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Differential geometry with applications to mechanics and physics /

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Bibliographic Details
Main Author: Talpaert, Yves (euthor.)
Corporate Author: Taylor & Francis
Format: eBook
Language:English
Published: Boca Raton : CRC Press, Taylor & Francis Group, [2001]
Series:Monographs and textbooks in pure and applied mathematics.
Subjects:
Geometry, Differential.
Géométrie différentielle.
MATHEMATICS > Geometry > General.
Geometry, Differential
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Cover
  • Half Title
  • Title Page
  • Copyright Page
  • Dedication
  • PREFACE
  • CONTENTS
  • LECTURE 0: TOPOLOGY AND DIFFERENTIAL CALCULUS REQUIREMENTS
  • 1. TOPOLOGY
  • 1.1 TOPOLOGICAL SPACE
  • 1.2 TOPOLOGICAL SPACE BASIS
  • 1.2.1 Definition
  • 1.2.2 Example of the metric space
  • 1.2.3 Separable space
  • 1.3 HAUSSDORFF SPACE
  • 1.4 HOMEOMORPIDSM
  • 1.5 CONNECTED SPACES
  • 1.6 COMPACT SPACES
  • 1.7 PARTITION OF UNITY
  • 2. DIFFERENTIAL CALCULUS IN BANACH SPACES
  • 2.1 BANACH SPACE
  • 2.1.1 Norm and normed vector space
  • 2.1.2 Banach space
  • 2.1.3 Isomorphism of normed vector spaces
  • 2.2 DIFFERENTIAL CALCULUS IN BANACH SPACES
  • 2.2.1 Tangent mapping
  • 2.2.2. Differentiable mapping at a point
  • 2.2.3 Differentiable mapping
  • 2.2.4 Cq diffeomorphism (q≥ 1)
  • 2.2.5 Inverse mapping and implicit function theorems
  • 2.2.6 Tangent mapping
  • 2.2.7 Immersion 1 and submersion
  • 2.3 DIFFERENTIATION OF Rn INTO BANACH
  • 2.4 DIFFERENTIATION OF Rn INTO R
  • 2.4.1 Directional derivative
  • 2.4.2 Theorem of differentiation
  • 2.4.3 Linear differential forms on Rn
  • 2.5 DIFFERENTIATION OF Rn INTO Rm
  • 2.5.1 Differential and Jacobian matrix
  • 2.5.2 Image (in Rm) of a basis vector (of Rn) under dfx
  • 2.5.3 Theorems
  • 2.5.4 Diffeomorphism and Jacobian
  • 2.5.5 Inverse mapping theorem
  • 2.5.6 Implicit function theorem
  • 2.5.7 Differentiable composite mapping theorem
  • 2.5.8 Constant rank theorem
  • 2.5.9 Immersion- Submersion
  • 3. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5
  • Exercise 6.
  • Exercise 7.
  • Exercise 8.
  • Exercise 9.
  • Exercise 10.
  • LECTURE 1: MANIFOLDS
  • INTRODUCTION
  • 1. Coordinates on S2
  • 2. Stereograpbic projection
  • 1. DIFFERENTIABLE MANIFOLDS
  • 1.1 CHART AND LOCAL COORDINATES
  • 1.1.1 Chart
  • 1.1.2 Local coordinates
  • 1.2 DIFFERENTIABLE MANIFOLD STRUCTURE
  • 1.2.1 Atlas.
  • 1.2.2 Differentiable manifold structure
  • 1.2.3 Change of charts
  • 1.3 DIFFERENTIABLE MANIFOLDS
  • 1.3.1 Definitions
  • 1.3.2 Product manifold
  • 1.3.3 Examples of manifolds
  • 1.3.4 Orientable manifolds
  • 2. DIFFERENTIABLE MAPPINGS
  • 2.1 GENERALITIES ON DIFFERENTIABLE MAPPINGS
  • 2.1.1 Differentiable mapping between manifolds
  • 2.1.2 Properties of differentiable manifolds
  • 2.2 PARTICULAR DIFFERENTIABLE MAPPINGS
  • 2.2.1 Diffeomorphism and local diffeomorphism
  • 2.2.2 Immersion- Submersion- Embedding
  • 2.3 PULL-BACK OF FUNCTION
  • 2.3.1 Real-valued function on manifold
  • 2.3.2 Pull-back offunction under differentiable mapping
  • 3. SUBMANIFOLDS
  • 3.1 SUBMANIFOLDS OF Rn
  • 3.2 SUBMANIFOLD OF MANIFOLD
  • 4. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • Exercise 6.
  • Exercise 7.
