Computer Algebra Recipes

5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 231
5.2.3 Benny's Solution
A good scientist is a person with original ideas. A good engineer is
a person who makes a design that works with as few original ideas
as possible. There are no prima donnas in engineering.
Freeman Dyson, British-born U.S. physicist (1923{)
Greg Arious Nerd is currently teaching the mathematical physics course to a
mixture of future engineers and physicists at Erehwon's most famous academic
institution, EIT (Erehwon Institute of Technology). The students are being
instructed in the use of the Elpam computer algebra system in solving their
mathematical physics problems. As a classroom example, Greg selects a somewhat
arti¯cial, but pedagogically useful, two-dimensional static potential problem.
A circular annulus has an angular potential distribution Á(10;μ)=15cosμ
speci¯ed on the inner radius r1 =10cmandapotentialÁ(20;μ)=30sinμ on
the outer radius r2 = 20 cm. The question to be answered is, \What is the
potential distribution in the annular region, and what do the equipotentials
look like in this region?"
The following recipe for solving this problem has been submitted by one of
Greg's engineering students, Benjamin Beetlebrox III. Although, as a descendent
of one of the founding families on Erehwon, he doesn't like his ¯rst name
shortened, we shall take Freeman Dyson's words to heart and call him Benny.
In addition to the plots package needed for plotting the equipotentials,
Benny loads the VectorCalculus package, because it contains the Laplacian
command, which will enable him to easily generate Laplace's equation for the
potential Á(r; μ) in polar coordinates. The radial (r) and angular (μ) polarcoordinates
are related to the Cartesian coordinates (x; y) through the relations
x = r cos μ, y = r sin μ.
> restart: with(plots): with(VectorCalculus):
Curious about the coordinate systems that the Elpam system supports, Benny
enters the following command line. On executing this line, a list of the available
two- and three-dimensional coordinate systems appears in a Help page.
> ?coords;
Through the hyperlinks at the bottom of the Help page, Benny is led to the
coordplot and coordplot3d commands for plotting representative curves and
surfaces in two and three dimensions, corresponding to holding each coordinate
equal to a constant value. Closing the Help window, Benny uses coordplot
to plot the lines r =constantandμ = constant in polar coordinates. The
grid option is used to control the number of constant values and therefore
the number of lines drawn. The default is grid=[12,12]. The values of the
constants are added to the graph by including labeling=true. More detailed
explanations and other options may be found on the coordplot Help page.
> coordplot(polar,grid=[5,7],labelling=true,
scaling=constrained);