Advanced Calculus is intended as a text for courses that furnish
the backbone of the student's undergraduate education in
mathematical analysis. The goal is to rigorously present the
fundamental concepts within the context of illuminating examples
and stimulating exercises. This book is self-contained and starts
with the creation of basic tools using the completeness axiom. The
continuity, differentiability, integrability, and power series
representation properties of functions of a single variable are
established. The next few chapters describe the topological and
metric properties of Euclidean space. These are the basis of a
rigorous treatment of differential calculus (including the Implicit
Function Theorem and Lagrange Multipliers) for mappings between
Euclidean spaces and integration for functions of several real
variables. Special attention has been paid to the motivation for
proofs. Selected topics, such as the Picard Existence Theorem for
differential equations, have been included in such a way that
selections may be made while preserving a fluid presentation of the
essential material. Supplemented with numerous exercises,
""Advanced Calculus"" is a perfect book for undergraduate students
of analysis.
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