Chapter 6 • The First and Second Fundamental Forms

6.1: The First Fundamental Form

Example 6.1.3: The xy-Plane

figure1     figure2     figure3


Example 6.1.4: Cylinder

figure4     figure5


Example 6.1.5: Spheres

figure7     figure8


Page 148: Arc Length for General Curves on Surfaces

figure12     figure13     figure14


Example 6.1.6: Loxodromes

figure15     figure16     figure17


Proposition 6.1.7: Area

figure18     figure19     figure20


Example 6.1.8

figure21     figure22     figure23


6.2: The Gauss Map

Definition 6.2.1: Gauss Map and Example 6.2.2: Gauss Map on a Sphere

figure24     figure25     figure26


Example 6.2.3: Elliptic Paraboloid

figure27     figure28     figure29


Example 6.2.4: Hyperbolic Paraboloid

figure33     figure34     figure35


Page 165: Differential of the Gauss Map

6.3: The Second Fundamental Form

Definition 6.3.2: Second Fundamental Form

figure36     figure37     figure38


Example 6.3.3: Spheres

figure39     figure40


Page 169: Definition of Osculating Paraboloid

figure43     figure44


Example 6.3.5: Torus

figure45     figure46


Example 6.3.6: Monkey Saddle

figure47     figure48     figure49


6.4: Normal and Principal Curvatures

Definition 6.4.1: Normal Curvature

figure50     figure51     figure52


Definition 6.4.5: Principal Curvatures and Principal Directions

6.5: Gaussian and Mean Curvature

Definition 6.5.1: Gaussian Curvature

Definition 6.5.1: Mean Curvature

Example 6.5.4: Function Graphs

Problem 6.5.9: Normal Variations (i.e., Parallel Surfaces)

6.6: Ruled Surfaces and Minimal Surfaces

Problem 6.6.9: Enneper's Surface

Problem 6.6.11: Deformations of Minimal Surfaces

Return to home page
Continue to Chapter 7