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An Introduction to Differential Systems and Their Characteristics
  • 仕様:カスタム縦235×横158 並製
  • 438ページ
  • ISBN978-4-88359-371-2
  • 発行日:2020/04/14

An Introduction to
Differential Systems
and
Their Characteristics

Kunio KAKIÉ

定価4,400円(本体4,000円+税)

在庫:あり

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概要

微分式系とその特性概念についての入門書。微分式系とは、外微分形式系と偏微分方程式系の総称。微分式系の解の存在を示すには、決定系の場合には現れなかった困難の克服が必要である。カルタンは、「包合性」という概念を導入し解の存在を示した。この著作の前半は、包合系の理論の詳しい解説に充てられている。後半は、微分式系の三種類の特性概念についての解説がなされている。偏微分方程式の古典理論も踏まえたうえで、「包合性」の性質及び特性概念の構造とその役割が詳しく説明されている。

目次

Preface
  • Convention and Notation
Chapter 0. Introduction
  • 1. Exterior differential systems
  • 2. Systems of partial differential equations
  • 3. Characteristics
  • 4. The characteristic module
  • 5. Prolongations
Chapter 1. Exterior Differential Systems
  • 1. Integral manifolds and integral elements
  • 2. Regular integral elements and Kähler lemma
  • 3. Involutiveness and Cartan’s test
  • 4. The Cartan-Kähler theorem
Chapter 2. Involutiveness of Symbols
  • 1. Symbols
  • 2. Involutive symbols
  • 3. Examples
Chapter 3. Systems of Partial Differential Equations
  • 1. Fibered manifolds and jet bundles
  • 2. Partial differential equations
  • 3. Reduction to a first order equation
  • 4. Involutive partial differential equations
  • 5. The Cartan-Kähler theorem for differential equations
  • 6. A prolongation theorem
  • 7. Competely integrable systems
  • 8. Illustrative examples
  • 9. Concluding Notes
Chapter 4. Exterior Differential Systems (continued)
  • 1. Exterior differential systems in a generalized sense
  • 2. The symbol of an exterior differential system
  • 3. Pfaffian systems
  • 4. Canonical systems on Grassmann bundles
  • 5. Prolongation in exterior differential systems
  • 6. Canonical Pfaffian systems on Grassmann bundles
  • 7. Supplement
Chapter 5. Contact Differential Systems
  • 1. Contact systems on jet bundles
  • 2. A canonical injection
  • 3. Contact differential systems
  • 4. Relation between two kinds of symbols
Chapter 6. The Characteristic Variety
  • 1. The characteristic covectors of a symbol
  • 2. The characteristic variety of an exterior differential system
  • 3. The characteristic variety of a partial differential equation
  • 4. Application
Chapter 7. The Characteristic Module
  • 1. Primary decompsosition of a homogeneous submodule
  • 2. An operation H acting on homogeneous submodules
  • 3. The characteristic polynomial of a graded module
  • 4. The characteristic module of a symbol
  • 5. Construction of involutive subsymbols
  • 6. The characteristic module of a differential system
Chapter 8. The Monge Characteristics
  • 1. The characteristic vectors of a symbol
  • 2. The characteristic vectors in a partial differential equation
  • 3. The Monge characteristic systems
  • 4. An illustrative example
  • 5. Application
Chapter 9. Pfaffian Systems and Their Monge Characteristics
  • 1. Pfaffian systems with independence condition
  • 2. The symbol and the characteristic variety
  • 3. The Monge characteristics in linear Pfaffian systems
  • 4. Pfaffian systems with independence condition of rank 2
  • 5. The Monge characteristics in contact differential systems
  • 6. The Monge characteristics in prolonged systems
Chapter 10. Cauchy Characteristics
  • 1. Cauchy characteristic spaces
  • 2. Cartan characteristic spaces
  • 3. Cauchy characteristics for partial differential equations
  • 4. Cauchy characteristics in Pfaffian systems
Chapter 11. Prolongation Theory
  • 1. Prolongation theorems for symbols
  • 2. The Cartan-Kuranishi prolongation theorem
  • 3. Generalized prolongation procedures
  • 4. Prolongation procedures for exterior differential systems
Appendix A. Tensor Algebra
  • 1. Tensor product and Symmetric product
  • 2. Exterior product
Appendix B. Existence Theorems for Differential Equations
  • 1. An existence theorem for partial differential equations
  • 2. An existence theorem for ordinary differential equations
Appendix C. Two Formulae in Differential Geometry
Bibliography
Index

著者紹介

Kunio KAKIÉ(かきえ くにお)

装幀