Welcome to Exploring Multivariable Calculus! This website is dedicated to exploring calculus visually using the computer. In particular, it's focused on helping you explore multivariable calculus and some three-dimensional topics within single variable calculus. (In the left panel, you will find a link to another area of my website that is focused on single variable calculus as well as a link to my instructor home page.)
This web project is being developed with support from the National Science Foundation under the grant, DUE-CCLI 0736968.
In the left panel, you will also find a link to the main Multivariable Exploration applet and links to various focused applets in this collection.
Why use visualizations like those found here to explore multivariable calculus? As you begin to explore the Java applets found on this site, I think you will find a richer understanding of the geometric aspects of the concepts of multivariable calculus. My goal is to enhance the geometric intuition of calculus students so that they are able to visualize the concepts and actually "see" the rich visual relationships and interactions described by the calculus concepts.
As an instructor, I often found it difficult to draw the three-dimensional concepts clearly on the chalkboard, and found myself waving my hands to try to get students to see what I was seeing. Now that I have these computer visualization tools, I feel I can show students a much clearer picture of what I have been describing verbally.
If the applet does not work, you most likely need to Click here to install the newest version of the Java Plugin on your browser.
I also encourage anyone who is interested in this project to take the time to write something on the discussion board. You will need to become a member to do this, but I am the only one who will see this member information. It is a private website, and membership makes it easier to keep spam from being placed on the discussion board. Please consider sharing ways you have used the project materials with your class, any special projects you have used with your multivariable calculus class, any real-life applications or examples you have found especially useful in this class, links to other useful materials for multivariable calculus on the web, etc. My goal for this discussion is to make it a place for us to exchange ideas and further enrich the resources of this website for teaching and learning multivariable calculus.
Instructors: If you are an instructor using this project in any way, please send me an email to let me know of your interest. I would love to see more people using the materials from this project, and it is important that I be able to report how the project is doing to the NSF.
The following PDF documents give more information about the goals and current state of the project.
9-27-2011 Graph implicit equations in spherical or cylindrical coordinates like rho^2 = -sec(2ϕ) or r^2*theta^2 = r +theta.
- Implicit surfaces/equations can now be graphed in spherical and cylindrical coordinates as well as in cartesian coordinates.
- You can automatically change coordinate systems by typing in an equation in the other mode.
- You can also use the menu on the Implicit Surface dialog to change coordinate systems.
- Buttons have been added to make it easier to get phi, theta and rho, but you can also type these words in the equations.
- Example Surfaces have been added to the Example menu on the Implicit Surface dialog that use spherical and cylindrical coordinate systems.
10-29-2010 I added several new links that bring up the CalcPlot3D applet with particular topical examples all set up. These links can be found on the right sidebar, just below the main link to the CalcPlot3D applet. These examples are all based on scripts that are on the Example Scripts list in the applet, so you can try them out using that list without clicking on each of these links. Please let me know if you have suggestions for other examples.
9-17-2010 You can now graph Implicit Surfaces of the form, f(x, y, z) = c, or more generally of the form, f(x, y, z) = g(x, y, z). Find them on the Graph menu! This allows you to easily graph quadric surfaces, cylindrical surfaces, and many other topographically interesting surfaces. A large set of example surfaces is included under the Examples menu on the Implicit Surface dialog. Currently the algorithm used for plotting implicit surfaces is the Marching Cubes algorithm. I am preparing two additional algorithms which should provide even better results for some surfaces. Note that if no equal sign is entered, the expression is assumed to be equal to 0, i.e., it assumes you entered f(x, y, z) where f(x, y, z) = 0.
9-1-2010 I added a feature to allow graphing of complex functions (real or imaginary parts, etc.) of the form z = f(w) where z is the real z-coordinate and w is a complex number. The real part is taken by default, so you can enter z = w^2, for example. If you want the imaginary part, enter z = Im(w^2). Some interesting examples are included in the function list. This feature and the geometric mean and arithmetic mean examples were suggested by Dr. Joseph Straight, a professor at SUNY Fredonia.
