### 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/r2
• 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