syllabus. Here is a link to university policies, and the math department policies.
Ideas for the project. It's time to start thinking about the project!
You can now get a copy of Mathematica on your own computer! Details are at http://nau.edu/CEFNS/IT/Wolfram-Mathematica/.
Monday, Nov. 18: Here is a mathematica notebook about the eigenvalue problem for the time-independent Schrodinger equation. This notebook also features sparse matrix calculations, which allow very large matrices (for example, 100,000 x 100,000).
Friday, Nov. 15: Here is a mathematica notebook about the L matrix that approximates the 1-D Laplacian.
Friday, Nov. 1:
More Desmos calculator solutions of the wave equation utt = c2 uxx: The initial conditions
are u(x,0) = f(x), ut(x,0) = g(x). The "guitar string" BCs are u(0,t) = u(1,t) = 0.
(The string is 0 < x < 1, but the solutions are shown for a larger interval.)
Note: Do not leave lots of tabs running animations. It might over-work the electrons in your computer.
wave equation f(x) = bump, g(x) = 0. (Click triangle to start animation.) f(x) can be changed.
wave equation f = 0, g(x) = sech2(x).
(Actually, you input F(x), and then g(x) = -1/2 F'(x) is the inital velocity, shown with the dotted line.)
wave equation f = 0, g(x) = x exp(-x2). Similar to the previous one.
plucked quitar string (f is an asymmetric tent, g = 0).
bowed violin string with f(x) = 0, g(x) = 4x.
The animation shows a bow pulling down on the string at x = 0.8. The string moves slowly when it sticks to the bow, then the string slips and moves up quickly.
Monday, Oct. 28: Plucked guitar string
Desmos animations using the Fourier Series and the
d'Alembert solution,
both with inital position a symmetric tent function and zero inital velocity.
Friday, Oct. 19:
As I learned today, you can graph a sum using Sigma notation in Desmos graphing calculator.
In the calculation I did in class, I made a sign error right at the end of class. This edited picture of the board shows the corrections.
You can see a nice pdf of the calculation, FCSofSine.pdf, along with the mathematica notebook that made the figures, FCSofSine.nb.
Friday, Oct. 11: Fourier Series
Here are some Desmos Calculator graphs, with sliders for the coefficients in the Fourier Series,
Fourier Sine Series,
and Fourier Cosine Series.
Here are simple Mathematica notebooks for computing the
FS,
FSS, and
FCS.
A FSS that diverges, and a FSS that appears to converge
even though f(x) is not piecewise smooth.
Wednesday, Oct. 9: Fourier Series
Here are some Mathematica notebooks, showing the Gibbs Phenomenon.
Wednesday, Oct. 2: Computer-generated contour map of the solution found on Monday to Laplace's equation
in a rectangle with u = 0 on sides, u = 1 on the bottom, and insulating top.
Computer-generated contour map of the solution to Laplace's equation in the disk with u = 1 on the top and u = -1 on the bottom.
Friday, Sept. 27: Some animated gifs showing solutions to the heat equation
(with k = 1 and L = pi) with various BCs and ICs.
Problem 2.3.3(d) (zero temp. boundaries) with time advancing slowly and
quickly.
Problem 2.4.1(a) (insulating boundaries) with time advancing
slowly and
quickly.
Problem 2.4.1(c) (insulating boundaries) with time advancing
slowly and
quickly.
Tuesday, Sept. 18: Orthogonality of Sines pdf. Proof of Equation (2.3.32).
Friday, Sept. 13:
Mathematica Notebook for plotting z = u(r, theta). PlotFunctionPolarCoordinates.nb.
Here is a scan of Friday's board with help on problem 1.5.3: prob1.5.3help.pdf.
Here are some images of the globe I had in class: Kudos to those who can figure out what year this globe was made: ColonialAfrica.jpg
and SouthEastAsia.jpg
.
Wednesday, Sept. 11: Mathematica Notebook on gradient vector fields, gradientVectorField.nb.
Friday, Sept. 6:
Equilibrium solutions with certain boundary conditions (using Desmos graphing calculator):
Problem 1.4.1(e): u(0) = T1 (set to 0), u(L) = T2, u''(x) = -1.
Problem 1.4.1(h): u'(0) - (u(0) - T) = 0, u'(L) = alpha, u''(x) = 0.
Another example with Newton's Law with cooling: k u'(0) - u(0) = 0 (k = K0/H), u(1) = T, u''(x) = 0.
Friday, August 30:
In-class project:
Suppose that a contigo thermos filled with hot coffee is placed on the desk in our classroom at t = 0.
The coffee is originally at 95 degrees C, and the room is at 20
degrees C. Considering only the heat capacity of the coffee and the air in the room,
and the conservation of energy, estimate the temperature in the room after waiting a long time.
Assume that the windows and doors of the room are closed, and that the walls are insulating.
There is a range of ``right'' answers. You will need to research
the specific heat and density of air and water/coffee. You will also need
to come up with reasonable estimates for the volume of the room and
of the thermos.
We will start with a prediction of the temperature rise, and discuss the results of the calculation. How could you control the experiment to make the change in temperature measurable? (Think about windows, doors, and people.)
Wednesday, August 28:
Formula sheet from my previous ODE final.
Paul's Notes on ODEs.
