Chapter 6 • The First and Second Fundamental Forms




6.1: The First Fundamental Form


Example 6.1.3: The xy-Plane


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Example 6.1.4: Cylinder


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Example 6.1.5: Spheres


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Page 168: Arc Length for General Curves on Surfaces


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Example 6.1.6: Loxodromes


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Proposition 6.1.7: Area


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Example 6.1.8


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Problem 6.1.5: Torus


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6.2: Map Projections


Subsection 6.2.2: Azimuthal Map Projections


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Subsection 6.2.3: Cylindrical Map Projections


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6.3: The Gauss Map


Definition 6.3.1: Gauss Map and Example 6.3.2: Gauss Map on a Sphere


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Example 6.3.3: Elliptic Paraboloid


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Example 6.3.4: Hyperbolic Paraboloid


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Page 196: Differential of the Gauss Map






6.4: The Second Fundamental Form


Definition 6.4.2: Second Fundamental Form


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Example 6.4.3: Spheres


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Page 201: Osculating Paraboloid


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Example 6.4.5: Torus


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Example 6.4.6: Monkey Saddle


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6.5: Normal and Principal Curvatures



Definition 6.5.1: Normal Curvature



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Definition 6.5.5: Principal Curvatures and Principal Directions



p.216: Euler's Curvature Formula






6.6: Gaussian and Mean Curvature


Definition 6.6.1: Gaussian Curvature



Definition 6.6.1: Mean Curvature



Example 6.6.4: Function Graphs



Problem 6.5.12: Normal Variations (e.g., Parallel Surfaces)






6.7: Ruled Surfaces and Minimal Surfaces


Problem 6.7.11: Enneper's Surface



Problem 6.7.14: Deformations of Minimal Surfaces



Problem 6.7.17: Parallel Surfaces





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