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 168: 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


Problem 6.1.5: Torus


6.2: Map Projections

Subsection 6.2.2: Azimuthal Map Projections

figure64     figure65


Subsection 6.2.3: Cylindrical Map Projections

figure66     figure67
figure68     figure69


6.3: The Gauss Map

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

figure24     figure25     figure26


Example 6.3.3: Elliptic Paraboloid

figure27     figure28     figure29


Example 6.3.4: Hyperbolic Paraboloid

figure33     figure34     figure35


Page 196: Differential of the Gauss Map

6.4: The Second Fundamental Form

Definition 6.4.2: Second Fundamental Form

figure36     figure37     figure38


Example 6.4.3: Spheres

figure39     figure40


Page 201: Osculating Paraboloid

figure43     figure44


Example 6.4.5: Torus

figure45     figure46


Example 6.4.6: Monkey Saddle

figure47     figure48     figure49


6.5: Normal and Principal Curvatures

Definition 6.5.1: Normal Curvature

figure50     figure51     figure52


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

Return to home page
Continue to Chapter 7