br/><br/>piece-wise continuous, 34<br/><br/>sectionally continuous, 34<br/><br/>single-valued, 22<br/><br/>uniformly continuous, 38<br/><br/>Functional dependence, 423, 433<br/><br/>167, 200, 421, 438<br/><br/>Functional determinant, 153, 157,<br/><br/>Fundamental theorem of integral calculus, 110, 118, 120<br/><br/>G<br/><br/>Gamma functions, 372<br/><br/>Gauss's test for series, 230<br/><br/>Gauss's theorem, 169n. Geometric series, 214<br/><br/>Gradient, 78<br/><br/>Gravitational potential (see Poten-tial)<br/><br/>Gravity, center of, 137, 159, 165<br/><br/>Gmen's theorem, 167, 170, 172, 181<br/><br/>H<br/><br/>Helix, 85, 86<br/><br/>Homogeneous functions, 75<br/><br/>Hyperbolic functions, 306<br/><br/>Hyperbolic paraboloid, 325<br/><br/>I<br/><br/>Implicit functions, 68, 415-440<br/><br/>ferentiation of, 71, 416<br/><br/>existence theorem on, 425<br/><br/>ger derivatives of, 80<br/><br/>Improper integrala, 336-377<br/><br/>multiple, 365<br/><br/>Indefinite integrals, 119<br/><br/>Indeterminate forma, 4<br/><br/>Induction, mathematical, 9, 431<br/><br/>differentiable, 43<br/><br/>Infinite integrals, 335, 347<br/><br/>443<br/><br/>testa for convergence of, 341, 349,<br/><br/>367<br/><br/>(See also Improper integrals)<br/><br/>Infinite series, 200-266<br/><br/>absolute convergence of, 236, 240,<br/><br/>253, 268<br/><br/>addition of, 212, 240, 281<br/><br/>alternating, 234<br/><br/>of arbitrary terms, 233<br/><br/>conditional convergence of, 236<br/><br/>convergence of, 200, 233, 236<br/><br/>differentiation of, 261<br/><br/>division of, 285<br/><br/>double, 246<br/><br/>expansion in, 208, 383<br/><br/>of functions, 247, 275<br/><br/>geometric, 214<br/><br/>integration of, 258, 309<br/><br/>multiplication of, 242, 281<br/><br/>of orthogonal functions, 380<br/><br/>of positive terms, 233<br/><br/>of power functions, 267-334<br/><br/>algebra, 280<br/><br/>applications, 291-334<br/><br/>calculations with, 285<br/><br/>expansion in, 291<br/><br/>integration, 300<br/><br/>reversion, 289<br/><br/>tests for convergence, 269<br/><br/>uniqueness theorem on, 279<br/><br/>remainder in, 234<br/><br/>sum of, 209, 395<br/><br/>tests for convergence, 215, 225<br/><br/>237, 262, 269<br/><br/>uniform convergence of, 247, 252<br/><br/>275<br/><br/>(See also Fourier series)<br/><br/>Infinitesimal, 648.<br/><br/>Infinity, 16<br/><br/>Integrable function, 109, 335m<br/><br/>Integral equation, 413<br/>440<br/><br/>ADVANCED CALCULU<br/><br/>nal<br/><br/>Taylor's formula, 202, 293, 317 applications of, 208, 322<br/><br/>Taylor's series, 206<br/><br/>Tests, for integrals, 341, 349, 367 for series, 215, 225, 237, 262, 264<br/><br/>Total differential, 64 (See also Exact differential)<br/><br/>Transformation, of coordinates, 153 of integrals, 167, 170, 172, 181, 196 of inversion, 440<br/><br/>Trigonometric series (see Fourier series)<br/><br/>Triple integrals, 142 improper, 367 U<br/><br/>Undetermined multipliers, 328 Uniform continuity, 37, 60<br/><br/>Uniform convergence, of integrals, 355, 367 of series, 247, 262, 264, 408<br/><br/>Upper bound, of a function, 106 V<br/><br/>Velocity potential, 206 Vicinity, 8 Volume, element of, 156, 158, 159 of revolution, 128, 143, 155 W<br/><br/>Wallis's formula, 311 Weierstrass test, for integrals, 355 for series, 262 Work, as a line integral, 201<br/><br/>Y<br/><br/>Young's theorem, 80m.<br/><br/>Z<br/><br/>Zeta-function, 364, 365 
<br/><br/> PREFACE<br/><br/>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.<br/><br/>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.<br/><br/>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.<br/> 
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<br/><br/><label>ISBN: </label>9786078273188 
<br/><br/><label>LC Class. No.: </label>LCC