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Over 100 years ago Harald Bohr identified a deep problem about the
convergence of Dirichlet series, and introduced an ingenious idea
relating Dirichlet series and holomorphic functions in high
dimensions. Elaborating on this work, almost twnety years later
Bohnenblust and Hille solved the problem posed by Bohr. In recent
years there has been a substantial revival of interest in the
research area opened up by these early contributions. This involves
the intertwining of the classical work with modern functional
analysis, harmonic analysis, infinite dimensional holomorphy and
probability theory as well as analytic number theory. New
challenging research problems have crystallized and been solved in
recent decades. The goal of this book is to describe in detail some
of the key elements of this new research area to a wide audience.
The approach is based on three pillars: Dirichlet series, infinite
dimensional holomorphy and harmonic analysis.
The aim of this book is to facilitate the use of Stokes' Theorem in
applications. The text takes a differential geometric point of view
and provides for the student a bridge between pure and applied
mathematics by carefully building a formal rigorous development of
the topic and following this through to concrete applications in
two and three variables. Key topics include vectors and vector
fields, line integrals, regular k-surfaces, flux of a vector field,
orientation of a surface, differential forms, Stokes' theorem, and
divergence theorem. This book is intended for upper undergraduate
students who have completed a standard introduction to differential
and integral calculus for functions of several variables. The book
can also be useful to engineering and physics students who know how
to handle the theorems of Green, Stokes and Gauss, but would like
to explore the topic further.
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