Chapter 1. Curves and Transformations in Three Dimensions
1.1 Curves and Parameterizations
1.2 Properties of Curves in R3
1.3 Curvature and Torsion
1.4 Helices
1.5 Plane Curves
1.6 Quaternions and Rotations in Three Dimensions
Appendix A The Catenary
Appendix B Astrobiology and Curves in R3
Appendix C Maple
Chapter 2. Introduction to Classical Riemannian Geometry
2.1 Surfaces in R3
2.2 Tangent Planes
2.3 First Fundamental Form of a Surface
2.4 Other Fundamental Forms
2.5 Manifolds in Rm
2.6 Tensors
Appendix A Maple and Matlab Programs
Chapter 3. Tensor Analysis on Riemann Manifolds
3.1 Geodesics
3.2 Examples of Geodesics
3.3 Covariant Differentiation
3.4 Riemann and Ricci Tensors
35 Parallel Transportation of Vectors
3.6Applications to General Relativity
3.7The Torus in R3
Appendix A Maple and MatLab programs
Chapter 4. Basic Topology and Analysis
4.1 Basic Notions of Topology
4.2 Basic notions from Analysis
4.3 Summary
Chapter 5. Differential Manifolds
5.1 Introduction
5.2 Charts and Atlases
5.3 Orientation
5.4 Differentiable Mappings
5.5 The Riemann Sphere-Stereographic mappings
5.6 Stereographic Atlas
5.7 Grassmann Manifolds
Chapter 6. Differentiation on Manifolds
6.1 Differentiation
6.2 Tangent Vectors
6.3 Derived Mappings
Chapter 7. Vectors and Bundles
7.1 The Tangent Bundle
7.2 Cotangent Bundle
7.3 Brief Linear Algebra Review
7.4 Cotagent Bundle on a Manifold
7.5 Vector Fields
Chapter 8. Differential forms
8.1 Exterior Products
8.2 Differential forms
8.3 Hodge Star Operator
8.4 Inverse of Poincare Lemma
8.5 Applications
8.6 Differential Geometry of Manifolds in R3
8.7 Frames in R3
8.8 Smooth Manifolds in R3
8.9 The Laplace Operator
8.10 Maxwell Equations in Free Space Appendix A Functionals over a Vector Space
Chapter 9. Integration on Manifolds in R3
9.1 Integration in one-dimensions
9.2 Volumes in R3
Chapter 10. Integration on Manifolds
10.1 Simplicies in Euclidean Space
10.2 Simplicies and Chains on Manifolds
10.3 Integration of Forms on Manifolds
10.4 Integral Theorems for Surfaces in R3
10.5 Relative Tensors and Integration on Non-Orientable Manifolds
Chapter 11. Symmetry and Lie Groups
11.1 Definition of a Group
11.2 Introduction to Symmetry
11.3 Lie Groups: (“operational definition”)
11.4 Spaces with Indefinite Metrics
11.5 Lie Algebras
11.6 Manifolds and Lie Groups
11.7 The Exponential Map
11.8 Lie transformation Groups
11.9 Applications
Biography
Mayer Humi is Professor of Mathematics at Worcester Polytechnic University. He holds a Ph.D. in Mathematical Physics and Mathematical Modeling. He is an Associate Editor of the International Journal of Differential Equations and has published over 90 journal papers. His research on the development and application of mathematical methods to atmospheric research and satellites orbits. Other research topics include mathematical physics, celestial mechanics, atmospheric flow, Lie groups, and differential equations.
