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Table of contents
Differential Geometry from First Principles
A beginner-friendly path from curves and surfaces to manifolds, curvature, and geometric thinking
Read each section in order. Every title can be opened as a TheoryTrace document.
- Cover1
- Copyright2
- How to read this book3
- Introduction4
- Chapter 1: What Differential Geometry Studies5
- Chapter 2: The Essential Linear Algebra6
- Chapter 3: The Essential Multivariable Calculus7
- Chapter 4: Curves in Euclidean Space8
- Chapter 5: Curvature and Torsion of Curves9
- Chapter 6: Surfaces as Parametrized Shapes10
- Chapter 7: The First Fundamental Form11
- Chapter 8: The Gauss Map and the Second Fundamental Form12
- Chapter 9: Gaussian and Mean Curvature13
- Chapter 10: Geodesics and Shortest Paths14
- Chapter 11: Intrinsic Geometry and Gauss’s Theorema Egregium15
- Chapter 12: Covariant Derivatives and Parallel Transport on Surfaces16
- Chapter 13: The Gauss–Bonnet Theorem17
- Chapter 14: From Surfaces to Manifolds18
- Chapter 15: Tangent Spaces and Smooth Maps19
- Chapter 16: Vector Fields, Flows, and Lie Brackets20
- Chapter 17: Differential Forms and Integration21
- Chapter 18: Stokes’s Theorem as a Unifying Principle22
- Chapter 19: Riemannian Metrics23
- Chapter 20: Connections, Geodesics, and Curvature on Manifolds24
- Chapter 21: Ricci Curvature, Scalar Curvature, and Geometric Meaning25
- Chapter 22: Classical Examples and Computations26
- Chapter 23: How Differential Geometry Connects to Modern Mathematics27
- Chapter 24: A Guided Problem-Solving Workshop28
- Conclusion29