Analysis in Vector Spaces : A Course in Advanced Calculus

✍ Scribed by Mustafa A. Akcoglu ; Paul F.A. Bartha ; Dzung Minh Ha.

• Publisher: Wiley
• Year: 2009
• Tongue: English
• Leaves: 479
• Category: Library

✦ Synopsis

The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples.

✦ Table of Contents

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 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. 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 Derivatives. 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. HigherOrder. 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 Local Contents on Manifolds. 9.3 Integration on Manifolds. 9.4 Surface Integrals of Vector Fields. 9.5 Geometric Content. 10. Stokes' Theorem. 10.1 Flows. 10.2 Rate of Change of Volume in Flows. 10.3 Stokes? Regions. PART: IV APPENDICES. Appendix A: Construction of 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. 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.