Differential Geometry of Manifolds.
Differential Geometry of Manifolds, Second Edition presents the extension of differential geometry from curves and surfaces to manifolds in general. The book provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together classical and modern formulations. It in...
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Full Text (via Taylor & Francis) |
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| Format: | eBook |
| Language: | English |
| Published: |
Milton :
CRC Press LLC,
2019.
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| Edition: | 2nd ed. |
| Series: | Textbooks in mathematics (Boca Raton, Fla.)
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| Subjects: |
Table of Contents:
- Cover
- Half Title
- Series Page
- Title Page
- Copyright Page
- Table of Contents
- Preface
- Acknowledgments
- 1: Analysis of Multivariable Functions
- 1.1 Functions from ℝn to ℝm
- 1.2 Continuity, Limits, and Differentiability
- 1.3 Differentiation Rules
- Functions of Class Cr
- 1.4 Inverse and Implicit Function Theorems
- 2: Variable Frames
- 2.1 Frames Associated to Coordinate Systems
- 2.2 Frames Associated to Trajectories
- 2.3 Variable Frames and Matrix Functions
- 3: Differentiable Manifolds
- 3.1 Definitions and Examples.
- 3.2 Differentiable Maps between Manifolds
- 3.3 Tangent Spaces
- 3.4 The Differential of a Differentiable Map
- 3.5 Manifolds with Boundaries
- 3.6 Immersions, Submersions, and Submanifolds
- 3.7 Orientability
- 4: Multilinear Algebra
- 4.1 Hom Space and Dual
- 4.2 Bilinear Forms and Inner Products
- 4.3 Adjoint, Self-Adjoint, and Automorphisms
- 4.4 Tensor Product
- 4.5 Components of Tensors over V
- 4.6 Symmetric and Alternating Products
- 4.7 Algebra over a Field
- 5: Analysis on Manifolds
- 5.1 Vector Bundles on Manifolds
- 5.2 Vector and Tensor Fields on Manifolds.
- 5.3 Lie Bracket and Lie Derivative
- 5.4 Differential Forms
- 5.5 Pull-Backs of Covariant Tensor Fields
- 5.6 Lie Derivative of Tensor Fields
- 5.7 Integration on Manifolds
- Definition
- 5.8 Integration on Manifolds
- Applications
- 5.9 Stokes' Theorem
- 6: Introduction to Riemannian Geometry
- 6.1 Riemannian Metrics
- 6.2 Connections and Covariant Differentiation
- 6.3 Vector Fields along Curves
- Geodesics
- 6.4 Curvature Tensor
- 6.5 Ricci Curvature and Einstein Tensor
- 7: Applications of Manifolds to Physics
- 7.1 Hamiltonian Mechanics
- 7.2 Special Relativity.
- Pseudo-Riemannian Manifolds
- 7.3 Electromagnetism
- 7.4 Geometric Concepts in String Theory
- 7.5 Brief Introduction to General Relativity
- A: Point Set Topology
- A.1 Metric Spaces
- A.2 Topological Spaces
- B: Calculus of Variations
- B.1 Formulation of Several Problems
- B.2 Euler-Lagrange Equation
- B.3 Several Dependent Variables
- B.4 Isoperimetric Problems and Lagrange Multipliers
- C: Further Topics in Multilinear Algebra
- C.1 Binet-Cauchy and k-Volume of Parallelepipeds
- C.2 Volume Form Revisited
- C.3 Hodge Star Operator
- Bibliography
- Index.