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This book is a comprehensive, unifying introduction to the field of
mathematical analysis and the mathematics of computing. It develops
the relevant theory at a modern level and it directly relates
modern mathematical ideas to their diverse applications. The
authors develop the whole theory. Starting with a simple axiom
system for the real numbers, they then lay the foundations,
developing the theory, exemplifying where it's applicable, in turn
motivating further development of the theory. They progress from
sets, structures, and numbers to metric spaces, continuous
functions in metric spaces, linear normed spaces and linear
mappings; and then differential calculus and its applications, the
integral calculus, the gamma function, and linear integral
operators. They then present important aspects of approximation
theory, including numerical integration. The remaining parts of the
book are devoted to ordinary differential equations, the
discretization of operator equations, and numerical solutions of
ordinary differential equations. This textbook contains many
exercises of varying degrees of difficulty, suitable for
self-study, and at the end of each chapter the authors present more
advanced problems that shed light on interesting features, suitable
for classroom seminars or study groups. It will be valuable for
undergraduate and graduate students in mathematics, computer
science, and related fields such as engineering. This is a rich
field that has experienced enormous development in recent decades,
and the book will also act as a reference for graduate students and
practitioners who require a deeper understanding of the
methodologies, techniques, and foundations.
This book is a comprehensive, unifying introduction to the field of
mathematical analysis and the mathematics of computing. It develops
the relevant theory at a modern level and it directly relates
modern mathematical ideas to their diverse applications. The
authors develop the whole theory. Starting with a simple axiom
system for the real numbers, they then lay the foundations,
developing the theory, exemplifying where it's applicable, in turn
motivating further development of the theory. They progress from
sets, structures, and numbers to metric spaces, continuous
functions in metric spaces, linear normed spaces and linear
mappings; and then differential calculus and its applications, the
integral calculus, the gamma function, and linear integral
operators. They then present important aspects of approximation
theory, including numerical integration. The remaining parts of the
book are devoted to ordinary differential equations, the
discretization of operator equations, and numerical solutions of
ordinary differential equations. This textbook contains many
exercises of varying degrees of difficulty, suitable for
self-study, and at the end of each chapter the authors present more
advanced problems that shed light on interesting features, suitable
for classroom seminars or study groups. It will be valuable for
undergraduate and graduate students in mathematics, computer
science, and related fields such as engineering. This is a rich
field that has experienced enormous development in recent decades,
and the book will also act as a reference for graduate students and
practitioners who require a deeper understanding of the
methodologies, techniques, and foundations.
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