An Introduction to Multivariable Analysis from Vector to Manifold

Softcover Nachdruck of the original 1. Auflage 2002
ISBN: 978-1-4612-6600-6
Verlag: Birkhäuser Boston

Buch, Englisch, 295 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 470 g

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 and 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.

Zielgruppe

Research

Autoren/Hrsg.

Taylor, Michael D.

Mikusinski, Piotr

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

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.- 7.3Integrals 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.