"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
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. Re-sizing a graph that is pasted into Paint is not recommended, as both the graph and its lables can become distorted. Resize your graph by changing the size of the DPGraph window which can be made quite small on your computer screen. Use the Page UP/Down buttons on the keyboard to enlarge the graph itself within the smaller window, if it is too small to make out details. Finally, paste this re-sized graph into Paint.
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 remember which directory the installation file (M307MPP.exe, about 260K) is saved in (it is best to create a new folder called "MPP" before downloading, and then download the installation file into the MPP folder). After download finishes, go to that directory and click (or double-click) on the file name to automatically install the MPP files in the current directory. Once installation is complete, close the black DOS installation window. Clicking or double-clicking on the file MPP.exe starts the MPP; similarly, MPP3D.exe starts the 3 dimensional version.
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 Paste in the edit menu. Once the graph appears, click Image and then Invert colors to change the black background to white, which makes it easier for you to lable your printout later if need be (you will also save black ink!). For additional information, see "Printing with Paint" in the DPGraph section above.
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.
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.
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