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.
Author(s): Edouard Goursat
Publisher: Dover Publications
Year: 1959
Language: English
Pages: 568
City: New York
Tags: Advanced Calculus, Real Analysis,Differential Integral Calculus
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