6-1-2010 New parametric surface examples added including Umbilic Torus, Dupin Cyclide, Pisot Triaxial, Limpet Torus, Triaxial Hexatorus, and Scherk's Minimal Surface. Enjoy!
5-20-2010 Added a large number of new parametric surface examples to the list in the applet. These include several new versions of the Klein Bottle, Boy's Surface, several minimal surfaces including an Enneper's surface, Catalan's surface, and Henneberg's surface, numerous shells, horns, tori, etc. I particularly like the Fresnel Surface #2, Boy's Surface, Maeder's Owl, and the new Klein Surfaces using 2 and 4 parts. I also found one to represent a regular octahedron and a hyperbolic octahedron. Can anyone tell me the parametric equations of the other Platonic Solids (if they exist)? And of the other hyperbolic versions of the Platonic Solids?
9-10-2009 Added the ability to view the TNB-Frame for space curves. Also the Osculating Circle and curvature value can be displayed. See the View menu on the Add a Space Curve dialog.
8-1-2009 Added a scripting feature that allows you to create a dynamic slideshow by saving slides of objects you have created in the applet. This slideshow can then be saved locally as an editable script file. You can then load the script in at a later time (perhaps during class) as part of a demonstration or by students as a guided exploration. If you are interested in more information on this feature, please ask me for the documentation (as it has not yet been posted). This scripting feature was the focus of a minicourse I presented at MathFest 2009 in Portland, OR. I will also be presenting this minicourse at MathFest 2010 and at the 2011 JMM (see below).
3-17-2009 Added a new Guided Tour/Tutorial for CalcPlot3D. You can access it here or in the CalcPlot3D applet from the Help menu.
3-5-2009 Parametric Surfaces can now be graphed. Find them on the Graph menu! See the list of examples on the Examples menu in the Parametric Surface dialog. Have fun exploring!
2-10-2009 You can now use 3D glasses to get a more convincing 3D view of the 3D plots in CalcPlot3D. You can use Red-Cyan/Red-Blue 3D glasses for almost all of the options, but there is one option for the Amber-Blue 3D glasses that were recently distributed for the Super Bowl halftime feature, and there is an option for Red-Green 3D glasses too. There are also two options for viewing the images in 3D without 3D glasses: Stereo Pair and Cross-eyed. Try these new viewing options out by selecting them from the View Settings menu in the applet. Let me know what you think! Click here for the 3D View Help to see more details.
Jan. 2014: I will present a 2-hour poster session on CalcPlot3D and several 15-minute sessions on the project at the 2014 JMM conference in Baltimore, MD.
August 2014: I plan to present on my project at MathFest 2014 in Portland, OR.
August 12-16, 2013 I presented a 5-day workshop on this project in Puerto Vallarta, Mexico, translated into Spanish.
March 21 - 24, 2013: I presented a 2-hour computer minicourse on using CalcPlot3D & other Java applets in calculus at the 2013 ICTCM conference in Boston, MA.
Jan. 2013: I presented a 2-hour poster session and several 15-min sessions on this project at the 2013 JMM (Joint Math Meetings) in San Diego, CA.
August 13-17, 2012 I presented a 5-day workshop on this project in Puerto Vallarta, Mexico, translated into Spanish.
March 23 - 24, 2012: I presented a 2-hour computer minicourse on using CalcPlot3D in multivariable calculus at the 2012 ICTCM conference in Orlando, FL.
January 4-8, 2012 I presented a 2-hour poster session and a 15-min session on this project at the 2012 JMM (Joint Math Meetings) in Boston, MA.
November 10-13, 2011 I presented a 2-hour workshop and a regular 45-min session on this project at AMATYC 2011 in Austin, TX.
August 22-26, 2011 I presented a 4-day workshop on this project in Mazatlan, Mexico, translated into Spanish.
June 29, 2011I presented my first Webinar for AMATYC, titled Making Calculus Come Alive using Dynamic Visualization. During this webinar I demonstrated applets I have created for single variable calculus as well as CalcPlot3D (for multivariable calculus). To see the recorded webinar (once it is online) click here. You may also want to see the June 2011 AMATYC Webinar webpage I created.
