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Practical Guide to the Invariant Calculus [Hardback]

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(University of Kent, Canterbury)
  • Formāts: Hardback, 260 pages, height x width x depth: 234x157x17 mm, weight: 550 g, Worked examples or Exercises; 5 Halftones, unspecified; 45 Line drawings, unspecified
  • Sērija : Cambridge Monographs on Applied and Computational Mathematics
  • Izdošanas datums: 29-Apr-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521857015
  • ISBN-13: 9780521857017
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  • Cena: 105,43 €
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Practical Guide to the Invariant CalculusPalielināt
  • Formāts: Hardback, 260 pages, height x width x depth: 234x157x17 mm, weight: 550 g, Worked examples or Exercises; 5 Halftones, unspecified; 45 Line drawings, unspecified
  • Sērija : Cambridge Monographs on Applied and Computational Mathematics
  • Izdošanas datums: 29-Apr-2010
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521857015
  • ISBN-13: 9780521857017
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521857017.html
Keywords:
This book explains recent results in the theory of moving frames that concern the symbolic manipulation of invariants of Lie group actions. In particular, theorems concerning the calculation of generators of algebras of differential invariants, and the relations they satisfy, are discussed in detail. The author demonstrates how these new ideas lead to significant progress in two main applications: the solution of invariant ordinary differential equations and the structure of Euler Lagrange equations and conservation laws of variational problems.

The expository language used here is primarily that of undergraduate calculus rather than of differential geometry, making the topic more accessible to a student audience. More sophisticated ideas from differential topology and Lie theory are explained from scratch, using illustrative examples and exercises. This book is ideal for graduate students and researchers working in differential equations, symbolic computation, applications of Lie groups and, to a lesser extent, differential geometry.

Recenzijas

'This book is a very good introduction to Cartan's method of moving frames, using primarily undergraduate calculus. full of good examples.' Mathematical Reviews

Papildus informācija

Moving frames explained in the language of undergraduate calculus, suitable for graduate students.
Preface ix
Introduction to invariant and equivariant problems 1(11)
The curve completion problem
1(3)
Curvature flows and the Korteweg-de Vries equation
4(1)
The essential simplicity of the main idea
5(4)
Overview of this book
9(2)
How to read this book
11(1)
1 Actions galore
12(39)
1.1 Introductory examples
12(6)
1.2 Actions
18(6)
1.2.1 Semi-direct products
23(1)
1.3 New actions from old
24(9)
1.3.1 Induced actions on functions
24(1)
1.3.2 Induced actions on products
25(1)
1.3.3 Induced actions on curves
26(1)
1.3.4 Induced action on derivatives: the prolonged action
27(4)
1.3.5 Some typical group actions in geometry and algebra
31(2)
1.4 Properties of actions
33(4)
1.5 One parameter Lie groups
37(2)
1.6 The infinitesimal vector fields
39(12)
1.6.1 The prolongation formula
44(2)
1.6.2 From infinitesimals to actions
46(5)
2 Calculus on Lie groups
51(22)
2.1 Local coordinates
51(4)
2.2 Tangent vectors on Lie groups
55(7)
2.2.1 Tangent vectors for matrix Lie groups
58(2)
2.2.2 Some standard notations for vectors and tangent maps in coordinates
60(2)
2.3 Vector fields and integral curves
62(5)
2.3.1 Integral curves in terms of the exponential of a vector field
66(1)
2.4 Tangent vectors at the identity versus one parameter subgroups
67(1)
2.5 The exponential map
68(1)
2.6 Associated concepts for transformation groups
69(4)
3 From Lie group to Lie algebra
73(41)
3.1 The Lie bracket of two vector fields on Rn
74(13)
3.1.1 Frobenius' Theorem
82(5)
3.2 The Lie algebra bracket on TeG
87(18)
3.2.1 The Lie algebra bracket for matrix Lie groups
90(5)
3.2.2 The Lie algebra bracket for transformation groups, and Lie's Three Theorems
95(10)
3.3 The Adjoint and adjoint actions for transformation groups
105(9)
4 Moving frames
114(37)
4.1 Moving frames
114(8)
4.2 Transversality and the converse to Theorem 4.1.3
122(4)
4.3 Frames for SL(2) actions
126(1)
4.4 Invariants
127(5)
4.5 Invariant differentiation
132(8)
4.5.1 Invariant differentiation for linear actions of matrix Lie groups
139(1)
4.6 Recursive construction of frames
140(8)
4.7 Joint invariants
148(3)
5 On syzygies and curvature matrices
151(34)
5.1 Computations with differential invariants
152(9)
5.1.1 Syzygies
159(2)
5.2 Curvature matrices
161(6)
5.3 Notes for symbolic computation
167(1)
5.4 The Serret-Frenet frame
168(7)
5.5 Curvature matrices for linear actions
175(5)
5.6 Curvature flows
180(5)
6 Invariant ordinary differential equations
185(21)
6.1 The symmetry group of an ordinary differential equation
187(2)
6.2 Solving invariant ordinary differential equations using moving frames
189(3)
6.3 First order ordinary differential equations
192(3)
6.4 SL(2) invariant ordinary differential equations
195(4)
6.4.1 Schwarz' Theorem
195(2)
6.4.2 The Chazy equation
197(2)
6.5 Equations with solvable symmetry groups
199(3)
6.6 Notes on symbolic and numeric computation
202(1)
6.7 Using only the infinitesimal vector fields
202(4)
7 Variational problems with symmetry
206(35)
7.1 Introduction to the Calculus of Variations
206(10)
7.1.1 Results and non-results for Lagrangians involving curvature
212(4)
7.2 Group actions on Lagrangians and Noether's First Theorem
216(6)
7.2.1 Moving frames and Noether's Theorem, the appetizer
220(2)
7.3 Calculating invariantised Euler-Lagrange equations directly
222(14)
7.3.1 The case of invariant, unconstrained independent variables
224(3)
7.3.2 The case of non-invariant independent variables
227(3)
7.3.3 The case of constrained independent variables such as arc length
230(2)
7.3.4 The `mumbo jumbo'-free rigid body
232(4)
7.4 Moving frames and Noether's Theorem, the main course
236(5)
References 241(3)
Index 244
Elizabeth Louise Mansfield is Professor of Mathematics at the University of Kent, Canterbury.