偏微分方程组中的李结构法

所属分类:数学  
出版时间:2004-8   出版时间:清华大学   作者:斯托尔马克   页数:572  
Tag标签:数学,Mathematics,PDE,英语  

内容概要

This book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations. It is the first book to present substantial results on local solvability of general and, in particular, nonlinear PDE systems without using power series techniques. The book describes a general approach to systems of partial differential equations based on ideas developed by Lie, Cartan and Vessiot. The most basic question is that of local solvability, but the methods used also yield classifications of various families of PDE systems. The central idea is the exploitation of singular vector field systems and their first integrals. These considerations naturally lead to local Lie groups, Lie pseudogroups and the equivalence problem, all of which are covered in detail. This book will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.

书籍目录

Preface
1
Introduction
and
summary
2
PDE
systems
,pfaffian
systems
and
vector
field
systems
2.1
ODE
systems,vector
fields
and
1-parameter
groups
2.2
first
order
PDE
systems
in
one
dependent
variable,pfaffian
equations
and
contact
transformantions
2.3
Jet
bundles
and
contact
pfaffian
systems
2.4
The
theorem
of
Frobenius
2.5
Mayer's
blowing-up
method
for
proving
3
Cartan's
local
existence
theorem
3.1
Involutions
and
characters
3.2
From
involutions
to
complete
systems
3.3
How
general
is
the
general
solutions
?
3.4
Cauchy
characteristics
3.5
Maximal
involutions
and
integrable
vector-field
systems4
Involutivity
and
the
prolongation
theorem
4.1
Independence
condition
and
involutivity
4.2
Prolongations
4.3
Explanation
of
the
prolongation
theorem5
Drach's
Classification,seceond
order
PDEs
in
one
dependent
variable,and
Monge
characteristics
5.1
The
classification
of
Drach
5.2
Second
order
PDEs
in
on
unknown
and
their
singular
vector
fields
5.3
Monge
characteristic
subsystems
6
Integration
of
vector
field
systems
satisfying
6.1
Maximal
involutions
6.2
Complete
subsystems
6.3
The
generalized
contact
bracket
6.4
Reduction
to
a
canonical
form
and
systems
of
contact
coordinates
6.5
How
to
find
all
maximal
completes
subsystems
of
6.6
Contact
transformations
and
Lie
pseudogroups
6.7
Explicitly
integrable
systems7
Higher
order
contact
transformations
8
Local
Lie
groups
9
Structual
classification
of
3-dimensional
Lie
algebras
over
the
comples
numbers
10
Lie
euqations
and
Lie
vector
field
systems
11
Second
order
PDEs
in
one
dependent
and
two
independent
variables12
Hyperbolic
PDEs
with
Monge
systems
admitting
two
or
three
first
integrals13
Classification
of
hyperbolic
Goursat
equations
14
Cartan's
theory
of
Lie
pseudogroups
15
The
equivalence
problem16
Parabolic
PDEs
and
associated
PDEs
systems
17
The
equivallence
problem
for
general
3-dimensional
pfaffian
systems
in
five
variables18
Involutive
second
order
PDE
systems
in
one
dependent
and
three
independent
variables
,solved
by
he
method
of
Monge
BiliographyIndex

图书封面

图书标签Tags

数学,Mathematics,PDE,英语


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