Doug Hamilton's Problem Solving Hints
Doug Hamilton's Problem Solving Hints
This page is meant to give you advice to help you improve your problem
solving skills and your homework writeups. I encourage you to employ
these ideas in all of your technical classes.
Draw a Picture. The best way to start
most problems is to draw a picture. Label any key variables, and try to
visualize how the system works. This very important first step allows
you to begin to build up insight into what is going on, can allow you
to visualize properties that the answer must have, and often suggests
avenues to pursue toward a solution.
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
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 the Professor can follow what you have done in
a derivation (especially if you go off on a wild tangent!) and will
help him or her 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 argument of all 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 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 and always be suspicious
of an answer! If a derived equation or numerical value looks funny,
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
for the Professor to see.
Break problems down into small chunks.
A common reason for mistakes in derivations is that you skip
steps, perhaps because you think they're easy or you don't want
to write them down. If you find yourself making frequent errors,
though, break your solution down into smaller steps. Maybe you
shouldn't shift terms from one side to another in an equation,
cancel out quantities, and divide by a common factor all at once.
Maybe you should write out your substitution in an integral
explicitly rather than incorporating it in a single step. These
hints will help reduce errors and will allow you to identify
errors more easily when they happen.
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 skill and prepare for in
class tests. But by all means get help from other people (other
students, or the Professor) 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 any hints that
the Professor might give.
Go over Homework Solution Sets. When you get
homeworks back from the Professor, 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. Someone spends a lot of
time writing up solution sets so that you can all improve you problem
solving abilities. Take advantage of the opportunity!
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