### Test 1: Curves

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 surfaces.

**At the Exam**

You may make yourself some reference notes on the very small card I hand out.
The mini reference card must be **handwritten**.
Think of the card as a way to include some important concepts, computations, or derivations
that you aren't as comfortable with. You won't have room for much, so you should try to
internalize as much as you can.
One scientific calculator or graphing calculator allowed (but no cell phone nor other calculators bundled in combination with additional technologies). I don't see that you would need this, but I know some people like to have it with
them.
You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be stand alone--no cell phone, internet or other technological connections)
Partial credit will be given, so (if you have time) showing your reasoning or thoughts on questions you are unsure of can help your grade.

There will be three parts to the exam.

Part 1: Fill in the blank/short answer

Part 2: Calculations and Interpretations

Part 3: Short Derivations/Proofs
I suggest that you review your class notes, the
class highlights page,
and go over ASULearn
solutions to the projects, and the ASULearn Curve Glossary.

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

definitions related to any of the topics in the
glossary on curves
parametrizations, curvature or torsion of "basic" curves
such as a circle, line, plane curves y=f(x), or a helix or strake
questions similar to previous clicker questions (see the
class highlights page),
where you fill in a blank rather than answer as a
multiple choice. For instance, instead of asking a multiple choice
question, it would be fill in the blank like

-curvature T + torsion B = ____________________

(with N' as a good answer), or I could ask

N' = ________________________

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.
other questions on material from class

**Part 2: Calculations and Interpretations**
There will be some by-hand computations and interpretations,
like
Solving for the scalar curvature of a plane curve
Finding T(t), T(s) and curvature (vector and scalar)
for a curve
Finding B and tau, given T and N
Finding N, given T
Interpreting results, like recognizing that
a line is the shortest distance between two points in Euclidean geometry,
tau=0 is planar, k=0 is a line,
constant positive scalar curvature and planar is part of a circle, constant
tau/scalar curvature is a circular helix...
**Part 3: Short Derivations/Proofs**
There will be some
short proofs - the same as we've seen before. Review the
following:

For a regular curve, show that s(t) has an inverse (and showing how the mean value theorem applies).
the derivative of a unit vector is perpendicular to the original vector if neither are the 0 vector
prove that B is a unit vector
The proofs of the Frenet equations. You would be given one short
part, such as

prove T' has no component in
the B direction

prove that T' has a scalar curvature component in the N direction

or a similar part of a proof for T', N' or B'
curvature of a curve is 0 iff the curve is a line
the Darboux derivations from the homework

You should know the *results* of other statements we proved in
class, which could be asked about in the first two sections of the exam, but I won't ask
you for any other complete proofs, other than those listed here.