March 17-19, 2011I presented a 2-hour minicourse on this project at ICTCM (the International Conference on Technology in Collegiate Mathematics) in Denver. I also presented a second talk on my project.
Links: ICTCM conference link - http://ictcm.pearsontc.net/
My Minicourse link - http://tinyurl.com/2fbe2tk
My other talk link - http://tinyurl.com/2agnpop
January 6-9, 2011 I presented my 4-hour minicourse again at the 2011 Joint Math Meetings of the MAA/AMS in New Orleans, LA. See the description below. I hope to give this minicourse again at a future JMM or MathFest.
Minicourse #13: Creating Demonstrations and Guided Explorations for Multivariable Calculus using CalcPlot3D, organized by Paul Seeburger, Monroe Community College.
Part 1: Friday, 1:00 p.m. - 3:00 p.m.;
Part 2: Sunday, 1:00 p.m. 3:00 p.m.
It is often difficult for students to develop an accurate and intuitive understanding of the geometric relationships of calculus from static diagrams alone. This course explores a collection of freely available Java applets designed to help students make these connections. Our primary focus will be visualizing multivariable calculus using CalcPlot3D, a versatile new applet developed by the presenter through NSF-DUE-0736968. Participants will also learn how to customize this applet to create demonstrations and guided exploration activities for student use. Images created in this applet can be pasted into participants' documents. See http://web.monroecc.edu/calcNSF/. Some basic HTML experience is helpful. All participants are expected to bring a laptop computer to the minicourse.
This minicourse includes 2-hours on how to use the main project applet, CalcPlot3D, and 2 hours on how to create and edit scripts using CalcPlot3D. Scripts allow you to create demonstrations, student explorations, or customized versions of the applet that start with the functions and curves you desire.
For more information on the 2011 Joint Math Meeting conference see: http://www.ams.org/meetings/national/jmm/2125_intro.html
For the official announcement of this minicourse, see Minicourse #13 at:
I also presented two short paper sessions and a 2 hour poster session on my project at the 2011 JMM.
November 11-14, 2010 I presented a 2-hour workshop and a 2-hour poster session on this project at AMATYC 2010 in Boston, MA. I also presented a 15-minute talk on visualizing intersections of surfaces with CalcPlot3D.
Sept. 30 -October 2, 2010 I presented a 45 session on using CalcPlot3D to teach Lagrange multiplier optimization and I presented a 2-hour workshop on using the CalcPlot3D applet at the 2010 Kansas City Regional Mathematics Technology EXPO.
August 5-7, 2010 I presented a 4-hour minicourse on this project at MathFest 2010 in Pittsburgh, PA.
#6 — Creating Demonstrations and Guided Explorations for Multivariable Calculus using CalcPlot3D
Paul Seeburger, Monroe Community College
Part 1: Friday, August 6, 3:30 p.m. – 5:30 p.m.
Part 2: Saturday, August 7, 3:30 p.m. – 5:30 p.m.
It is often difficult for students to develop an accurate and intuitive understanding of the geometric relationships of calculus from static diagrams alone. This course explores a collection of freely available Java applets designed to help students make these connections. Our primary focus will be visualizing multivariable calculus using CalcPlot3D, a versatile new applet developed by the presenter through NSF-DUE-0736968. Participants will also learn how to customize this applet to create demonstrations and guided exploration activities for student use. Images created in this applet can be pasted into participants’ documents. See http://web.monroecc.edu/calcNSF/. Some basic HTML experience is helpful. All participants are expected to bring a laptop computer to the minicourse.
This minicourse includes 2-hours on how to use the main project applet, CalcPlot3D, and 2 hours on how to create and edit scripts using CalcPlot3D. Scripts allow you to create demonstrations, student explorations, or customized versions of the applet that start with the functions and curves you desire. For more information on the conference see: http://www.maa.org/mathfest/mathfest.cfm
For the official announcement of this minicourse, see Minicourse #6 at:
I also presented one other contributed paper session related to this project at MathFest 2010.
March 12-13, 2010 I presented on this project at ICTCM (the International Conference on Technology in Collegiate Mathematics) in Chicago.