Lecture |
Topics |
1 |
Introduction; review
of lines, planes, vectors, spheres |
2 |
Plane
curves; speed, arc-length, tangent and normal vectors |
3 |
Curvature;
Frenet formulas for plane curves; geometric and physical interpretations
of curvature |
4 |
Space
curves; tangent, normal, and binormal vectors; Frenet formulas for space
curves |
5 |
Total curvature
of closed plane curves; rotation index |
6 |
Convex curves; curvature
and convexity |
7 |
Isoperimetric inequality;
four vertex theorem |
8 |
Fenchel's theorem
on total curvature of space curves |
9 |
Crofton's formula
for plane curves and for curves on the sphere |
10 |
The Fary-Milnor
theorem on curvature of knots |
11 |
Introduction to
surfaces; coordinate patches; regular surfaces |
12 |
The first fundamental
form; element of area; arc-length of curves; unit normal vector to a surface |
13 |
Normal curvature
and Meusnier's Theorem; the second fundamental form |
14 |
Examples; surfaces
of revolution |
15 |
Geodesic curvature;
the Christoffel symbols |
16 |
Euler's equations
for a geodesic; examples |
17 |
More on geodescis;
proof that straight lines are shortest paths in the plane |
18 |
The Gauss map; principal
curvatures; umbilic points; lines of curvature |
19 |
Mean curvature and
Gaussian curvature; examples |
20 |
Minimal surfaces
(soap films) |
21 |
Sign of the Gaussian
curvature; elliptic, parabolic, and hyperbolic points |
22 |
Gauss's Theorema
Egregium ("Gaussian curvature is intrinsic") |
23 |
Consequences of
the Theorema Egregium; non-existence of accurate flat maps of the world;
statement of the Gauss-Bonnet theorem |
