Synopses & Reviews
Mathematical development, the author of this text observes, comes about through specific, easily understood problems that require difficult solutions and demand the use of new methods. Richard Courant employs this instructive approach in a text that balances the individuality of mathematical objects with the generality of mathematical methods.
Beginning with a discussion of Dirichlet's principle and the boundary-value problem of potential theory, the text proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Succeeding chapters explore the general problem of Douglas and conformal mapping of multiply connected domains, concluding with a survey of minimal surfaces with free boundaries and unstable minimal surfaces.
Synopsis
An examination of approaches to easy-to-understand but difficult-to-solve mathematical problems, this classic text explores the balance and tension between the individuality of mathematical objects and the generality of mathematical methods.
Synopsis
An examination of approaches to easy-to-understand but difficult-to-solve mathematical problems, this classic text begins with a discussion of Dirichlet's principle and the boundary value problem of potential theory, then proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Also explores minimal surfaces with free boundaries and unstable minimal surfaces. 1950 edition.
Synopsis
An examination of approaches to easy-to-understand but difficult-to-solve mathematical problems, this classic text begins with a discussion of Dirichlet's principle and the boundary value problem of potential theory, then proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Also explores minimal surfaces with free boundaries and unstable minimal surfaces. 1950 edition.
Synopsis
Originally published: New York: Interscience Publishers, 1950, in series: Pure and applied mathematics (Interscience Publishers); v. 3.
Table of Contents
Introduction
I. Dirichlet's Principle and the Boundary Value Problem of Potential Theory
and#160; 1. Dirichlet's Principle
and#160; 2. Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk
and#160; 3. Dirichlet's integral and quadratic functionals
and#160; 4. Further preparation
and#160; 5. Proof of Dirichlet's Principle for general domains
and#160; 6. Alternative Proof of Dirichlet's Principle
and#160; 7. Conformal mapping of simply and doubly connected domains
and#160; 8. Dirichlet's Principle for free boundary values. Natural boundary conditions
II. Conformal Mapping on Parallel-Slit Domains
and#160; 1. Introduction
and#160; 2. Solution of variational problem II
and#160; 3. Conformal mapping of plane domains on slit domains
and#160; 4. Riemann domains
and#160; 5. General Riemann domains. Uniformization
and#160; 6. Riemann domains defined by non-overlapping cells
and#160; 7. Conformal mapping of domains not of genus zero
III. Plateau's Problem
and#160; 1. Introduction
and#160; 2. Formulation and solution of basic variational problems
and#160; 3. Proof by conformal mapping that solution is a minimal surface
and#160; 4. First variation of Dirichlet's integral
and#160; 5. Additional remarks
and#160; 6. Unsolved problems
and#160; 7. First variation and method of descent
and#160; 8. Dependence of area on boundary
IV. The General Problem of Douglas
and#160; 1. Introduction
and#160; 2. Solution of variational problem for k-fold connected domains
and#160; 3. Further discussion of solution
and#160; 4. Generalization to higher topological structure
V. Conformal Mapping of Multiply Connected Domains
and#160; 1. Introduction
and#160; 2. Conformal mapping on circular domains
and#160; 3. Mapping theorems for a general class of normal domains
and#160; 4. Conformal mapping on Riemann surfaces bounded by unit circles
and#160; 5. Uniqueness theorems
and#160; 6. Supplementary remarks
and#160; 7. Existence of solution for variational problem in two dimensions
VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces
and#160; 1. Introduction
and#160; 2. Free boundaries. Preparations
and#160; 3. Minimal surfaces with partly free boundaries
and#160; 4. Minimal surfaces spanning closed manifolds
and#160; 5. Properties of the free boundary. Transversality
and#160; 6. Unstable minimal surfaces with prescribed polygonal boundaries
and#160; 7. Unstable minimal surfaces in rectifiable contours
and#160; 8. Continuity of Dirichlet's integral under transformation of r-space
Bibliography, Chapters I to VI
Appendix. Some Recent Developments in the Theory of Conformal Mapping. by M. Schiffer
and#160; 1. Green's function and boundary value problems
and#160; 2. Dirichlet integrals for harmonic functions
and#160; 3. Variation of the Green's formula
Bibliography to Appendix
Index