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ASTR430 Handout 1:
Problem Solving Hints
This handout is meant to give you advice to help you improve your
problem solving skills and your homework writeups. You should follow
these points for ASTR430 homeworks, and you are encouraged to employ
them in your other science classes as well. Write up neat homeworks.
Take pride in your homework writeups and do the best job that you can
on them. Take the time to solve the homework problems roughly on
scratch paper, and then copy them over neatly, filling in additional
details on your final copy.
- Show your work. Give written descriptions of what you
are doing, and why you are doing it. This is often especially
useful at the beginning of a problem where it will force you to
think about the problem physically and formulate your approach
mathematically. Descriptions will also maximize the chances that a
grader can follow what you have done in a derivation (especially if
you go off on a wild tangent!) and will help the grader to give you
constructive comments on your work. Give enough detail, and show
enough mathematical steps, that students less advanced than you
could understand your derivation!
- Check units. Any equation that you write must be
dimensionally correct. Check your equations occasionally as you go
through a derivation. It takes just a second to do so, and you can
quickly catch many common errors. Remember this general rule: in
all physically valid solutions, the arguments of all transcendental
functions (e.g. trigonometric functions, exponentials, logs,
hyperbolic functions, etc.) must be dimensionless. Taking the
cosine of something with units of mass or length makes no physical
sense.
- Check limits. Check all of your final answers and
important intermediate results to see if they behave correctly in as
many different limits as you can think of. Sometimes you will know
how a general expression should behave if a particular variable is
set to zero, infinity, or some other value. Make sure that your
general expression actually displays the expected behavior!
- Take advantage of symmetries. Symmetries are
fundamental in physics (and astronomy!). Problems can have symmetry
about a point (spherical symmetry), a line (cylindrical or axial
symmetry), or a plane (mirror symmetry). You can use symmetries in
two ways: 1) to check your final answer to a problem; or, with a
little more effort, 2) to simplify the derivation of that final
answer. As an example, time-independent central forces (like
gravity) have spherical symmetry because the magnitude of the force
depends only on the distance from the origin. In this case,
spherical symmetry means that once we find one solution (e.g. a
particular ellipse for gravity), all other possible orientations of
this solution in space are also solutions.
- Use common sense. Usually you will have some physical
insight into how the solution to a problem should look. Compare your
derived solution to a problem to what you expect from physical
insight. Trust your instincts! If a derived equation or numerical
value looks strange, go back through the derivation and look for an
error. If you can't find an error, make a note of your concerns near
your final solution so the grader can comment on them.
- Get help from others. Work on the homework problems on
your own first and get as far as you can on them. This is the best
way to improve your problem-solving skills and prepare for in-class
tests. But certainly get help from other people (other students, or
the prof) when you are stuck! By trying the problems first, you
will be able to ask more intelligent questions and better understand
the ideas of other students and/or the hints the prof might give.
- Go over homework solution sets. When you get homeworks
back, go over the solution sets and your corrected homework
together. Use the solution set to see how to get past points where
you were stuck, and make sure that you could easily do a similar
problem if given the chance, say on a midterm. Even if you get a
particular problem correct, there is always much to learn by
following through someone else's solution, particularly the prof's!
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Derek C. Richardson
2005-09-14