An Introduction to Multivariable Analysis from Vector to Manifold

An Introduction to Multivariable Analysis from Vector to Manifold by Piotr Mikusinski, Michael D. Taylor.

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 multivaria...

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Bibliographic Details
Main Authors: Mikusinski, Piotr (Author), Taylor, Michael D. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2002.
Edition:1st ed. 2002.
Series:Springer eBook Collection.
Subjects:
Geometry.
Mathematical analysis.
Analysis (Mathematics).
Functions of complex variables.
Applied mathematics.
Engineering mathematics.
Differential geometry.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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.

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