An Introduction to Multivariable Analysis from Vector to ManifoldMultivariable analysis is an important subject for mathematicians, both pure and applied. Apart from mathematicians, we expect that physicists, mechanical engi neers, electrical engineers, systems engineers, mathematical biologists, mathemati cal economists, and statisticians engaged in multivariate analysis will find this book extremely useful. The material presented in this work is fundamental for studies in differential geometry and for analysis in N dimensions and on manifolds. It is also of interest to anyone working in the areas of general relativity, dynamical systems, fluid mechanics, electromagnetic phenomena, plasma dynamics, control theory, and optimization, to name only several. An earlier work entitled An Introduction to Analysis: from Number to Integral by Jan and Piotr Mikusinski was devoted to analyzing functions of a single variable. As indicated by the title, this present book concentrates on multivariable analysis and is completely self-contained. Our motivation and approach to this useful subject are discussed below. A careful study of analysis is difficult enough for the average student; that of multi variable analysis is an even greater challenge. Somehow the intuitions that served so well in dimension I grow weak, even useless, as one moves into the alien territory of dimension N. Worse yet, the very useful machinery of differential forms on manifolds presents particular difficulties; as one reviewer noted, it seems as though the more precisely one presents this machinery, the harder it is to understand. |
Contents
II | 1 |
III | 3 |
IV | 6 |
V | 8 |
VI | 10 |
VII | 13 |
VIII | 22 |
IX | 30 |
XXVIII | 132 |
XXIX | 135 |
XXX | 137 |
XXXI | 140 |
XXXII | 143 |
XXXIII | 147 |
XXXIV | 153 |
XXXV | 155 |
Other editions - View all
An Introduction to Multivariable Analysis from Vector to Manifold Piotr Mikusinski,Michael D. Taylor No preview available - 2002 |
An Introduction to Multivariable Analysis from Vector to Manifold Piotr Mikusinski,Michael Taylor No preview available - 2012 |
Common terms and phrases
a₁ b₁ basis brick function Cauchy sequence change-of-variables characteristic function compact set consider coordinate patch countable defined definition denote det(A det(g diffeomorphism differential forms domain dx₁ equation Example Exercises exists function f ƒ and g ƒ ƒ ƒ is integrable induced orientation integrable function inverse function theorem K-cell K-form K-manifold K-vector in RN Lebesgue integral Lemma Let f Let ƒ linear subspace linear transformation linearly independent matrix measure zero metric space natural numbers nonempty nonzero normed space one-to-one open set open subset orthogonal orthonormal parallelepiped permutation Proof Prove Theorem real numbers real-valued function RK+M scalar set of measure Show step functions subset of RN Suppose ƒ tangent vector transformation f U₁ vector space wedge product y₁
References to this book
Introduction to Hilbert Spaces with Applications Lokenath Debnath,Piotr Mikusinski Limited preview - 2005 |