Students will work through the following material (the first column has links to pdf files, and the second, to TeX sources):

Why learn Maple?
Almost every aspect of the practice of mathematics, both pure and applied, has been improved and amplified in recent decades by the widespread availability of "computer algebra systems", CAS. This technology is much more than just algebra, of course. It is a collection of systematic and powerful programs that permit algebraic manipulation

• What is the coefficient of x⁶y⁴z² in (x+y+z)¹²?

• What is an approximation to the only root of 3x+cos(2x²)=0?

• What do the points (x,y,z) which satisfy the equation z²+(x²+y²-1)(x²+y²-2)=0 look like?

The freedom to work with "exact" symbolic computation, with numerical approximation (with specified accuracy) and with visual display of data (human beings learn much more from pictures than from lists of numbers!) is very useful. Maple provides an environment which allows all of these, plus the freedom to move among these representations of mathematical ideas.

Much teaching and research is now improved by access to powerful programs which allow experimentation. Examples can be discovered and explored which are useful for instruction. These programs can also be used to further understand complicated phenomena which are not easily explained.

Computer help
Many students have graphing calculators. These are useful, but are limited by speed and memory size. Simple errors may occur. There are large computer programs with powerful numerical, symbolic, and graphical capabilities. These still may have the potential for errors (as some of the contents of the link discuss), but much effort has gone into their programming. The most widely distributed programs are Maple, Mathematica, and Derive. Here Maple will be favored, since almost every large computer system at Rutgers has Maple installed. These programs are not infallible but they can be very helpful. Other programs are available with special capabilities. For example, Matlab, a program originally directed at problems of linear algebra, is widely used at the Engineering School.

How to get those answers
The answers to the questions above were obtained with the following Maple instructions. Please: these instructions are not given to impress you, but rather to show how easy it is to get the answers.

• coeff(coeff(coeff((x+y+z)^12,x^6),y^4),z^2);
  The command coeff(P,monomial) finds the coefficient of the monomial in the expression P. Layering three repetitions of coeff finds the desired coefficient.

• fsolve(3*x+cos(2*x^2)=0,x);
  fsolve is a general "floating point" approximate equation solver. Care must be used if there's more than one root. There are also symbolic solvers, useful when there is a nice formula for the solution.

• with(plots):
  V:=((x^2+y^2)-1):
  W:=((x^2+y^2)-2):
  implicitplot3d(-V*W=z^2,x=-2..2,y=-2..2,z=-2..2,grid=[30,30,30],axes=normal);
  The implicitplot3d command sketches graphs which are defined implicitly by equations. Since Maple has so many functions and libraries available, many need to be specifically loaded before use. The command with(plots); loads a variety of plotting commands. The implicitplot3d command has a wide variety of options. The grid option gives control over the spacing of sample points. Of course increasing the number of sample points "costs" computational time and storage space but does given finer detail.

Other references and programming in Maple
There's a nice "reference card" with common Maple commands which students may find helpful: please look at this University of Michigan web page. There are many available online tutorials. For example, here is a link to one at Brown University.

Maple is also a programming environment. Maple programs are called procedures. The Maple language has many statements supporting program flow such as if ...then and while and do etc., and also has a variety of data types. There's no time in this course to teach this material, but students should know that programming is possible.

There are a number of books on Maple programming which can be found with an easy web search. My current favorite is Maple: A comprehensive introduction by Roy Nicolaides and Noel Walkington, Cambridge University Press ($75, 484 pages, available for less in places on the web). There are also many web pages which discuss programming in Maple. For example, here's one online tutorial. Warning: such pages are only for the enthusiast!

Maintained by greenfie@math.rutgers.edu and last modified 9/2/2008. On 7/1/2009 some modifications were made to the General Introduction page.