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Differential geometry with applications to mechanics and physics

Author: Yves Talpaert
Publisher: Boca Raton : CRC Press, Taylor & Francis Group, [2001] ©2001
Series: Monographs and textbooks in pure and applied mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Yves Talpaert
ISBN: 9781420029840 1420029843
OCLC Number: 1034988732
Description: 1 online resource : illustrations.
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.
Series Title: Monographs and textbooks in pure and applied mathematics.
Responsibility: Yves Talpaert.

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