Format:
Online-Ressource (VIII, 275p. 20 illus, digital)
ISBN:
9780817682712
Series Statement:
SpringerLink
Content:
Preface.-1 Manifolds.- 2 Lie Derivatives -- 3 Differential Forms -- 4 Integral Manifolds -- 5 Connections -- 6. Riemannian Manifolds -- 7 Lie Groups -- 8 Hamiltonian Classical Mechanics -- References.-Index.
Content:
This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics. The work’s first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics. Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.
Note:
Description based upon print version of record
,
""Differentiable Manifolds""; ""Preface""; ""Contents""; ""Chapter 1: Manifolds""; ""1.1 Differentiable Manifolds""; ""Differentiability of Maps""; ""1.2 The Tangent Space""; ""The Tangent Bundle of a Manifold""; ""1.3 Vector Fields""; ""1.4 1-Forms and Tensor Fields""; ""Tensor Fields""; ""Chapter 2: Lie Derivatives""; ""2.1 One-Parameter Groups of Transformations and Flows""; ""Integral Curves of a Vector Field""; ""Second-Order ODEs""; ""Canonical Lift of a Vector Field""; ""2.2 Lie Derivative of Functions and Vector Fields""; ""2.3 Lie Derivative of 1-Forms and Tensor Fields""
,
""Chapter 3: Differential Forms""""3.1 The Algebra of Forms""; ""3.2 The Exterior Derivative""; ""Poincaré's Lemma""; ""Chapter 4: Integral Manifolds""; ""4.1 The Recti cation Lemma""; ""4.2 Distributions and the Frobenius Theorem""; ""Involutive Distributions""; ""4.3 Symmetries and Integrating Factors""; ""Symmetries of a Second-Order Ordinary Differential Equation""; ""Chapter 5: Connections""; ""5.1 Covariant Differentiation""; ""Parallel Transport""; ""Covariant Derivative of Tensor Fields""; ""5.2 Torsion and Curvature""; ""Parallel Transport in Terms of the Tangent Bundle""
,
""5.3 The Cartan Structural Equations""""5.4 Tensor-Valued Forms and Covariant Exterior Derivative""; ""Chapter 6: Riemannian Manifolds""; ""6.1 The Metric Tensor""; ""Isometries. Killing Vector Fields""; ""Conformal Mappings""; ""6.2 The Riemannian Connection""; ""Rigid Bases""; ""Geodesics of a Riemannian Manifold""; ""6.3 Curvature of a Riemannian Manifold""; ""Ricci Tensor, Conformal, and Scalar Curvature""; ""6.4 Volume Element, Divergence, and Duality of Differential Forms""; ""Divergence of a Vector Field""; ""Duality of Differential Forms""
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""6.5 Elementary Treatment of the Geometry of Surfaces""""Chapter 7: Lie Groups""; ""7.1 Basic Concepts""; ""7.2 The Lie Algebra of the Group""; ""The Structure Constants""; ""Lie Group Homomorphisms""; ""Lie Subgroups""; ""7.3 Invariant Differential Forms""; ""The Maurer-Cartan Equations""; ""Invariant Forms on Subgroups of GL(n,R)""; ""7.4 One-Parameter Subgroups and the Exponential Map""; ""7.5 The Lie Algebra of the Right-Invariant Vector Fields""; ""7.6 Lie Groups of Transformations""; ""The Adjoint Representation""; ""Chapter 8: Hamiltonian Classical Mechanics""
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""8.1 The Cotangent Bundle""""The Fundamental 1-Form""; ""8.2 Hamiltonian Vector Fields and the Poisson Bracket""; ""The Canonical Lift of a Vector Field to the Cotangent Bundle""; ""Symplectic Manifolds""; ""8.3 The Phase Space and the Hamilton Equations""; ""Constants of Motion and Symmetries""; ""8.4 Geodesics, the Fermat Principle, and Geometrical Optics""; ""Jacobi's Principle""; ""Geometrical Optics""; ""8.5 Dynamical Symmetry Groups""; ""Lifted Actions""; ""Hidden Symmetries""; ""8.6 The Rigid Body and the Euler Equations""; ""Euler Angles""; ""Dynamics of a Rigid Body""
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""8.7 Time-Dependent Formalism""
Additional Edition:
ISBN 9780817682705
Additional Edition:
Buchausg. u.d.T. Torres del Castillo, Gerardo F., 1956 - Differentiable manifolds New York : Birkhäuser/Springer, 2012 ISBN 0817682708
Additional Edition:
ISBN 9780817682705
Language:
English
Subjects:
Mathematics
Keywords:
Differentialgeometrie
;
Mathematische Physik
;
Lie-Gruppe
;
Hamilton-Formalismus
;
Differentialgeometrie
;
Mathematische Physik
;
Lie-Gruppe
;
Hamilton-Formalismus
DOI:
10.1007/978-0-8176-8271-2
URL:
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URL:
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