Geometric measure theory : a beginner's guide / Frank Morgan.

Format:

Book

Author/Creator:

Morgan, Frank (Professor of Mathematics, Williams College), author.

Language:

English

Subjects (All):

Geometric measure theory.

Physical Description:

1 online resource (154 p.)

Place of Publication:

San Diego, California ; London : Academic Press, Inc., 1988.

Language Note:

English

Summary:

Geometric Measure Theory

Contents:

Front Cover; Geometrie Measure Theory: A Beginner's Guide; Copyright Page; Table of Contents; Preface; CHAPTER 1. Geometric Measure Theory; 1.1. Archetypical Problem; 1.2. Surfaces as a Mappings; 1.3. The Direct Method; 1.4. Rectifiable Currents; 1.5. The Compactness Theorem; 1.6. Advantages of Rectifiable Currents; 1.7. The Regularity of Area-minimizing Rectifiable Currents; CHAPTER 2. Measures; 2.1. Definitions; 2.2. Lebesgue Measure; 2.3. Hausdorff Measure [GMT 2.10]; 2.4. Integralgeometric Measure; 2.5. Densities [GMT 2.9.12, 2.10.19]; 2.6. Approximate Limits [GMT 2.9.12]

2.7. Besicovitch Covering Theorem [GMT 2.8.15]2.8.Corollary. Hn = Ln on Rn; 2.9. Corollary; 2.10. Corollary; EXERCISES; CHAPTER 3. Lipschitz Functions and Rectifiable Sets; 3.1. Lipschitz Functions; 3.2. Rademacher's Theorem [GMT 3.1.6]; 3.3. Approximation of a Lipschitz Function by aC1 Function [GMT 3.1.15].; 3.4.Lemma (Whitney's Extention Theorem) [GMT 3.1.14]; 3.5. Proposition [GMT 2.10.11]; 3.6. Jacobians; 3.7. The Area Formula [GMT 3.2, 3]; 3.8. The Coarea Formula [GMT 3.2.11]; 3.9. Tangent Cones; 3.10. Rectifiable Sets [GMT 3.2.14]; 3.11. Proposition [cf. GMT 3.2.18, 3.2.29]

3.12. Proposition [GMT 3.2.19]3.13. General Area-coarea Formula [GMT 3.2.22]; 3.14. Product of measures [GMT 3.2.23]; 3.16. Crofton's Formula [GMT 3.2.26]; 3.17. Structure Theorem [GMT 3.3.13]; EXERCISES; CHAPTER 4. Normal and Rectifiable Currents; 4.1. Vectors and Differential Forms [GMT, Chapter 1 and 4.1]; 4.2. Currents [GMT 4.1.1, 4.1.7]; 4.3. Important Spaces of Currents [GMT 4.1.24, 4.1.22, 4.1.7, 4.1.5]; 4.4. Theorem [GMT 4.1.28]; 4.5. Normal Currents [GMT 4.1.7, 4.1.12]; 4.6. Proposition [GMT 4.1.17]; 4.7. Theorem [GMT 4.1.20]; 4.8. Theorem [GMT 4.1.23]

4.9. Constancy Theorem [GMT 4.1.31]4.10. Cartesian Products; 4.11. Slicing [GMT 4.2.1]; 4.12. Lemma [GMT 4.2.15]; EXERCISES; CHAPTER 5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces; 5.1. The Deformation Theorem [GMT 4.2.9]; 5.2. Corollary; 5.3. The Isoperimetric Inequality [GMT 4.2.10]; 5.4. The Closure Theorem [GMT 4.2.16]; 5.5. The Compactness Theorem [GMT 4.2.17]; 5.6. The Existence of Area-minimizing Surfaces; 5.7. The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds [GMT 5.1.6]; EXERCISES; CHAPTER 6. Examples of Area-Minimizing Surfaces

6.1. The Minimal Surface Equation [GMT 5.4.18]6.2. Remarks on Higher Dimensions; 6.3. Complex Analytic Varieties [GMT 5.4.19]; EXERCISES; CHAPTER 7. The Approximation Theorem; 7.1. The Approximation Theorem [GMT 4.1.24]; CHAPTER 8. Survey of Regularity Results; 8.1. Theorem; 8.2. Theorem [Fe]; 8.3. Theorem; 8.4. Boundary Regularity; 8.5. General Ambients and Other Integrands; EXERCISES; CHAPTER 9. Monotonicity and Oriented Tangent Cones; 9.1. Locally Integral Flat Chains [GMT 4.1.24, 4.3.16]; 9.2. Monotonicity of the Mass Ratio; 9.3. Theorem [GMT 5.4.3]; 9.4. Corollary; 9.5. Corollary

9.7. Oriented Tangent Cones [GMT 4.3.16]

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

Description based on print version record.

ISBN:

1-4832-7780-1