A Practical Guide to the Invariant Calculus

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 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 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.

ISBN 0521857015   Pages 260 ISBN13 9780521857017 (What's this?)   Weight (grammes) 550 Publisher Cambridge University Press   Published in Cambridge Imprint Cambridge University Press   Series title Cambridge Monographs on Applied and Computational Mathematics Format Hardback   Height (mm) 228 Publication date 29 Apr 2010   Width (mm) 152 DEWEY 515.3   Spine width (mm) 17 DEWEY edition DC22   Academic level Undergraduate

		Preface
		Introduction to invariant and equivariant problems		1
		The curve completion problem		1
		Curvature flows and the Korteweg-de Vries equation		4
		The essential simplicity of the main idea		5
		Overview of this book		9
		How to read this book		11
1		Actions galore		12
1.1		Introductory examples		12
1.2		Actions		18
1.2.1		Semi-direct products		23
1.3		New actions from old		24
1.3.1		Induced actions on functions		24
1.3.2		Induced actions on products		25
1.3.3		Induced actions on curves		26
1.3.4		Induced action on derivatives: the prolonged action		27
1.3.5		Some typical group actions in geometry and algebra		31
1.4		Properties of actions		33
1.5		One parameter Lie groups		37
1.6		The infinitesimal vector fields		39
1.6.1		The prolongation formula		44
1.6.2		From infinitesimals to actions		46
2		Calculus on Lie groups		51
2.1		Local coordinates		51
2.2		Tangent vectors on Lie groups		55
2.2.1		Tangent vectors for matrix Lie groups		58
2.2.2		Some standard notations for vectors and tangent maps in coordinates		60
		More...

"This book is full of good examples. Mansfield is clearly in love with the subject; her enthusiasm is apparent. While learning about moving frames is still not easy, this book helps, especially if the reader follows the advice of the author when she writes, "How to read this book... is with pencil and paper, and symbolic computation software. The only way to see the magic is to do it." Thomas Garrity, Mathematical Reviews

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