Author | Krantz, Steven G. author |
---|---|
Title | The Geometry of Domains in Space [electronic resource] / by Steven G. Krantz, Harold R. Parks |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1574-5 |
Descript | X, 309 p. online resource |
1. Elementary Topics -- 1.1 Smooth Functions -- 1.2 The Concept of Defining Function -- 1.3 Measure Theory -- 2. Domains with Smooth Boundaries -- 2.1 The Tangent Bundle and Normal Bundle of the Boundary -- 2.2 The Second Fundamental Form and Curvature -- 2.3 Surfaces with Constant Mean Curvature -- 3. Measures -- 3.1 The Carathรฉodory Construction -- 3.2 Rectifiability -- 3.3 Minkowski Content -- 3.4 A Space-Filling Curve -- 3.5 Covering Lemmas -- 3.6 Functions of Bounded Variation -- 3.7 Domains with Finite Perimeter -- 3.8 The Area Formula -- 3.9 The Co-Area Formula -- 4. Sobolev Spaces -- 4.1 Basic Definitions and Results -- 4.2 Restriction and Trace Theorems for Sobolev Spaces -- 4.3 Domain Extension Theorems for Sobolev Spaces -- 5. Smooth Mappings -- 5.1 Sardโs Theorem -- 5.2 Extension Theorems -- 5.3 Proof of the Whitney Extension Theorem -- 5.4 Application of the Whitney Extension Theorem -- 5.5 Multidimensional Versions of the Fundamental Theorem of Calculus -- 6. Convexity -- 6.1 The Classical Notion of Convexity -- 6.2 Other Characterizations of Convexity -- 6.3 Exhaustion Functions -- 6.4 Convexity of Order k -- 7. Steiner Symmetrization -- 7.1 Basic Properties -- 7.2 The Isodiametric, Isoperimetric and Brunn-Minkowski Inequalities -- 7.3 Equality in the Isoperimetric Inequality -- 8. Topics Related to Complex Analysis -- 8.1 Quasiconformal Mappings -- 8.2 Weylโs Theorem on Eigenvalue Asymptotics of a Domain in Space -- A.1. Metrics on the Collection of Subsets of Euclidean Space -- A.2. The Constants Associated to Euclidean Space -- Guide to Notation