*"I am rather an
addict of doing things on the computer, because that gives you an
explicit criterion of what's going on. I have a visual way of
thinking, and I'm happy if I can see a picture of what I'm
working with."* - John Milnor, mathematician

**Required Software: DPGraph.**

** Description.** This software easily graphs (and
animates when there are parameters) one or more equations in 2 or
3 variables within the same coordinate system. It is well-suited
for graphing curves and surfaces (implicitly or explicitly),
solid regions that are domains of triple integrals as well as
solids whose volumes or surface areas are to be evaluated. In
addition, the "scrollbar" feature allows a visual or
geometric analysis of surface slices (or traces) or finding
maxima and minima of functions of two or three variables with or
without a constraint. DPGraph is the primary software for this
course.

** Access.** You may download and
install a copy of DPGraph on your computer by clicking here and
following the instructions (VCU students may do this free of
charge; click "V" for VCU's name - see or e-mail me if
you need help with this). If you do not have a computer please
see me as soon as possible.

* Basic Operation*. Click here for a simple first example
containing basic insturctions and tips on how to get started with
the DPGraph. For easy reference, consider printing these
insturctions using your browser's print button. Once you learn
these basic features, check the following links for more complex
examples from multivariable calculus and other useful ideas:

Additional Examples (**used
in graded assignments ***- learn to do math
problems with DPGraph)*

DPGraph Home (*find
complete resources for the software*)

** Printing with MS Paint.** Windows standard
software "Paint" is widely available, versatile and
easy to use for printing most DPGraph outputs. After pasting the
graph you intend to print to the DPGraph clipboard, open the
Paint and simply paste the graph into it from the Edit menu. Note
that it may be necessary to get the Paint out of drawing mode
(cursor looks like a pencil) before you can paste from the
clipboard.

You may add text, lables, numbers, etc. then save and print your file (see the Printing section in General Comments below regarding colors). In some cases, you can overlay two or more graphs on top of each other, by clicking "Image" in Paint and unchecking "draw opaque". This is useful when you want to show several 2D curves in one coordinate system, or several slices of the same graph together in one diagram.

**Optional Software: The Mathematics Plotting
Program (MPP).**

** Description.** This software is multipurpose
(graphing/numeric). The MPP comes in two packages: MPP for 2
dimensional graphing and MPP3D. These programs are more limited
in their scope than Maple, but easier to use, and do everything
noted above in the section on Maple. Further, there are nice
features that are unique to the MPP; these include getting graphs
of regions of double integration set up automatically simply by
entering the double integral for numerical evalutaion... or
seeing a Riemann Sum diagram drawn when evaluating a definite
integral. The MPP was developed by researchers at the US Naval
Academy, and its home page on the internet is accessed by
clicking here.

Most, though not all of the calculations needed to do project or homework problems can be done on a graphics caluculator such as the TI-83. Therefore, the MPP is a recommended software for this course (given the limitations of calculators, e.g., in doing multiple integrals). However, it is not required.

** Access.** Since MPP is shareware (or public
domain), you may download and install a free copy on your PC.
This is a DOS-based software which runs in Windows. Click here to download a combined copy of MPP
(including the 3D version) and install it on your computer or on
a floppy disk. If you do not have access to a computer, please
see me as soon as possible. When you download, choose the option
"save file to disk" and

* Printing MPP graphs with MS Paint*.
Although the software has its own print function (P or F in the
Help menu), incompatibility with some modern printers may cause
difficulties in printing your graphs directly. An indirect but
still easy and in some ways preferable way is to do the
following: Press [Alt] and [Print Screen] buttons together; this
will create a copy of the MPP screen containing your graph onto
the clipboard. Then open Paint and click

* Numerical Integration with the MPP*.
Single, double and triple integration are very easy and rewarding
to do in MPP. You may double-integrate (numerically) in
rectangular or polar coordinates by simply filling the blanks in
the on-screen integrals! Similarly, you can triple-integrate in
rectangular, cylindrical or spherical coordinates. Click here for step-by-step
instructions for double and triple integration.

* Graphing space curves and parametric sufaces with
the MPP*. Graphing parametric curves and surfaces in
space is easily accomplished with the MPP. Click here for step-by-step
instructions.

* The MPP Tip Sheet.* To see a list of
things you can do with the MPP, including some that go beyond
DPGraph, click here (print these
instructions for easy reference). Experimenting with the software
is the best way of getting to know its various features.

**HELP!**

*The "help" menu in these modern software is meant
to be used !* Help is not just for those times you're stuck in
something. In DPGraph the Help button gives basic instructions on
the software usage. In MPP, helpful comments are provided in the
first screen for each menu item selected.

**SAVING FILES.**

It is always a good idea to save your files as soon as you've
finished writing their cores, even if you're not completely done
(in DPGraph, simply click "save"). This avoids a sudden
loss of data due to mistakes or mishaps. If you work in the VCU
Computer Lab, be sure to take a floppy disk with enough empty
space on it to save your files.

Last Updated August, 2002