Vector Analysis with Mathematica and Java
At the University of Minnesota we have a Multivariable Calculus and Vector Analysis course which makes heavy use of technology. Students spend two hours per week working in a computer lab using Mathematica. Occasionally we get requests from other instructors who would like to use our material, so I'm trying to collect everything in one place for easy access. See below for the history of this material, as well as legal information about using it in your own courses.
Fall 2004 Update: After changing textbooks in our course, we've had to change the labs (again). This page has the older versions; if you're interested, you can see the reorganized labs at the course page.
The materials here are organized into three categories:
- Interactive Examples and Demonstrations
- Mathematica Notebooks for Lab Sessions
- History and Usage Information
Most of my java-based demonstrations and examples were created for use in class, whether in lecture or a lab session. They generally include a brief explanation of what you're looking at, but the focus is on the example. Most of Duane Nykamp's examples, on the other hand, are part of course readings that he has developed for students to read outside of class. Here are a few pictures, followed by a more comprehensive list. The list is generally out of date, so if you have any requests, please let me know; we may have already created it.
These materials all require Java, which is installed with most modern web browsers. If you have trouble viewing any of these pages, please let me know.
- Interactive Gallery of Quadric Surfaces
- Parametrized Surface: A Torus
- Animated normal vector on the cone -- is this a smooth surface?
- Continuously Varying Normal Vectors on a Paraboloid
- Geometric Interpretation of Partial Derivatives
- Directional Derivative Example
- Tangent Plane with Vectors
- Estimating Double Integrals
- Change of Variables: Polar to Rectangular Coordinates
- Change of Variables: A Nonlinear Transformation
- Stokes's Theorem: Infinitely many Surfaces with the same Boundary
- Lagrange Multipliers; this has no accompanying text, but if you know how Lagrange multipliers work you can probably figure out the images!
- Animated Position and Velocity vectors (This is adapted by Duane Nykamp from a mathematica lab I wrote.)
- Particles in Motion: Parametric Curves with Velocity Vectors
- 2D Curvature Examples: parabola; cubic; quartic; cusp; all (large file)
- 3D Curvature Examples: helix; elliptical helix; tornado; exponential spiral; twisted cubic; all (large file)
Nykamp:See Nykamp's course readings for Math 2374 for additional material (including some interactive examples I may have missed). In a few cases he and I have accidentally duplicated each other's work.
- Parametrization of a line
- Cross Product
- Triple Scalar Product
- Determining and Parametrizing a plane
- Translating, rescaling, and reflecting surfaces
- Level Curves
- Directional Derivative and Gradient
- Derivative of a path
- Scalar Path Integrals
- Path Integrals of Vector Fields
- Divergence and Curl
- Surfaces of Revolution
- Spherical Coordinates
- Parametric Surfaces (Helicoid)
- A Nonorientable Surface (Moebius Strip)
- The Idea of Stokes's Theorem
- Change of Variables: Triple Integrals
I've incorporated LiveGraphics3D into Mathematica for use in our labs. This allows students to create three-dimensional graphics which students can rotate and scale using the mouse. In order for you to use these commands, you'll have to download the math2374.nb file. Before using any of our labs, open this notebook and click on the button to define our customized commands and load various packages.
I've also written a document for our students explaining how to install LiveGraphics3D so it will work with Mathematica. You should follow these instructions if you want our custom commands to work.
The numbering scheme is based on the particular structure of our course. It is explained in the first lab, if you're interested. Occasionally the later labs refer to the earlier ones, so if you'd like to use these in a different order you'll have to sort out the cross-references.
I've included the "real" labs here -- the mathematica notebooks -- and web versions you can view without Mathematica. I wouldn't recommend using the web versions in class; these labs are meant to be interactive.
- Introduction to Mathematica notebook; web version
- Graphing Surfaces: notebook; web version
- Parametrizing Curves: notebook; web version
- Parametrizing Surfaces: notebook; web version
- Continuity and Differetiability: notebook; web version
- Directional Derivatives, Gradients,and Vector Fields: notebook; web version
- Tangent Planes: notebook; web version
- Arclength and Line Integrals of Scalar Functions: notebook; web version
- Line Integrals of Vector Fields: notebook; web version
- Applications of Integration: notebook; web version
- Surface Integrals of Scalar Functions: notebook; web version
- Surface Integrals of Vector Fields: notebook; web version
- Stokes's Theorem: notebook; web version
- Divergence Theorem: notebook; web version
History and Details
This material is from our Math 2374 course. The labs were originally written by Cindy Kaus, who is now at Metropolitan State University. I first taught the course in 2001, and eventually helped revamp the labs. James Swenson and Dan Drake were also a part of that project. Since then Duane Nykamp has also created a number of excellent online examples.
The computer based assignments are all Mathematica notebooks, but we've added special commands, such as Plot3DLive, which works like Plot3D except the output appears in a new window, where you can rotate the surface and view it from every angle. These commands use a nifty little Java applet called LiveGraphics3D. If you'd like to use our notebooks, make sure to read the tips about getting these commands to work on your system.
The html versions of the labs show unevaluated copies of the notebooks, unless otherwise noted. As time permits, I'll try to go back and post copies of the evaluated notebooks, where the output from the Live commands will be embedded as Java applets in the web page.
The interactive examples also use LiveGraphics3D and should work on any Java-enabled web browser. This includes nearly every browser, but if you have trouble, ask your system administrator; if you're using a home computer, you can download it at the Java website.
Unfortunately it seems necessary to protect this material. All of the materials here are copyrighted by their respective authors. We've all agreed to license them according to the Creative Commons Attribution-NoDerivs-NonCommercial License License. This means individuals are free to use it, and instructors can use it in their classroom, but you can't make and distribute a modified version without permission from us. (We're using the "no derivative clause" just to help us keep track of what's being used; some of the materials are undergoing revision, and we wouldn't want people to modify a "bad" version that we've since updated.)
Please let us know if you find this material useful, or have any suggestions. Thanks!
This page is http://www.math.umn.edu/~rogness/multivar/index.shtml and belongs to firstname.lastname@example.org The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.
Many thanks to css/edge for a lot of the ideas used in the creation of this page.