### 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.

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,
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
(Wiki-like entries are on ASULearn)
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 on -curvature T + torsion B, it could be fill in the blank (with N'
as a good answer), or I could ask what N'=?.
questions on material from class

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,
like
Solving for the scalar curvature of a plane curve using the formula we
derived for y=f(x) plane curves: (|f''|/|(1+f'^2)^(3/2))|
Finding T(t), T(s) and curvature (vector and scalar)
for a curve that is not of the form y=f(x).
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 postive scalar curvature and planar is part of a circle, constant
tau/scalar curvature is a circular helix...
**Part 3: Short Proofs**
There will be some
short proofs - the same as we've seen before. Review the
following:

prove
the
derivative of a unit vector is perpendicular to the original vector if it
is not the 0 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, but I won't ask
you for any other complete proofs, other than those listed here.