### Test 2 Study Guide

It is time for our first test in order to be sure that
everyone reviews some of the fundamental concepts before we move on to
geometry of space-time and applications to general relativity

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/short answer

Part 2: Calculations and Interpretations

Part 3: Short Proofs
I suggest that you review your class notes, the
class highlights page, including clicker questions,
and go over ASULearn
solutions to the projects, and the
surfaces glossary.

**Part 1: Fill in the blank/short answer**
There will be some short answer questions, such as providing:

definitions and big picture ideas related to any of the topics in the
glossary on surfaces
(Wiki-like entries are on ASULearn for the terms), including
the parametrization, shape and geometric properties of the 9
surfaces listed there (review class notes, homework and clickers for those).
questions similar to previous clicker questions (see the
class highlights page),
where you fill in a blank rather than answer as a
multiple choice.
questions on material from class and homework solutions

Note: there is often more than one answer possible
for fill in the blank
questions: choose one response.
Full credit responses demonstrate deep understanding of differential
geometry. Informal responses are fine as long as they are correct.

**Part 2: Calculations and Interpretations**
There will be some by-hand computations and interpretations
Be able to compute Xu and Xv for a surface
Be able to calculate the curvature vector dT/ds = T'(t)/speed,
a normal to a
surface Xu x Xv,
the projection of the curvature onto the normal
(the normal curvature), and the geodesic curvature vector (what is left
over) for a curve on the plane or the cylinder (where the computations
are quicker than they would be on other surfaces).
Be able to interpret whether curves are geodesics via a given
geodesic curvature as well as geometric/physical
arguments relating to whether
the curvature vector is completely in the normal direction or not.
[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 to say whether the curvature vector is
parallel to the normal and hence gives a geodesic, or not]
Be able to compute E, F and G and interpret whether the Pythagorean
theorem holds or not, or whether Xu and Xv are perpendicular.
**Part 3: Short Proofs**
There will be some
short proofs - the same as we've seen before. Review the
following:

Prove how E, F, and G and the metric equation arise from our usual
definition of arc length along a curve, as on
class slides
Prove that a geodesic must be a constant speed curve as
on class slides
Prove that the determinant of the metric form gives the area
of a flat Xu, Xv parallelogram as on
class slides
Assume that gamma and gamma'' have already shown to be parallel
for a geodesic parametrized by arc length on the sphere. Prove that
gamma had to be a great circle, as on the last bullet point of
the last class slide
In hyperbolic geometry models with radius r as the interior radius
of the annuli of width delta, with delta approaching 0, prove that if two
geodesics are d units apart along the base curve and we travel c units
away from the base curve along each geodesic, then they are at a distance
of d exp(-c/r).
We filled in the details, after starting from
here