  • LECTURE 2: TANGENT VECTOR SPACE
  • 1. TANGENT VECTOR
  • 1.1 TANGENT CURVES
  • 1.1.1 Curve
  • 1.1.2 "Reading" of a curve
  • 1.1.3 Tangent curves
  • 1.2 TANGENT VECTOR
  • 1.2.1 First definition of tangent vector
  • 1.2.2 Function along a curve and tangency
  • 1.2.3 Derivation in the Leibniz sense
  • 1.2.4 Second definition of a tangent vector
  • 2. TANGENT SPACE
  • 2.1 DEFINITION OF A TANGENT SPACE
  • 2.2 BASIS OF TANGENT SPACE
  • 2.3 CHANGE OF BASIS
  • 3. DIFFERENTIAL AT A POINT
  • 3.1 DEFINITIONS
  • 3.2 THE IMAGE IN LOCAL COORDINATES
  • 3.3 DIFFERENTIAL OF A FUNCTION
  • 4. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • LECTURE 3: TANGENT BUNDLE
  • VECTOR FIELD
  • ONE-PARAMETER GROUP LIE ALGEBRA
  • INTRODUCTION
  • 1. TANGENT BUNDLE
  • 1.1 NATURAL MANIFOLD TM
  • 1.2 EXTENSION AND COMMUTATIVE DIAGRAM
  • 2. VECTOR FIELD ON MANIFOLD
  • 2.1 DEFINITIONS
  • 2.2 PROPERTIES OF VECTOR FIELDS
  • 3. LIE ALGEBRA STRUCTURE
  • 3.1 BRACKET
  • 3.1.1 Vector field product
  • 3.1.2 Operation "bracket.
  • 3.1.3 Important theorem
  • 3.2 LIE ALGEBRA
  • 3.3 LIE DERIVATIVE
  • 4. ONE-PARAMETER GROUP OF DIFFEOMORPIDSMS
  • 4.1 DIFFERENTIAL EQUATIONS IN BANACH
  • 4.1.1 Integral curve
  • 4.1.2 Existence and uniqueness of solution
  • 4.1.3 Differential equation and vector field
  • 4.2 ONE-PARAMETER GROUP OF DIFFEOMORPIDSMS
  • 4.2.1 Local transformation of M
  • 4.2.2 One parameter (local) group of diffeomorpbisms
  • 4.2.3 One-parameter (global) group of diffeomorphisms
  • 4.2.4 Second order tangent bundle
  • 5. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • Exercise 6.
  • Exercise 7.
  • Exercise 8.
  • Exercise 9.
  • Exercise 10.
  • Exercise 11.
  • Exercise 12.
  • Exercise 13.
  • Exercise 14.
  • Exercise 15.
  • Exercise 16.
  • Exercise 17.
  • LECTURE 4: COTANGENT BUNDLE VECTOR BUNDLE OF TENSORS
  • 1. COTANGENT BUNDLE AND COVECTOR FIELD
  • 1.1 1-FORM
  • 1.1.1 Definition
  • 1.1.2 Expression of a 1-form
  • 1.1.3 Change of cobasis
  • 1.2 COTANGENT BUNDLE
  • 1.3 FIELD OF COVECTORS
  • 2. TENSOR ALGEBRA
  • 2.1 TENSOR AT A POINT AND TENSOR ALGEBRA
  • 2.1.1 Definition and examples
  • 2.1.2 Change of basis
  • 2.1.3 Tensor algebra
  • 2.1.4 Contraction
  • 2.2 TENSOR FIELDS AND TENSOR ALGEBRA
  • 2.2.1 Vector bundle of tensors
  • 2.2.2 Pull-back of a tensor of type (0 p)
  • 2.2.3 Covariant functor Tqp
  • 2.2.4 Tensor field and algebra
  • 3. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • Exercise 6.
  • Exercise 7.
  • Exercise 8.
  • Exercise 9.
  • Exercise 10.
  • Exercise 11.
  • Exercise 12.
  • LECTURE 5: EXTERIOR DIFFERENTIAL FORMS
  • 1. EXTERIOR FORM AT A POINT
  • 1.1 DEFINITION OF A p-FORM
  • 1.2 EXTERIOR PRODUCT OF 1-FORMS
  • 1.3 EXPRESSION OF A p-FORM
  • 1.3.1 Expression of a 2-form
  • 1.3.2 Expression of a p-form
  • 1.4 EXTERIOR PRODUCT OF FORMS
  • 1.5 EXTERIOR ALGEBRA.
  • 2. DIFFERENTIAL FORMS ON A MANIFOLD
  • 2.1 EXTERIOR ALGEBRA (GRASSMANN ALGEBRA)
  • 2.1.1 Differential form
  • 2.1.2 Algebra of exterior differential forms
  • 2.2 CHANGE OF BASIS
  • 2.2.1 Differentiable form of degree 2
  • 2.2.2 Differential form of degree p
  • 3. PULL-BACK OF A DIFFERENTIAL FORM
  • 3.1 DEFINITION AND REPRESENTATION
  • 3.2 PULL-BACK PROPERTIES
  • 4. EXTERIOR DIFFERENTIATION
  • 4.1 DEFINITION
  • 4.2 EXTERIOR DIFFERENTIAL AND PULL-BACK
  • 5. ORIENTABLE MANIFOLDS
  • 6. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • Exercise 6.
  • Exercise 7.
  • Exercise 8.
  • Exercise 9.
  • Exercise 10.
