Computer Algebra Systems at UNF
Program Pages:Below are links for pages specifically focusing on the four computer algebra systems used at UNF.
What are computer algebra systems?In the broadest sense, they are programs designed to do mathematics. The main duties involve:
- Numerical Computations: The programs are used to final numerical
approximations of solutions, derivatives, integrals, differential
equations, etc. Often problems that cannot be solved explicitly
can be solved numerically, and often times (especially in applications)
a numerical answer is all that is necessary.
- Data Analysis: Having
data is not enough; we need to extract useful information from
it. There are multitudes of algorithms designed for data
analysis, most of which involve too much work to be done by hand.
CAS's put all of these algorithms in one place, and they give an
environment where the algorithms are easy to implement.
- Data Visualization: CAS's
can graph 2-D and 3-D functions in a variety of ways. They are
also designed to graph vector fields and solutions to differential
- Symbolic Computations: Most of the CAS's have the ability to do symbolic manipulation of expressions: reducing, expanding, simplifying, derivatives, antiderivatives, etc. They can provide the exact answer to an equation (as opposed to a decimal approximation), and they can express your results in terms of a wide variety of previously defined functions.
- Scientific Workplace (sort of)
- Pari, Maccauly, Singular, Geomview, etc
- Stats programs
Applications:Think of any task in research or teaching where mathematical computation is needed. Think of any task in research in teaching where mathematical visualization would be useful. In all these tasks, a computer algebra system can be used. The following is a brief list of problems at UNF for which I have used CAS's.
Programming Examples:The following are a few programs I have used in various classes. These are designed to show how CAS's visualization abilities can be used to make some interesting projects for the students.
- The Coolest Problem in Numerical
Analysis. Ever. (numerical analysis, Mathematica): This
notebook allows the user to create a fractal based on Newton's
Method. While the students don't learn much from this exercise,
it is a beautiful problem, and it puts some excitement into a somewhat
- The Umbilic Bracelet (calculus 3, Maple): This notebook gives a closer look at the umbilic bracelet, which is the figure on front of Larson's calculus textbooks.
- Plot Charges (vector calculus, Maple): This program draws the vector field associated with a system of point charges in the plane. This allows students to see the various types of vector fields, and it was used to get intuition about Green's theorem and winding numbers.
- Plotting with Color and Plotting Roots (complex analysis, Maple): This provided programs that drew representations of complex functions and complex roots. Students were then asked to analyze the graphs and come up with conjectures about the general structure of complex graphs. Again, introducing graphics into this subject adds beauty to a topic that could be presented as nothing but equations.
- Let's go 4-D! (calculus 3, Maple): This notebook has the students explore the analogies that lead to the understanding of higher dimensional objects. In particular, the students work out the volume of hyperspheres. While this is very cumbersome to do by hand, it is a breeze to do by CAS (assuming you can figure out the correct formula to type in).
Written by Daniel Dreibelbis as part of the Faculty Fellows project: Using Computer Algebra Systems in Teaching and Research
Last modified: March 2005