Akcoglu / Bartha / Ha Analysis in Vector Spaces

1. Auflage 2011
ISBN: 978-1-118-16459-4
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 480 Seiten, E-Book

A rigorous introduction to calculus in vector spaces
The concepts and theorems of advanced calculus combined withrelated computational methods are essential to understanding nearlyall areas of quantitative science. Analysis in Vector Spacespresents the central results of this classic subject throughrigorous arguments, discussions, and examples. The book aims tocultivate not only knowledge of the major theoretical results, butalso the geometric intuition needed for both mathematicalproblem-solving and modeling in the formal sciences.
The authors begin with an outline of key concepts, terminology,and notation and also provide a basic introduction to set theory,the properties of real numbers, and a review of linear algebra. Anelegant approach to eigenvector problems and the spectral theoremsets the stage for later results on volume and integration.Subsequent chapters present the major results of differential andintegral calculus of several variables as well as the theory ofmanifolds. Additional topical coverage includes:
* Sets and functions
* Real numbers
* Vector functions
* Normed vector spaces
* First- and higher-order derivatives
* Diffeomorphisms and manifolds
* Multiple integrals
* Integration on manifolds
* Stokes' theorem
* Basic point set topology
Numerous examples and exercises are provided in each chapter toreinforce new concepts and to illustrate how results can be appliedto additional problems. Furthermore, proofs and examples arepresented in a clear style that emphasizes the underlying intuitiveideas. Counterexamples are provided throughout the book to warnagainst possible mistakes, and extensive appendices outline theconstruction of real numbers, include a fundamental result aboutdimension, and present general results about determinants.
Assuming only a fundamental understanding of linear algebra andsingle variable calculus, Analysis in Vector Spaces is anexcellent book for a second course in analysis for mathematics,physics, computer science, and engineering majors at theundergraduate and graduate levels. It also serves as a valuablereference for further study in any discipline that requires a firmunderstanding of mathematical techniques and concepts.

Autoren/Hrsg.

Akcoglu, Mustafa A.

Bartha, Paul F. A.

Ha, Dzung Minh

Inhaltsverzeichnis

Preface.
PART I BACKGROUND MATERIAL.
1 Sets and Functions.
1.1 Sets in General.
1.2 Sets of Numbers.
1.3 Functions.
2 Real Numbers.
2.1 Review of the Order Relations.
2.2 Completeness of Real Numbers.
2.3 Sequences of Real Numbers.
2.4 Subsequences.
2.5 Series of Real Numbers.
2.6 Intervals and Connected Sets.
3 Vector Functions.
3.1 Vector Spaces: The Basics.
3.2 Bilinear Functions.
3.3 Multilinear Functions.
3.4 Inner Products.
3.5 Orthogonal Projections.
3.6 Spectral Theorem.
PART II DIFFERENTIATION.
4 Normed Vector Spaces.
4.1 Preliminaries.
4.2 Convergence in Normed Spaces.
4.3 Norms of Linear and Multilinear Transformations.
4.4 Continuity in Normed Spaces.
4.5 Topology of Normed Spaces.
5 Derivatives.
5.1 Functions of a Real Variable.
5.2 Differentiable Functions.
5.3 Existence of Derivatives.
5.4 Partial Derivatives.
5.5 Rules of Differentiation.
5.6 Differentiation of Products.
6 Diffeomorphisms and Manifolds.
6.1 The Inverse Function Theorem.
6.2 Graphs.
6.3 Manifolds in Parametric Representations.
6.4 Manifolds in Implicit Representations.
6.5 Differentiation on Manifolds.
7 Higher-Order Derivatives.
7.1 Definitions.
7.2 Change of Order in Differentiation.
7.3 Sequences of Polynomials.
7.4 Local Extremal Values.
PART III INTEGRATION.
8 Multiple Integrals.
8.1 Jordan Sets and Volume.
8.2 Integrals.
8.3 Images of Jordan Sets.
8.4 Change of Variables.
9 Integration on Manifolds.
9.1 Euclidean Volumes.
9.2 Integration on Manifolds.
9.3 Oriented Manifolds.
9.4 Integrals of Vector Fields.
9.5 Integrals of Tensor Fields.
9.6 Integration on Graphs.
10 Stokes' Theorem.
10.1 Basic Stokes' Theorem.
10.2 Flows.
10.3 Flux and Change of Volume in a Flow.
10.4 Exterior Derivatives.
10.5 Regular and Almost Regular Sets.
10.6 Stokes' Theorem on Manifolds.
PART IV APPENDICES.
Appendix A: Construction of the Real Numbers.
A.1 Field and Order Axioms in Q.
A.2 Equivalence Classes of Cauchy Sequences in Q.
A.3 Completeness of R.
Appendix B: Dimension of a Vector Space.
B.1 Bases and Linearly Independent Subsets.
Appendix C: Determinants.
C.1 Permutations.
C.2 Determinants of Square Matrices.
C.3 Determinant Functions.
C.4 Determinant of a Linear Transformation.
C.5 Determinants on Cartesian Products.
C.6 Determinants in Euclidean Spaces.
C.7 Trace of an Operator.
Appendix D: Partitions of Unity.
D.1 Partitions of Unity.
Index.

Über Autor(innen)

MUSTAFA A. AKCOGLU, PhD, is Professor Emeritus in theDepartment of Mathematics at the University of Toronto, Canada. Hehas authored or coauthored over sixty journal articles on thetopics of ergodic theory, functional analysis, and harmonicanalysis.
PAUL F.A. BARTHA, PhD, is Associate Professor in theDepartment of Philosophy at The University of British Columbia,Canada. He has authored or coauthored journal articles on topicssuch as probability and symmetry, probabilistic paradoxes, and thegeneral philosophy of science.
DZUNG MINH HA, PhD, is Associate Professor in theDepartment of Mathematics at Ryerson University, Canada. Dr. Hafocuses his research in the areas of ergodic theory and operatortheory.