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Stormark O. — Lie's Structural Approach to PDE Systems
Stormark O. — Lie's Structural Approach to PDE Systems

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Название: Lie's Structural Approach to PDE Systems

Автор: Stormark O.


Here is a lucid and comprehensive introduction to the differential geometric study of partial differential equations (PDE). The first book to present substantial results on local solvability of general and nonlinear PDE systems without using power series techniques, it describes a general approach to PDE systems based on ideas developed by Lie, Cartan and Vessiot. The central theme 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.

Язык: en

Рубрика: Математика/Анализ/Дифференциальные уравнения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 572

Добавлена в каталог: 16.04.2005

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Предметный указатель
1-graph      16
1-parameter subgroup      189
Anti-derived system      35 454
Arbitrariness of the general solution      60
Auxiliary variables      379 412
Backlund's theorems      168
Cartan form      382
Cartan — Kaehler theory      54
Cartan's equivalence problem      422
Cartan's local existence theorem      52
Cartan's test for involutivity      89
Cauchy characteristic pfaffian system      62
Cauchy characteristic vector field      40 60
Cauchy characteristic vector field system      60
Cauchy characteristics      65
Cauchy — Kowalewski theorem      43
Classification of 3-dimensional Lie algebras      229
Complete integral      29
Complete pfaffian system      26
Complete vector field system      26
Completion of a vector field system      132
Contact bracket      144
Contact coordinates      145
Contact form      17
Contact pfaffian system      21 173
Contact transformation      18 154
Contact transformations of order q      168
Covariant derivative      425
Darboux's method      290
Defining PDE system of a Lie pseudogroup      156
Derived system      24
Dimension of a pfaffian system      24
Dimension of a vector field system      24
Drach's classification      115
Dual pfaffian system      24
Dual vector field system      24
Equivalence problem      368 421
Equivalent Lie pseudogroups      418
Equivalent PDE systems      272
Essential invariants      403
EXP      193
Explicitly integrable vector field system      161
First fundamental theorem for Lie croups      202
First fundamental theorem for Lie pseudogroups      382
First fundamental theorem for Lie pseudogroups, second version      400
First integral      10
First integral of the first order      299
Frobenius's theorem      27 28
General involution      42
Generalized contact bracket      138
Generalized g-graphs      23
Genus (mod $\mathcal P$)      68
Hadamard's lemma      47
Hyperbolic Goursat equation      304 346
Hyperbolic PDE      121 280
In involution with      39
Inessential invariants      403
Infinitesimal right transformations      183
Integrability condition      97
Integrable vector field system      75 318
Integral curve      65
Integral manifold      17
Integral manifold of a pfaffian system      22
Integral manifold of a vector field system      24
Integrate a vector field or pfaffian system      30
Integration problem for a pfaffian system      35
Integration problem for a vector field system      36
Involution      37
Involutive genus      42
Involutive PDE system      99
Involutive structure      90
Involutive with respect to the independent variables      80
Isomorphic Lie pseudogroups      417
Isomorphic prolongation of a Lie pseudogroup      381
Isomorphic prolongation of a pseudogroup      170
Jacobian form      27
Jet bundle      20 21
Killing algebra      218
Killing vector field      214
Left multiplication      180 183
Left-invariant diifeomorphism      180
Left-invariant parameter group      180
Left-invariant vector fields      183
Lie algebra      184
Lie equation      241
Lie pseudogroup      156
Lie structure of a pfaffian system      35
Lie structure of a vector field system      xiv 36
Lie vector field associated to a Lie algebra      240
Lie vector field system      252
Lie's rectification theorem for first order PDE systems      167
Lifted equivalence problem      424
Local 1-parameter group      13
Local equivalence      368
Local foliation      27
Local Lie group      156 175
Local rectification lemma for vector fields      10
Locally isomorphic Lie groups      177
Maurer — Cartan forms      184 202
Monge characteristic subsystem      124
Monge characteristics      124
Monge's method      509
Normal prolongation of a Lie pseudogroup      413
Parabolic Goursat equation      458
Parabolic PDE      122 280
Pfaffian system      8
Polar functions      145
Principal subsystem      298
Principal vector field      288
Projective group      13
Prolongation of a contact transformation      170
Prolongation of a Lie pseudogroup      380
Prolongation of a PDE system      96
Prolongation of a pfaffian system      83
Prolongation of a vector field system      83 102
Prolongation theorem for local diffeomorphisms      173
Prolongation theorem for PDE systems      99
Prolongation theorem for pfaffian systems      107
Pseudogroup      155
Reciprocal lie algebras      197
Reciprocal Lie groups      180
Reduced characters      88
Reduced pfaffian system      65
Reduced vector field system      61
Regular involution      42
Regular vector field      40
Resolved generators of a vector field system      27
Right multiplication      180
Right-invariant diffeomorphism      180
Right-invariant parameter group      180
Right-invariant vector field      185
Second fundamental theorem for Lie groups      206
Second fundamental theorem for Lie pseudogroups      386
Simple Lie algebra      239
Simply transitive Lie algebra      246
Singular involution      42
Singular subsystem      124
Singular vector field      40
sol(S)      23
Solvable Lie algebra      239
Source projection      21
Source transformation      373
Stability algebra      398
Stability group      397
Structurally equivalent Lie algebras      222
Structurally equivalent PDE systems      370
Structure constants of a Lie parameter group      178
Symmetry group of a PDE system      276
Systatic system      401
Target projection      21
Third fundamental theorem for Lie groups      207
Third fundamental theorem for Lie pseudogroups      394
Total derivative      95
Vector field system      8
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