Mathematicalexperiments |
Computerhelp |
Maple in Math291 on eden |
The Maplefield trip |
How to getthose answers |
Programmingin Maple |

Mathematical experimentsAlmost 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 systematic and powerful programs that permit algebraic manipulation
- What is the coefficient of
*x*^{6}*y*^{4}*z*^{2}in (*x*+*y*+*z*)^{12}?
numerical approximation
- What is an approximation to the only root of
3
*x*+cos(2*x*^{2})=0?
graphing
- 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 (humans learn much more from pictures than from lists of numbers!) is very useful.
Almost all of my teaching
, and
Mathematica. In this course DeriveMaple 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 available with special capabilities. For
example, Matlab, a program originally directed at problems of
linear algebra, is installed on systems of the Engineering School.
Also, I usually use
A year ago Professor Costin wrote a nice sequence of "labs" to
teach students about the capabilities of
The class will meet in the instructional computer lab ARC 116 (an X-term "lab" inside the Busch
Campus Computing Center) on Thursday, January 30,
from 4:30 to 5:50 PM. I hope that students will work
through the following sequence of exercises (copies will be
provided). Here are links to PDF files. These are slightly changed
(and, I hope, improved!) versions of the handouts used last semester.
The last question asked on the fourth handout is difficult. I've
written a complete solution
which is quite long, but verifying
`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 allows implicit equations to be sketched. 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.
Maple is also a programming environment. Maple
programs are known as procedures. The 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 programmng is possible.
I have several books on |

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