A course in mathematical analysis - Applications to Geometry, Expansion in Series, Definite Integrals, Derivatives and Differentias 1

Édouard Goursat's three-volume A Course in Mathematical Analysis remains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. Subjects in this, the first of the three volumes, include derivatives and differentials; implicit functions; functional determinants; change of variable; Taylor's series; maxima and minima; definite and indefinite integrals; double and multiple integrals; integration of total differentials; infinite series; power series; trigonometric series; plane and skew curves; and surfaces. Volume 2 addresses functions of a complex variable and differential equations; and Volume 3 explores variations of solutions, partial differential equations of the second order, integral equations, and calculus of variations.Édouard Goursat's three-volume A Course in Mathematical Analysis remains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. Subjects in this, the first of the three volumes, include derivatives and differentials; implicit functions; functional determinants; change of variable; Taylor's series; maxima and minima; definite and indefinite integrals; double and multiple integrals; integration of total differentials; infinite series; power series; trigonometric series; plane and skew curves; and surfaces. Volume 2 addresses functions of a complex variable and differential equations; and Volume 3 explores variations of solutions, partial differential equations of the second order, integral equations, and calculus of variations.Édouard Goursat's three-volume A Course in Mathematical Analysis remains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. Subjects in this, the first of the three volumes, include derivatives and differentials; implicit functions; functional determinants; change of variable; Taylor's series; maxima and minima; definite and indefinite integrals; double and multiple integrals; integration of total differentials; infinite series; power series; trigonometric series; plane and skew curves; and surfaces. Volume 2 addresses functions of a complex variable and differential equations; and Volume 3 explores variations of solutions, partial differential equations of the second order, integral equations, and calculus of variations. C 0 N T E N T S CHAPTERS PAGE I. DERIVATIVES AND DIFFERENTIALS............................ 1 I. FUNCTIONS OF A SINGLE VARIABLE.......................... 1 II. FUNCTIONS OF SEVERAL VARIABLES......................... 11 III. THE DIFFERENTIAL NOTATION............................. 19 II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGE OF VARIABLES............................................ 35 I. IMPLICIT FUNCTIONS...................................... 35 II. FUNCTIONAL DETERMINANTS................................. 52 III. TRANSFORMATIONS......................................... 61 III. TAYLOR SERIES.ELEMENTARY APPLICATIONS. MAXIMA AND MINIMA.............................................. 80 I. TAYLOR'S SERIES WITH A REMAINDER. TAYLOR’S SERIES....... 89 II. SINGULAR POINTS. MAXIMA AND MINIMA...................... 110 IV. DEFINITE INTEGRALS....................................... 134 I. SPECIAL METHODS OF QUADRATURE............................ 134 II. DEFINITE INTEGRALS. ALLIED GEOMETRICAL CONCEPTS......... 140 III. CHANGE OF VARIABLE. INTEGRATION LAY PARTS.............. 166 IV. GENERALIZATIONS OF THE IDEA OF AN INTEGRAL. IMPROPER INTEGRALS. LINE INTEGRALS............................... 175 V. FUNCTIONS DEFINED BY INFINITE INTEGRALS................. 192 VI. APPROXIMATE EVALUATION OF DEFINITE INTEGRALS............ 196 V. INDEFINITE INTEGRALS.................................... 208 I. INTEGRATION OF RATIONAL FUNCTIONS....................... 268 II. ELLIPTIC AND HYPERELLIPTIC INTEGRALS.................... 226 III. INTEGRATION OF TRANSCENDENTAL FUNCTIONS................. 236 VI. DOUBLE INTEGRALS........................................ 250 I. DOUBLE INTEGRALS. METHODS OF EVALUATION. GREEN’S THEOREM.250 II. CHANGE OF VARIABLES. AREA OF A SURFACE.................. 264 III. GENERALIZATIONS OF DOUBLE INTEGRALS. IMPROPER INTEGRALS. SURFACE INTEGRALS....................................... 277 IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS................. 284 VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFERENTIALS. 296 I. MULTIPLE INTEGRALS. CHANGE OF VARIABLES................ 296 II. INTEGRATION OF TOTAL DIFFERENTIALS..................... 313 VIII. INFINITE SERIES...................................... 327 I. SERIES OF REAL CONSTANT. TERMS. GENERAL PROPERTIES. TESTS FOR CONVERGENCE ............................... 327 II. SERIES OF COMPLEX TERMS.MULTIPLE SERIES.............. 350 III. SERIES OF VARIABLE TERMS. UNIFORM CONVERGENCE........ 366 IX. POWER SERIES. TRIGONOMETRIC SERIES................. 375 I. POWER SERIES OF A SINGLE VARIABLE.................. 375 II. POWER SERIES OF A SEVERAL VARIABLE................. 394 III. IMPLICIT FUNCTIONS. ANALYTIC CURVES AND SURFACES... 399 IV. TRIGONOMETRIC SERIES. MISCELLANEOUS SERIES......... 411 X. PLANE CURVES......................................... 426 I. ENVELOPES............................................ 426 II. CURVATURE............................................ 433 III. CONTACT OF PLANE CURVES.............................. 443 XI. SPACE CURVES ........................................ 453 I. OSCULATING PLANE..................................... 453 II. ENVELOPES OF SURFACES................................ 459 III. CURVATURE AND TORSION OF SKEW CURVES................. 466 IV. CONTACT BETWEEN SKEW CURVES. CONTACT BETWEEN CURVES AND SURFACES......................................... 486 XII. SURFACES............................................ 497 I. CURVATURE OF CURVES DRAWN ON A SURFACE...............497 II. ASYMPTOTIC LINES - CONJUGATE LINES.................. 506 III. LINES OF CURVATURE.................................. 514 IV. FAMILIES OF STRAIGHT LINES.......................... 520 INDEX..................................................... 541 v. 1. Derivatives and differentials. Definite integrals. Expansion in series. Applications to geometry / translated by Earle Raymond Hedrick v. 2. pt. 1. Functions of a complex variable / translated by Earle Raymond Hedrick and Otto Dunkel. pt. 2. Differential equations / translated by Earle Raymond Hedrick and Otto Dunkel v. 3. pt. 1. Variation of solutions. Partial differential equations of the second order / translated by Howard G. Bergmann. pt. 2. Integral equations. Calculus of variations / translated by Howard G. Bergmann.

Classic three-volume study. Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials. Volume 2 explores functions of a complex variable and differential equations. Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.