  • Exercise 11. Curl
  • Exercise 12. Divergence
  • Exercise 13.
  • Exercise 14. Curl and gradient
  • Exercise 15.
  • Exercise 16.
  • LECTURE 6: LIE DERIVATIVE-LIE GROUP
  • 1. LIE DERIVATIVE
  • 1.1 FIRST PRESENTATION OF LIE DERIVATIVE
  • 1.1.1 Lie derivative of a function
  • 1.1.2 Lie derivative of vector field
  • 1.1.3 Lie derivative of tensor fields and forms
  • 1.2 ALTERNATIVE INTERPRETATION OF LIE DERIVATIVE
  • 1.2.1 An alternative definition
  • 1.2.2 Lie derivative of vector field
  • 1.2.3 Lie derivative of covector field
  • 1.2.4 Lie derivative of tensor field of types (02)
  • 1.2.5 Lie derivative of tensors of type (0p) and (q0)
  • 1.2.6 Lie derivative of a tensor field of type (qp)
  • 1.2.7 Lie derivative of a p-form
  • 2. INNER PRODUCT AND LIE DERIVATIVE
  • 2.1 DEFINITION AND PROPERTIES
  • 2.2 FUNDAMENTAL THEOREM
  • 3. FROBENIUS THEOREM
  • 4. EXTERIOR DIFFERENTIAL SYSTEMS
  • 4.1 GENERALITIES
  • 4.2 PFAFF SYSTEMS AND FROBENIUS THEOREM
  • 5. INV ARIANCE OF TENSOR FIELDS
  • 5.1 DEFINITIONS
  • 5.2 INVARIANCE OF DIFFERENTIAL FORMS
  • 5.3 LIE ALGEBRA
  • 6. LIE GROUP AND ALGEBRA
  • 6.1 LIE GROUP DEFINITION
  • 6.2 LIE ALGEBRA OF LIE GROUP
  • 6.3 INVARIANT DIFFERENTIAL FORMS ON G.
  • 6.4 ONE-PARAMETER SUBGROUP OF A LIE GROUP
  • 7. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • Exercise 4.
  • Exercise 5.
  • Exercise 6.
  • Exercise 7.
  • Exercise 8.
  • Exercise 9.
  • Exercise l0.
  • Exercise 11.
  • Exercise 12.
  • Exercise 13.
  • Exercise 14.
  • Exercise 15.
  • Exercise 16.
  • Exercise 17.
  • Exercise 18.
  • LECTURE 7: INTEGRATION OF FORMS STOKES' THEOREM, COHOMOLOGY AND INTEGRAL INVARIANTS
  • 1. n-FORM INTEGRATION ON n-MANIFOLD
  • 1.1 INTEGRATION DEFINITION
  • 1.1.1 n-form being zero outside a compact
  • 1.1.2 Differential form of degree non M
  • 1.2 PULL-BACK OF A FORM AND INTEGRAL EVALUATION
  • 2. INTEGRAL OVER A CHAIN
  • 2.1 INTEGRAL OVER A CHAIN ELEMENT
  • 2.2 INTEGRAL OVER A CHAIN
  • 3. STOKES' THEOREM
  • 3.1 STOKES' FORMULA FOR A CLOSED p-INTERVAL
  • 3.2 STOKES' FORMULA FOR A CHIAN
  • 4. AN INTRODUCTION TO COHOMOLOGY THEORY
  • 4.1 CLOSED AND EXACT FORMS
  • COHOMOLOGY
  • 4.2 POINCARE LEMMA
  • 4.3 CYCLE
  • BOUNDARY
  • HOMOWGY
  • 5. INTEGRAL INVARIANTS
  • 5.1 ABSOLUTE INTEGRAL INVARIANT
  • 5.2 RELATIVE INTEGRAL INVARIANT
  • 6. EXERCISES
  • Exercise 1.
  • Exercise 2.
  • Exercise 3.
  • LECTURE 8: RIEMANNIAN GEOMETRY
  • 1. RIEMANNIAN MANIFOLDS
  • 1.1 METRIC TENSOR AND MANIFOLDS
  • 1.1.1 Pseudo-Riemannian and Riemannian manifolds
  • 1.1.2 Metric signature
  • 1.1.3 Scalar product
  • 1.1.4 Norm and angle
  • 1.2 CANONICAL ISOMORPHISM AND CONJUGATE TENSOR
  • 1.2.1 Canonical isomorphism existence
  • 1.2.2 Conjugate tensor
  • 1.2.3 Calculation of metric and conjugate tensors
  • 1.3 ORTHONORMAL BASES
  • 1.3.1 Orthonormal bases
  • 1.3.2 Orthogonal group
  • 1.4 HYPERBOLIC MANIFOLD AND SPECIAL RELATIVITY
  • 1.4.1 Minkowski spacetime
  • 1.4.2 Special relativity and special Lorentz transforms
  • 1.4.3 Lorentz group
  • 1.4.4 Time dilation and length contraction
  • 1.5 KILLING VECTOR FIELD
  • 1.6 VOLUME.