Analysis in Vector Spaces: Course in Advanced Calculus (Hardcover)
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AUSFÜHRLICHERE BESCHREIBUNG: This text is intended for a second course in analysis, concentrating on differentiation and integration of functions of several variables. The material is presented as a unified course on vector spaces. The book successfully provides a firm mathematical foundation for further study and established an indispensable background for further studies in mathematics or related disciplines that involve heavy use of mathematical techniques and concepts. The book also provides exposure to a variety of other mathematical topics such as basic point-set topology, measure theory, differential geometry, and the theory of manifolds. INHALT: 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. BIOGRAFIEN: Akcoglu, Mustafa A.: Mustafa A. Akcoglu, PhD, is Professor Emeritus in the Department of Mathematics at the University of Toronto, Canada. He has authored or coauthored over sixty journal articles on the topics of ergodic theory, functional analysis, and harmonic analysis.Bartha, Paul F. A.: Paul F. A. Bartha, PhD, is Associate Professor in the Department of Philosophy at The University of British Columbia, Canada. He has authored or coauthored journal articles on topics such as probability and symmetry, probabilistic paradoxes, and the general philosophy of science.Ha, Dzung Minh: Dzung Minh Ha, PhD, is Associate Professor in the Department of Mathematics at Ryerson University, Canada. Dr. Ha focuses his research in the areas of ergodic theory and operator theory. Buch, gebundene Ausgabe.

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NEW HARDCOVER // // Condition: NEW. Binding/format is: Hardcover. xii, 465 p. : 9.50 H x 1.25 D x 6.25 W inches 1.70 lbs. Language: English. ISBN 9780470148242.

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