### Study Guide for Test 2

This test will be closed to notes/books, but a calculator will be allowed.
There will be three parts to the test.

Part 1: Fill in the blank

Part 2: Calculations and Interpretations

Part 3: Derivations

I suggest that you review your class notes and go over ASULearn solutions to
the projects. Here are the topics to focus on:
Geodesics by symmetry arguments
Geodesics by covering arguments for the cone and the flat torus
(the flat Klein bottle would be similar and may appear on the test).
We looked at equations of geodesics as well as answered questions about the
number of geodesics and intersections of geodesics.
Parametrizations of surfaces we examined in the Maple documents:
for instance [x*cos(2*Pi*y), x*sin(2*Pi*y), y] is the helicoid. We
also looked at a catenoid, sphere, cylinder, strake, plane, and cone.
Curvature for a curve on a surface:
Be able to calculate the curvature vector dT/ds, a normal to a surface |X1 x X2|, the projection of the curvature onto the normal (the normal curvature), and the geodesic curvature vector. We did this in class for the cylinder.
First fundamental form
Surface area of surfaces
Metric form
Gauss curvature of a surface

Know the following derivations/proofs:
Duplicate the limit argument in hyperbolic geometry with radius r, from
class, that if two geodesics are d units apart along the base curve and we
travel c units away from the base curve, then they are distance d exp(-c/r).
Recall that this was very useful and eventually allowed us to show that
the GK of hyperbolic space was - 1/r^{2}
As we did in class, show how
E, F, and G and the metric equation
arise from our usual definition of arc length along a curve.
You can find this in notes, or in a slightly different version as the first
page of Bob
Gardner's pages
Geodesics on a sphere must be great circles from Thursday Mar 8 (obviously
I wouldn't give you the entire proof, but any subcomponent is fair game)
A geodesic must be a constant speed curve
Derivations from test 1

Additional Review Suggestions:
Given coordinates, like [x*cos(2*Pi*y), x*sin(2*Pi*y), y], identify
the surface.
A variety of concepts applied to
the cylinder, the strake,
the cone, the sphere, the torus, and the hyperbolic model.
Be able to calculate the curvature vector dT/ds, a normal to a surface |X1 x X2|, the projection of the curvature onto the normal (the normal curvature), and the geodesic curvature vector. We did this in class for the cylinder.
Be able to geometrically argue about the curvature vector, the normal vector, and the geodesic curvature vector. [For example, for a circle on a surface, we know the curvature vector of any circle points in to the center of the circle. Combine this with intuition about the normal to a surface...]
Review the Maple worksheets we used for surfaces

1) Maple file on geodesic and normal curvatures

2) Applications of the first fundamental form
(Local Isometry: Catenoid and Helicoid, and Area: Sphere,
Strake, Hyperbolic Geometry, Cone.)

3) Maple worksheet on Gauss curvature and the sphere, helicoid, and
catenoid.

I might present some Maple commands and
output and ask you to explain what they show us.