The first or second recitation meeting in Math 251 is held in a
*microcomputer lab* rather than in the usual recitation classroom.
Specific times and places for these meetings will be announced. In the
computer lab, students use the program
`Maple` to work through material on
arithmetic, algebra, calculus, and graphing, following the outlines given
in the pdf files linked here:

Arithmetic |
Algebra |
Calculus |
Graphing |

Below we give some further information about using
`Maple` at Rutgers, and why one might want to
do so.

`Maple` at Rutgers

The simplest way to begin using
`Maple` is at a Rutgers computer lab
(equipped with either Windows or Mac OSX; if you don't like
the flavor you get you can reboot and choose the other).
These computers are all equipped with the current version of
`Maple`.
So login, look for `Maple`, and start it running.

Other ways of getting access to `Maple` are discussed on the Rutgers software site
*software.rutgers.edu*. Here is a direct link to the relevant page:

Alternatively, it is possible to run the program remotely, in a browser, through the site apps.rutgers.edu.. However, this site experienced problems in in the past, and students are advised not to rely on it for last-minute preparation of labs.

**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

- What is the coefficient of
*x*^{6}*y*^{4}*z*^{2}in (*x*+*y*+*z*)^{12}?

- What is an approximation to the only root of
3
*x*+cos(2*x*^{2})=0?

- What do the points
(
*x*,*y*,*z*) which satisfy the equation*z*^{2}+(*x*^{2}+*y*^{2}-1)(*x*^{2}+*y*^{2}-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` 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).

*Created by Steven Greenfield.
Maintained by the course coordinator and the computer coordinator.
(for 2016/17 Eugene Speer, speer@math, and
Gregory Cherlin, cherlin@math); if in doubt contact the undergraduate office.
*