Mikusinski P., Taylor M.D. An Introduction to Multivariable Analysis from Vector to Manifold

Boston: Birkhäuser, 2002. - 295 p.Multivariable analysis is an important subject for mathematicians, both pure and applied. Apart from mathematicians, we expect that physicists, mechanical engineers, electrical engineers, systems engineers, mathematical biologists, mathematical 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.
The presentation of material in this book falls into roughly three parts: There is first a brief introduction to linear algebra and the elements of metric space theory; this provides our foundation for the study of multivariable analysis. The second section runs through Chapters Three, Four and part of Five, and covers standard multivariable fare in Rn: differentials as linear transformations, the inverse and implicit function theorems, Taylor's theorem, the change of variables theorem for multiple integrals, etc. The third section, starting in Chapter Five and going through Chapters Six and Seven, moves out of Rn to manifolds and analysis on manifolds, covering the wedge product, differential forms, and the generalized Stokes' theorem. The material is supported by numerous examples and exercises ranging from the computational to the theoretical, all aimed at bringing the important ideas more fully to life.True PDF