Ebook: An Introduction to Multivariable Analysis from Vector to Manifold

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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The subject of multivariable analysis is of interest to pure and applied mathematicians, physicists, electrical, mechanical and systems engineers, mathematical economists, biologists, and statisticians. This introductory text provides students and researchers in the above fields with various ways of handling some of the useful but difficult concepts encountered in dealing with the machinery of multivariable analysis and differential forms on manifolds. The approach here is to make such concepts as concrete as possible.

Highlights and key features:

* systematic exposition, supported by numerous examples and exercises from the computational to the theoretical

* brief development of linear algebra in Rn

* review of the elements of metric space theory

* treatment of standard multivariable material: differentials as linear transformations, the inverse and implicit function theorems, Taylor's theorem, the change of variables for multiple integrals (the most complex proof in the book)

* Lebesgue integration introduced in concrete way rather than via measure theory

* latar chapters move beyond Rn to manifolds and analysis on manifolds, covering the wedge product, differential forms, and the generalized Stokes' theorem

* bibliography and comprehensive index

Core topics in multivariable analysis that are basic for senior undergraduates and graduate studies in differential geometry and for analysis in N dimensions and on manifolds are covered. Aside from mathematical maturity, prerequisites are a one-semester undergraduate course in advanced calculus or analysis, and linear algebra. Additionally, researchers working in the areas of dynamical systems, control theory and optimization, general relativity and electromagnetic phenomena may use the book as a self-study resource.

 

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The subject of multivariable analysis is of interest to pure and applied mathematicians, physicists, electrical, mechanical and systems engineers, mathematical economists, biologists, and statisticians. This introductory text provides students and researchers in the above fields with various ways of handling some of the useful but difficult concepts encountered in dealing with the machinery of multivariable analysis and differential forms on manifolds. The approach here is to make such concepts as concrete as possible.

Highlights and key features:

* systematic exposition, supported by numerous examples and exercises from the computational to the theoretical

* brief development of linear algebra in Rn

* review of the elements of metric space theory

* treatment of standard multivariable material: differentials as linear transformations, the inverse and implicit function theorems, Taylor's theorem, the change of variables for multiple integrals (the most complex proof in the book)

* Lebesgue integration introduced in concrete way rather than via measure theory

* latar chapters move beyond Rn to manifolds and analysis on manifolds, covering the wedge product, differential forms, and the generalized Stokes' theorem

* bibliography and comprehensive index

Core topics in multivariable analysis that are basic for senior undergraduates and graduate studies in differential geometry and for analysis in N dimensions and on manifolds are covered. Aside from mathematical maturity, prerequisites are a one-semester undergraduate course in advanced calculus or analysis, and linear algebra. Additionally, researchers working in the areas of dynamical systems, control theory and optimization, general relativity and electromagnetic phenomena may use the book as a self-study resource.

 

Content:
Front Matter....Pages i-x
Vectors and Volumes....Pages 1-41
Metric Spaces....Pages 43-73
Differentiation....Pages 75-112
The Lebesgue Integral....Pages 113-151
Integrals On Manifolds....Pages 153-188
K-Vectors and Wedge Products....Pages 189-217
Vector Analysis on Manifolds....Pages 219-290
Back Matter....Pages 291-295

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The subject of multivariable analysis is of interest to pure and applied mathematicians, physicists, electrical, mechanical and systems engineers, mathematical economists, biologists, and statisticians. This introductory text provides students and researchers in the above fields with various ways of handling some of the useful but difficult concepts encountered in dealing with the machinery of multivariable analysis and differential forms on manifolds. The approach here is to make such concepts as concrete as possible.

Highlights and key features:

* systematic exposition, supported by numerous examples and exercises from the computational to the theoretical

* brief development of linear algebra in Rn

* review of the elements of metric space theory

* treatment of standard multivariable material: differentials as linear transformations, the inverse and implicit function theorems, Taylor's theorem, the change of variables for multiple integrals (the most complex proof in the book)

* Lebesgue integration introduced in concrete way rather than via measure theory

* latar chapters move beyond Rn to manifolds and analysis on manifolds, covering the wedge product, differential forms, and the generalized Stokes' theorem

* bibliography and comprehensive index

Core topics in multivariable analysis that are basic for senior undergraduates and graduate studies in differential geometry and for analysis in N dimensions and on manifolds are covered. Aside from mathematical maturity, prerequisites are a one-semester undergraduate course in advanced calculus or analysis, and linear algebra. Additionally, researchers working in the areas of dynamical systems, control theory and optimization, general relativity and electromagnetic phenomena may use the book as a self-study resource.

 

Content:
Front Matter....Pages i-x
Vectors and Volumes....Pages 1-41
Metric Spaces....Pages 43-73
Differentiation....Pages 75-112
The Lebesgue Integral....Pages 113-151
Integrals On Manifolds....Pages 153-188
K-Vectors and Wedge Products....Pages 189-217
Vector Analysis on Manifolds....Pages 219-290
Back Matter....Pages 291-295
....