Page 7 - Schaum's Outline of Theory and Problems of Advanced Calculus
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vi CONTENTS
CHAPTER 5 INTEGRALS 90
Introduction of the definite integral. Measure zero. Properties of definite
integrals. Mean value theorems for integrals. Connecting integral and
differential calculus. The fundamental theorem of the calculus. General-
ization of the limits of integration. Change of variable of integration.
Integrals of elementary functions. Special methods of integration.
Improper integrals. Numerical methods for evaluating definite integrals.
Applications. Arc length. Area. Volumes of revolution.
CHAPTER 6 PARTIAL DERIVATIVES 116
Functions of two or more variables. Three-dimensional rectangular
coordinate systems. Neighborhoods. Regions. Limits. Iterated limits.
Continuity. Uniform continuity. Partial derivatives. Higher order par-
tial derivatives. Differentials. Theorems on differentials. Differentiation
of composite functions. Euler’s theorem on homogeneous functions.
Implicit functions. Jacobians. Partial derivatives using Jacobians. The-
orems on Jacobians. Transformation. Curvilinear coordinates. Mean
value theorems.
CHAPTER 7 VECTORS 150
Vectors. Geometric properties. Algebraic properties of vectors. Linear
independence and linear dependence of a set of vectors. Unit vectors.
Rectangular (orthogonal unit) vectors. Components of a vector. Dot or
scalar product. Cross or vector product. Triple products. Axiomatic
approach to vector analysis. Vector functions. Limits, continuity, and
derivatives of vector functions. Geometric interpretation of a vector
derivative. Gradient, divergence, and curl. Formulas involving r. Vec-
tor interpretation of Jacobians, Orthogonal curvilinear coordinates.
Gradient, divergence, curl, and Laplacian in orthogonal curvilinear
coordinates. Special curvilinear coordinates.
CHAPTER 8 APPLICATIONS OF PARTIAL DERIVATIVES 183
Applications to geometry. Directional derivatives. Differentiation under
the integral sign. Integration under the integral sign. Maxima and
minima. Method of Lagrange multipliers for maxima and minima.
Applications to errors.
CHAPTER 9 MULTIPLE INTEGRALS 207
Double integrals. Iterated integrals. Triple integrals. Transformations
of multiple integrals. The differential element of area in polar
coordinates, differential elements of area in cylindrical and spherical
coordinates.
