Computer Algebra Systems at UNF

Program Pages:

Below are links for pages specifically focusing on the four computer algebra systems used at UNF.

• Maple
• Matlab
• MathCad
• Mathematica
  

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 equations.
  
• 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.

Examples:

• Maple
• MathCad
• Matlab
• Mathematica
• Derive
  
• TI-89/TI-92
• Scientific Workplace (sort of)
• Pari, Maccauly, Singular, Geomview, etc
• Stats programs

At UNF, we have Maple, MathCad, Matlab, Mathematica, TI-89/TI-92, Scientific Workplace, SASS, and SPSS.  The calculators (TI-89/TI-92) are a limited version of Maple or Mathematica.  Scientific Workplace is more of a mathematical word processor, but it does have some computational capabilities.  SASS and SPSS are both powerful statistics programs.  This page will focus on the first four programs, which best fit the idea of a CAS.

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.

Integrating/differentiating functions Numerical/exact solutions to ODE's, PDE's Solving equations/systems of equations Evaluating complicated functions Plotting functions, 2-D and 3-D Implicit plots, data plots Designing graphs for exams or publications Massive expression manipulation Programming Computational number theory/group theory

Matrix operations Visualizing 4-D objects/complex graphs Statistics Fourier transforms Signal/Image processing Cryptography Basis reduction Vector calculus Financial mathematics

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 dry subject.
  
• 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).

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Written by Daniel Dreibelbis as part of the Faculty Fellows project: Using Computer Algebra Systems in Teaching and Research
Last modified: March 2005