[AMath 383] Last lecture notes, homework, and projects
Dear class,
It's a busy day for email.
I've posted the last set of lecture notes that we'll use. Actually, we
won't even have time to work through the entire set in class. But I
strongly encourage you to read them, at least to get a feel for what is
going on. In a sense there aren't too many new concepts, but they are
presented in a different way.
On the last homework, due next Wednesday, I have decided to make part
(d) of problem number 2 optional. It is discussed in the lecture notes
I just posted, but you won't need to do that.
Of course, your projects are much more important than this homework.
You really should feel like you are basically done by this weekend, or
else you will find that your project won't get finished. Then you can
just polish up the writing. (Yes, it is a WRITING assignment, so you
are graded on your language.)
Speaking of projects, let me say a few words about my expectations.
Things that I don't especially like seeing:
(1) a lot of detailed work showing how you get to a result. For
example, don't show me the steps to nondimensionalize an equation.
Don't show me how to solve a differential equation using partial
fractions. Instead, use just a few words to explain your technique, and
give me the results. For example, in the nondimensionalization, tell me
the characteristic scales and the nondimensionalized equations. And
just let me know what the solution turned out to be. If I don't believe
you, I'll check it out.
(2) incomplete sentence. This is writing. You need to explain things
using sentences.
(3) facts without references. Don't quote me a figure that is data or
knowledge without letting me know where it comes from. If you use a
graphic from someone, give them credit. If you follow the form of
solution from another reference, give them credit. Include these in
your bibliography, and also cite them in the paper, so I know who said
what.
(4) lists. Like the one I'm writing now. Lists do have their place, and
if it is appropriate, feel free to include it in your project. But
often, lists are used because the author is too lazy to express the
thoughts in paragraph and sentence form. For example, don't create a
magic list of questions that you'll answer in the paper. Write a thesis
paragraph instead.
(5) math without interpretation. You will create a model. You will also
try to understand how the model behaves. This might be by looking for
equilibria, determining their stability, plotting actual solutions,
finding integral curves or conserved quantities, identifying parameters
where behavior changes, such as bifurcations. As you do this, you need
to remember to relate these mathematical ideas to the original problem
you were interested in. For example, let's say that you compute the
nullclines. Unless you can relate this calculation to the original
problem (such as finding equilibria), it's not helping your project. Or
if you graph a bifurcation diagram and see that an equilibrium changes
stability. Well, I'm going to want to know why that is important for
the physical system.
(6) arbitrary parameter values. If you don't know enough to have
reasonable ideas for parameter values, you may be better off leaving
them as symbols. Then you might explore the generic behaviors with a
nondimensional version. If the behavior doesn't change much as the free
parameters are adjusted, this is worth saying. If the behavior does
change when free parameters change, this is also important. I
especially don't like seeing the differential equations written down
immediately with the parameter values already substituted in. Start
with symbols, and then explain why you believe parameters are
appropriate.
That's probably enough for now.
Sincerely,
D. Brian Walton
AMath 383 Instructor
Guggenheim 408C
(206) 685-9298