Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction and summary
- 2 PDE systems, pfaffian systems and vector field systems
- 3 Cartan's local existence theorem
- 4 Involutivity and the prolongation theorem
- 5 Drach's classification, second order PDEs in one dependent variable, and Monge characteristics
- 6 Integration of vector field systems V satisfying dim V' = dim V + 1
- 7 Higher order contact transformations
- 8 Local Lie groups
- 9 Structural classification of 3-dimensional Lie algebras over the complex numbers
- 10 Lie equations and Lie vector field systems
- 11 Second order PDEs in one dependent and two independent variables
- 12 Hyperbolic PDEs with Monge systems admitting two or three first integrals
- 13 Classification of hyperbolic Goursat equations
- 14 Cartan's theory of Lie pseudogroups
- 15 The equivalence problem
- 16 Parabolic PDEs and associated PDE systems
- 17 The equivalence problem for general 3-dimensional pfaffian systems in five variables
- 18 Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
- Bibliography
- Index
Preface
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- 1 Introduction and summary
- 2 PDE systems, pfaffian systems and vector field systems
- 3 Cartan's local existence theorem
- 4 Involutivity and the prolongation theorem
- 5 Drach's classification, second order PDEs in one dependent variable, and Monge characteristics
- 6 Integration of vector field systems V satisfying dim V' = dim V + 1
- 7 Higher order contact transformations
- 8 Local Lie groups
- 9 Structural classification of 3-dimensional Lie algebras over the complex numbers
- 10 Lie equations and Lie vector field systems
- 11 Second order PDEs in one dependent and two independent variables
- 12 Hyperbolic PDEs with Monge systems admitting two or three first integrals
- 13 Classification of hyperbolic Goursat equations
- 14 Cartan's theory of Lie pseudogroups
- 15 The equivalence problem
- 16 Parabolic PDEs and associated PDE systems
- 17 The equivalence problem for general 3-dimensional pfaffian systems in five variables
- 18 Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
- Bibliography
- Index
Summary
Das Ziel der Wissenschaft ist einerseits, neue Tatsachen zu erobern, anderseits, bekannte unter hoheren Gesichtspunkten zusammenfassen.
S. LieOne of the principal objects of theoretical research is to find the point of view from which the subject appears in its greatest simplicity.
J.W. GibbsGood mathematics = down-to-earth mathematics.
A. AndreottiEveryone knows that the theory of PDE systems is enormously rich in results—but what about foundations? This monograph describes one approach, but there is no claim that it is the only one, or that it is the best possible; just that there is one, and moreover one that has been around for a long time without having been as recognized as it deserves.
The study is restricted to local solvability. If a PDE system S is defined in a domain D, and if it can be shown that S possesses local solutions at each point of D, the question of global solvability boils down to whether it is possible to glue together local solutions in order to form global ones—in analogy with the cohomology theory for coherent analytic sheaves. But first of all one has to know that there are local solutions.
A major idea is to regard PDE theory from the point of view of differential geometry—rather than basing it on analysis, say.
To illustrate this, let us first find a suitable angle from which ODE systems appear in a simple way.
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- Chapter
- Information
- Lie's Structural Approach to PDE Systems , pp. ix - xviPublisher: Cambridge University PressPrint publication year: 2000