Mathematica Resource Pages
I really need to put together an updated version of my Mathematica Resource Pages (after an initial burst of energy, I kind of ran out of steam and only added material needed for my Calculus IV and Differential Equations classes). When Wolfram Research came out with version 6 of Mathematica, I really became excited (in a sad, math teacher sort of way) about doing some new work with this. So that I spend more of my time working in Mathematica rather than trying to polish up a new version of my website, I plan to post those new demos here as-is for now. No warranties are implied (so if your computer blows up while using these, I just don't want to know about it) and I haven't spent a whole lot of time polishing the demos up for general release. If something is seriously not ready for prime time yet (or has some rough edges), I'll try to mention it here. When I have time and it is possible, I will also post Adobe Flash previews for the animations, so that you can view them on the web, before downloading if you wish. All of these files are Copyright 2008 © by Marcus McGuff, but anyone is free to make use of them for educational purposes, as long as you give proper attribution. (If you figure out a way to make money off of them, then I want a cut...) Constructive comments and suggestions are welcome; feel free to let me know of any problems you encounter (and I will add those to my list of things to fix when I have time). Enjoy...
Some screencasts (by Wolfram Research) on how to do a lot of interesting things in Mathematica can be found here. A few that might be of particular interest to the people likely to read this page might be:
Computer Lab Assignments for my classes can be found here.
General purpose
- The unTutorial - If you are looking for a tutorial on how to use Mathematica, you'll have to wait until I have more time (and get motivated). However, in the meantime, here are a set of quick examples showing you which commands to use to draw which types of graph and how to do other things you might want to do in an upper level community college math course. I will expand this as I have time and think of things that need to be added. (Right now, this is mostly graphs, but I will probably add to it as I get around to it...) Mathematica notebook Warning - BIG file (12 MB). Alternative: html version (kind of ugly; eventually, I will see if there is a way to format the web pages Mathematica outputs more nicely, but you can read through this version for now if you don't want to download the big notebook or you don't have Mathematica)
- Examples for "An Introduction to Mathematica" faculty workshop - These are some examples from the workshop I gave for faculty on how to use Mathematica in the classroom. They are meant to be examples of using Mathematica for slightly more complicated demonstrations than the very simple ones I discussed in the workshop, sort of a starting point for more sophisticated demonstrations (without requiring a lot of preexisting Mathematica knowledge). Mathematica notebook
- Drawing graphs for math classes - There are two graphing "widgets" here: One will help you to graph functions with vertical and horizontal asymptotes (with the nice dashed lines indicating where they are) and the other will help you draw piecewise-defined graphs, including open and closed points at appropriate locations. Mathematica automatically computes where these asymptotes and holes are and draws them for you. These are designed for taking some of the hassle out of preparing more complicated graphs for tests and worksheets (or for homework, if you are a student). You can copy and paste the graphs from these widgets into other programs (Word, for example). Let me know if you have any requests or comments about these. (You can also interactively zoom in and pan around your graphs.) Mathematica notebook
Calculus III and IV
- Visualizing level curves and sections/slices of surfaces - A couple of fancy demonstrations (I suggest most of you not bother looking at the code behind these; just use them as demos), plus a few simpler demonstrations of how to use Mathematica to show level curves on a surface and level surfaces in 3-space. One warning - On the demos where you can enter an arbitrary expression to graph, you must use standard Mathematica notation (square brackets, capitalize built-in functions, etc.). Mathematica notebook (Flash demos and html version are coming later)
- Investigating Limits of functions of 2 variables - A couple of fancy demonstrations that help you visualize and graphically investigate the limist of several weird functions of 2 variabes as (x,y) approaches (0,0). Mathematica notebook (Flash demos and html version are coming later)
- Tangent plane - Move the tangent plane around the surface (function). Mathematica notebook (Flash demos and html version are coming later)
- Visualizing the Gradient, Divergence, and Curl - Several demos to help you visualize the Div, Grad, and Curl. I'm still working on better versions of these, but they are a start. Mathematica notebook (Flash demos and html version are coming later)
- Visualizing parametric curves - Draw an arbitrary parametric curve in 3 dimensions; view how it is drawn and make it thicker so it is easy to see. (This one needs some tweaking to be really cool, but you might find it easier to see how parametric curves work in 3-D.) Mathematica notebook
