The translation of this Course was undertaken at the suggestion
of Professor W. F. Osgood, whose review of the original appeared
in the July number of the Bulletin of the American Mathematical
Society in 1903. The lack of standard texts on mathematical subjects
in the English language is too well known to require insistence.
I earnestly hope that this book will help to fill the need so generally
felt throughout the American mathematical world. It may be used
conveniently in our system of instruction as a text for a second course
in calculus, and as a book of reference it will be found valuable to
an American student throughout his work.
Few alterations have been made from the French text. Slight
changes of notation have been introduced occasionally for convenience, and several changes and .additions have been made at the suggestion of Professor Goursat, who has very kindly interested himself
in the work of translation. To him is due all the additional matter
not to be found in the French text, except the footnotes which are
signed, and even these, though not of his initiative, were always
edited by him. I take this opportunity to express my gratitude to
the author for the permission to translate the work and for the
sympathetic attitude which he has consistently assumed. I am also
indebted to Professor Osgood for counsel as the work progressed
and for aid in doubtful matters pertaining to the translation.
The publishers, Messrs. Ginn & Company, have spared no pains to
make the typography excellent. Their spirit has been far from commercial in the whole enterprise, and it is their hope, as it is mine,
that the publication of this book will contribute to the advance of
mathematics in America.
E R HENDRICK
Author(s): Édouard Goursat, Earle Raymond Hedrick and Otto Dunkel
Edition: 1
Publisher: Ginn & Company
Year: 1904
Language: English
Pages: 558
City: Boston, New York
Tags: Analysis, Differential Calculus, Integration
CONTENTS
CHAPTER 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 VARIABLE 35
I. Implicit Functions ........ 35
II. Functional Determinants ...... 52
III. Transformations ... .... 61
III. TAYLOR'S SERIES. ELEMENTARY APPLICATIONS. MAXIMA
AND MINIMA ........ 89
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 by Parts . . .166
IV. Generalizations of the Idea of an Integral. Improper
Integrals. Line Integrals ...... 175
V. Functions defined by Definite Integrals .... 192
VI. Approximate Evaluation of Definite Integrals . .196
V. INDEFINITE INTEGRALS 208
I. Integration of Rational Functions ..... 208
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
viii CONTENTS
CHAPTER PAGE
VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER
ENTIALS 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 . . 360
IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375
I. Power Series of a Single Variable . . . . . 375
II. Power Series in Several Variables ..... S94
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. SKEW CURVES 453
I. Osculating Plane ........ 453
II. Envelopes of Surfaces . . • . . . . . 459
III. Curvature and Torsion of Skew Curves .... 468
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 526
INDEX . 541
