Chapter 1: Calculus on Euclidean Space:
Euclidean Space. Tangent Vectors. Directional Derivatives. Curves
in R3. 1-forms. Differential Forms. Mappings.
Chapter 2: Frame Fields:
Dot Product. Curves. The Frenet Formulas. ArbitrarySpeed Curves.
Covariant Derivatives. Frame Fields. Connection Forms. The
Structural Equations.
Chapter 3: Euclidean Geometry:
Isometries of R3. The Tangent Map of an Isometry. Orientation.
Euclidean Geometry. Congruence of Curves.
Chapter 4: Calculus on a Surface:
Surfaces in R3. Patch Computations. Differentiable Functions and
Tangent Vectors. Differential Forms on a Surface. Mappings of
Surfaces. Integration of Forms. Topological Properties.
Manifolds.
Chapter 5: Shape Operators:
The Shape Operator of M R3. Normal Curvature. Gaussian Curvature.
Computational Techniques. The Implicit Case. Special Curves in a
Surface. Surfaces of Revolution.
Chapter 6: Geometry of Surfaces in R3:
The Fundamental Equations. Form Computations. Some Global Theorems.
Isometries and Local Isometries. Intrinsic Geometry of Surfaces in
R3. Orthogonal Coordinates. Integration and Orientation. Total
Curvature. Congruence of Surfaces.
Chapter 7: Riemannian Geometry: Geometric Surfaces. Gaussian
Curvature. Covariant Derivative. Geodesics. Clairaut
Parametrizations. The Gauss-Bonnet Theorem. Applications of
Gauss-Bonnet.
Chapter 8: Global Structures of Surfaces: Length-Minimizing
Properties of Geodesics. Complete Surfaces. Curvature and Conjugate
Points. Covering Surfaces. Mappings that Preserve Inner Products.
Surfaces of Constant Curvature. Theorems of Bonnet and Hadamard.
Includes fully updated computer commands in line with the latest software
Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.
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