Preface
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