Differential Equations - These have been split out into a separate web page with several new projects. Go here for more stuff
- Slope fields - This lets you enter a differntial equation with some optional parameters and allows you to examine what effect this has on its slope field and solution curves (you can set the intial conditions by clicking on the graph with the mouse). Mathematica notebook (Flash demos and html version are coming later)
- Phase portraits - This lets you graph systems of autonomous differential equations (linear and non-linear) to see their slope field, equilibrium (critical) points, eigenvalues/eigenvectors at the equilibrium points (well, the eigenvalues/eigenvectors of the "local linearization" at the equilibrium points), and plot solution curves just by clicking on the graph with the mouse; you can also zoom in/zoom out/move the graph around with the mouse (see instructions under graph). You can also enter parameters in your equations and see what effect varying these has on everything in real time. (I think this is a pretty cool demo, actually.) This will not work if your system gets too nonlinear, but powers of x and y should be fine (trig functions, nope). There are two versions:
- Full version (requires a full copy of Mathematica to run) - allows you to enter your own system of equations to try
- Light version (can be run on the free, downloadable Mathematica Player software) - will only work with the equations I entered for you to choose from (all linear systems, competing species model, predator/prey model, Duffing Equations, maybe more) Warning: if you click on the little triangles next to the equation pick-list and try to enter your own equations directly, it will not work in the free Mathematica Player version of this file.
Calculus I and II
- Slopes of secant and tangent lines - Enter a function and move the points around to see the slope of secant and tangent lines to that curve at those points. Mathematica notebook (Flash demos and html version are coming later)
- Interactive Demonstration of the Difference Quotient - This notebook graphically demonstrates the calculation of the
difference quotient of a user defined function. Contributed by David Woods (ACC Math Department). Mathematica notebook
- Numerical integration - Visual/numerical comparison of several different methods of numerically approximating the definte integral (left sum, right sum, trapezoid rule, midpoint rule, Simpson's rule). One of the few examples I have seen that actually shows the parabolas of Simpson's rule... (One confession: The "actual area" isn't really; it is the area you get if you use Mathematica's built-in numerical integration routine. I could have had Mathematica compute the indefinite integral and then evaluate it, but I decided that might be too slow or limiting for certain integrals. It wouldn't be hard to change it, so if my conscience bothers me enough, I might do that or make it a selectable option...) Mathematica notebook (Flash demos and html version are coming later)
- Taylor series - Mathematica notebook
- Interactive Taylor Polynomial Demonstration - This notebook demonstrates the convergence of Taylor polynomials to a user selected function. The function can be chosen from the pull - down menu or can be typed in by the user. Contributed by David Woods (ACC Math Department). Mathematica notebook
- Slope fields - This lets you enter a differntial equation with some optional parameters and allows you to examine what effect this has on its slope field and solution curves (you can set the intial conditions by clicking on the graph with the mouse). Mathematica notebook (Flash demos and html version are coming later)
Precalculus / Trigonometry / College Algebra / Calculus (a mixed bag)
- Function transformations - Choose a "base graph", then use your mouse to shift it, stretch it, and reflect it; then see the equation for your result. Also, choose an equation with parameters in it and see how the graph changes as you vary those parameters. Very useful in College Algebra and Precalculus when studying graphing transformations.
- Parametric grapher - This is a "graphing calculator" type application designed to help you visualize the graphing of sets of parametric equations (in 2 dimensions). This is actually a pretty cool application; it has a lot of flexibility. You can graph 2 sets of equations simulataneously, watch how the graph depends on its parameter, look at how points of intersection depend on the parameters, and investigate families of curves (or animate them) pretty easily. Mathematica notebook (Flash demos and html version are coming later)
- Polar grapher - This is a "graphing calculator" type application designed to help you visualize the graphing of polar plots. You can show 2 graphs simulataneously, watch how the graph depends on its angle, look at how points of intersection depend on the angle, and investigate families of curves (or animate them) pretty easily. There are several things I still want to add to this, but I decided to go ahead and put it up as-is for now. Mathematica notebook (Flash demos and html version are coming later)
If you want to view the material on my old Mathematica Resource Pages site, it is still available here.
