An Introduction to Multivariable Analysis from Vector to Manifold
Springer Science & Business Media, Nov 26, 2001 - Mathematics - 295 pages
Multivariable 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.
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1t follows 2-manifold 2-vector A"-dimensional A"-vector assume brick function Cauchy sequence characteristic function closed column vectors compact set consider continuous function convergence theorem coordinate patch coordinate system Corollary countable Cr function defined definition denote det(A diffeomorphism differential forms dimensional directed line segment domain equation equivalent Euclidean Example Exercises exists F1GURE function f induced orientation integrable function inverse function theorem Lebesgue integral Lemma linear subspace linear transformation linearly independent matrix measurable set measure zero metric space natural numbers nonempty nonzero normed space Note one-to-one open set open subset oriented manifold parallelepiped permutation plane positive number Prove Theorem pullback real numbers real-valued function satisfies scalar set of measure Show step functions Stokes Suppose tangent vector variables vector space volume wedge product write