PHGN311 Homework #8
Due Friday, Oct. 25, 2013 at the beginning of class
Reading Chapter 5 on multiple integrals. We will not do this entire chapter, but will
review some key topics like Jacobian’s. Show your work on all problems. Use
Mathematica to check results but not as a solution itself unless asked to.
1. Boas 7.12.21 (Fourier transform of a Gaussian is a Gaussian)
2. Boas 7.12.24
3. Boas 8.11.15 (don’t forget integration by parts)
4. Boas 7.12.34
5. Boas 7.12.35
6. This one will take some time so start early. Use separation of variables to find the
electrostatic potential inside the 2-d box with boundary conditions as given in the
figure below (the potential is zero on every boundary except the top where it
increases linearly with x). You will need to find a Fourier Series expansion at one
point, and you may have already done one like this in your prior homework. You
don’t have to find a closed form for the solution, you can just write out the first 3
terms as long as you show the math for how to get all the terms. For this part you can
use a computer application to check your results but do your own work.
You will end up with a sine (in x) times an
exponential (in y) series solution of the type we
developed in class. Plot the solution within the
box as a function of x and y for the first five terms
of the series, for 10 terms, and for “alot” of terms,
where alot is enough to see its converging to the
correct answer. Here you can use Mathematica or
equivalent to find the terms. Make sure they match
your result above. Do a 3d plot for each. For the
“alot” of terms solution, also do a contour plot.
Make any comments you can make about the convergence of the series to the answer.
7. Boas 5.4.14
8. Boas 5.4.19