TAILIEUCHUNG - AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY

This book is based on lectures delivered over the years by the author at the Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at City University of Hong Kong. Its two-fold aim is to give thorough introductions to the basic theorems of differential geometry and to elasticity theory in curvilinear coordinates. The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of analysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations | AN INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH APPLICATIONS TO ELASTICITY Philippe G. ClARLET City University of Hong Kong Contents Preface 5 1 Three-dimensional differential geometry 9 Introduction. 9 Curvilinear coordinates . 11 Metric tensor. 13 Volumes areas and lengths in curvilinear coordinates. 16 Covariant derivatives of a vector field. 19 Necessary conditions satisfied by the metric tensor the Riemann curvature tensor . 24 Existence of an immersion defined on an open set in R3 with a prescribed metric tensor. 25 Uniqueness up to isometries of immersions with the same metric tensor. 36 Continuity of an immersion as a function of its metric tensor . . 44 2 Differential geometry of surfaces 59 Introduction. 59 Curvilinear coordinates on a surface. 61 First fundamental form . 65 Areas and lengths on a surface . 67 Second fundamental form curvature on a surface. 69 Principal curvatures Gaussian curvature . 73 Covariant derivatives of a vector field defined on a surface the Gaufi and Weingarten formulas. 79 Necessary conditions satisfied by the first and second fundamen- tal forms the Gaufi and Codazzi-Mainardi equations Gaufi Theorema Egregium. 82 Existence of a surface with prescribed first and second fundamental forms. 85 Uniqueness up to proper isometries of surfaces with the same fundamental forms . 95 Continuity of a surface as a function of its fundamental forms . . 100

• Quản trị kinh doanh
• Differential geometry
• metric tensor
• Gaussian curvature
• Principal curvatures
• curvilinear coordinates
• shell theory
• Generating systems
• Differential invariant
• Pseudo Euclidean geometry
• Minkowski geometry
• Index p
• Kinematic Geometry of Surface Machining
• author of Kinematic Geometry of Surface Machining
• space theory
• time saving surface machining processes
• optimal machining operations
• optimal surface machining process
• geometry
• differential
• operations
• natural
• Ivan Kolar
• Peter W
• Michor
• Jan Slovak
• Handbook of mathematics for engineers
• Elementary algebra
• Elementary functions
• Elementary geometry
• Analytic geometry
• Diagnostic tests
• Basic algebra
• Differential equations
• Polar coordinates
• tangent vectors
• fundamental theorem
• frames
• higher rank
• derivatives
• manifolds
• partial derivatives
• vector functions
• the geometry of space
• infinite sequences
• further applications
• integration
• Differentiable 1!lanifolds
• Euclidean Space
• Lie Groups
• The Tangent Structure
• Hypersurfaces
• Submersion
• Averaging
• Conservation of mass
• Multiphase flow models
• Multiphase flow notation
• Size distribution functions
• cơ khí chế tạo máy
• sổ tay cơ khí
• tài liệu cơ khí
• đề thi cơ khí
• hướng dẫn cơ khí
• kỹ thuật cơ khí
• Hypercomplex Structures
• Geometric Structures
• Quaternionic Algebra
• Cohomology Rings
• Quaternionic Manifolds
• Định lí kiểu Bernstein
• Mặt cực đại hai chiều
• Không gian Minkowski R4 2
• Định thức Jacobi bị chặn
• Differential Geometry of Curves
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