Page 8 - Schaum's Outline of Theory and Problems of Advanced Calculus

CONTENTS vii

CHAPTER 10 LINE INTEGRALS, SURFACE INTEGRALS, AND
INTEGRAL THEOREMS 229
Line integrals. Evaluation of line integrals for plane curves. Properties
of line integrals expressed for plane curves. Simple closed curves, simply
and multiply connected regions. Green’s theorem in the plane. Condi-
tions for a line integral to be independent of the path. Surface integrals.
The divergence theorem. Stoke’s theorem.
CHAPTER 11 INFINITE SERIES 265
Definitions of infinite series and their convergence and divergence. Fun-
damental facts concerning infinite series. Special series. Tests for con-
vergence and divergence of series of constants. Theorems on absolutely
convergent series. Infinite sequences and series of functions, uniform
convergence. Special tests for uniform convergence of series. Theorems
on uniformly convergent series. Power series. Theorems on power series.
Operations with power series. Expansion of functions in power series.
Taylor’s theorem. Some important power series. Special topics. Taylor’s
theorem (for two variables).

CHAPTER 12 IMPROPER INTEGRALS 306
Definition of an improper integral. Improper integrals of the first kind
(unbounded intervals). Convergence or divergence of improper
integrals of the first kind. Special improper integers of the first kind.
Convergence tests for improper integrals of the first kind. Improper
integrals of the second kind. Cauchy principal value. Special improper
integrals of the second kind. Convergence tests for improper integrals
of the second kind. Improper integrals of the third kind. Improper
integrals containing a parameter, uniform convergence. Special tests
for uniform convergence of integrals. Theorems on uniformly conver-
gent integrals. Evaluation of definite integrals. Laplace transforms.
Linearity. Convergence. Application. Improper multiple integrals.

CHAPTER 13 FOURIER SERIES 336
Periodic functions. Fourier series. Orthogonality conditions for the sine
and cosine functions. Dirichlet conditions. Odd and even functions.
Half range Fourier sine or cosine series. Parseval’s identity. Differentia-
tion and integration of Fourier series. Complex notation for Fourier
series. Boundary-value problems. Orthogonal functions.