An Introduction to Multivariable Analysis from Vector to Manifold - Brossura

Mikusinski, Piotr; Taylor, Michael D.

Brossura

ISBN 10: 1461266009 ISBN 13: 9781461266006

Casa editrice: Birkhäuser, 2012

Riassunto

Multivariable analysis is of interest to pure and applied mathematicians, physicists, electrical, mechanical and systems engineers, mathematical economists, biologists, and statisticians. This book takes the student/researcher on a journey through the core topics of the subject. Systematic exposition, with numerous examples and exercises from the computational to the theoretical, makes difficult ideas as concrete as possible. Good bibliography and index. May be used as a classroom text or self-study resource. Prerequisites are a one-semester undergraduate course in advanced calculus or analysis, and linear algebra.

Su questo libro

Recensione

"This is a self-contained textbook devoted to multivariable analysis based on nonstandard geometrical methods. The book can be used either as a supplement to a course on single variable analysis or as a semester-long course introducing students to manifolds and differential forms."   ―Mathematical Reviews

"The authors strongly motivate the abstract notions by a lot of intuitive examples and pictures. The exercises at the end of each section range from computational to theoretical. The book is highly recommended for undergraduate or graduate courses in multivariable analysis for students in mathematics, physics, engineering, and economics."   ―Studia Universitatis Babes-Bolyai, Series Mathematica

"All this [the description on the book's back cover] is absolutely true, but omits any statement attesting to the high quality of the writing and the high level of mathematical scholarship. So, go and order a copy of this attractively produced, and nicely composed, scholarly tome. If you're not teaching this sort of mathematics, this book will inspire you to do so."   ―MAA Reviews

Contenuti

1 Vectors and Volumes.- 1.1 Vector Spaces.- 1.2 Some Geometric Machinery for RN.- 1.3 Transformations and Linear Transformations.- 1.4 A Little Matrix Algebra.- 1.5 Oriented Volume and Determinants.- 1.6 Properties of Determinants.- 1.7 Linear Independence, Linear Subspaces, and Bases.- 1.8 Orthogonal Transformations.- 1.9 K-dimensional Volume of Parallelepipeds in RN.- 2 Metric Spaces.- 2.1 Metric Spaces.- 2.2 Open and Closed Sets.- 2.3 Convergence.- 2.4 Continuous Mappings.- 2.5 Compact Sets.- 2.6 Complete Spaces.- 2.7 Normed Spaces.- 3 Differentiation.- 3.1 Rates of Change and Derivatives as Linear Transformations.- 3.2 Some Elementary Properties of Differentiation.- 3.3 Taylor’s Theorem, the Mean Value Theorem, and Related Results.- 3.4 Norm Properties.- 3.5 The Inverse Function Theorem.- 3.6 Some Consequences of the Inverse Function Theorem.- 3.7 Lagrange Multipliers.- 4 The Lebesgue Integral.- 4.1 A Bird’s-Eye View of the Lebesgue Integral.- 4.2 Integrable Functions.- 4.3 Absolutely Integrable Functions.- 4.4 Series of Integrable Functions.- 4.5 Convergence Almost Everywhere.- 4.6 Convergence in Norm.- 4.7 Important Convergence Theorems.- 4.8 Integrals Over a Set.- 4.9 Fubini’s Theorem.- 5 Integrals on Manifolds.- 5.1 Introduction.- 5.2 The Change of Variables Formula.- 5.3 Manifolds.- 5.4 Integrals of Real-valued Functions over Manifolds.- 5.5 Volumes in RN.- 6 K-Vectors and Wedge Products.- 6.1 K-Vectors in RN and the Wedge Product.- 6.2 Properties of A.- 6.3 Wedge Product and a Characterization of Simple K-Vectors.- 6.4 The Dot Product and the Star Operator.- 7 Vector Analysis on Manifolds.- 7.1 Oriented Manifolds and Differential Forms.- 7.2 Induced Orientation, the Differential Operator, and Stokes’ Theorem; What We Can Learn from Simple Cubes.- 7.3 Integrals and Pullbacks.- 7.4 Stokes’Theorem for Chains.- 7.5 Stokes’Theorem for Oriented Manifolds.- 7.6 Applications.- 7.7 Manifolds and Differential Forms: An Intrinsic Point of View.- References.

Dettagli dell'edizione

• Editore: Birkhäuser
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 1461266009
• ISBN 13: 9781461266006
• Rilegatura: Copertina flessibile
• Numero di pagine: 308
• Contatto del produttore: non disponibile
• Persona responsabile: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3,69115 Heidelberg, Germany; Heidelberg; 69115; Germania