Animated Demonstrations for Multivariable Calculus

John F. Putz
Mathematics and Computer Science Department
Alma College
Alma, Michigan 48801

  • Introduction
  • Quadric surfaces
  • Level curves and level surfaces
  • Velocity and acceleration vectors
  • Directional derivatives and gradient vectors
  • Conclusion

  • Introduction

    I once took my class outdoors and positioned the students at various points on a hillside. I asked them to point in the direction of greatest slope away from the point where they stood, extending their arms far from their bodies or near, according as they thought the maximum slope was large or small. To convey the idea of the gradient vector, I've also drawn arrows on basketballs, used transparencies provided by book publishers, and sketched elaborate diagrams in colored chalk on the blackboard. Like this one, several concepts in multivariable calculus present a teaching challenge because a student's ability to understand them is likely to have more to do with his or her facility with spatial relations than with mathematics.

    Maple's Release 4 permits what previous releases did not: displaying in the same window both an animation and a background, or "dead," plot. When I learned that Maple could this, I thought of several animated demonstrations that I might write for multivariable calculus showing, for example, a plane moving through a fixed surface. In this paper, I share demonstrations for the topics that I've found to be most illuminated by animations. Nothing else has worked as well.

    Quadric surfaces

    One way to understand why quadric surfaces have the shapes they do, is to consider their traces (cross-sections) in various planes parallel to the coordinate planes. The difficult part is putting the traces together in space to form the surface. The hyperboloid of two sheets

    is a good first example because, with a little analysis, it's easy to see that certain horizontal planes will not intersect the surface at all (Animation 1), and that all vertical ones will meet the surface in hyperbolas (Animation 2).

    The hyperboloid of one sheet (Animation 3)

    is a little harder to visualize because, in vertical planes, the hyperbolic traces change orientation when the plane gets farther from the origin than 2 in the direction of the y-axis or 3 in the direction of the x-axis. Animation 4 shows the surface and the plane x = k for varying values of k, each trace having equation

    The most difficult of the quadric surfaces for students to visualize, and certainly to draw, is the hyperbolic paraboloid. Like the hyperboloid of one sheet, the traces in certain planes are hyperbolas that change orientation after a point. As an example, I use horizontal planes intersecting the surface

    (See Animation 5.) I constructed the frames to include the plane that produces the degenerate hyperbola (two intersecting lines) lying between the two families of hyperbolas. The animation shows this reasonably well.

    Level curves and level surfaces

    Because it is a function of x and y, the previous example leads naturally to the topic of level curves (orthogonal projections onto the xy-plane of traces in horizontal planes). Given a real-valued function of two real variables, one way to understand the nature of its surface is to make a contour map, a plot in the plane of various level curves. The first animation I use shows the surface

    and the moving plane that creates the level curves (see Animation 6). The second shows the traces on the surface, then rotates into a view straight down the z-axis, which is the same as a contour map. (See Animation 7.) This shows very clearly how the contour map corresponds to the traces as they sit in space.

    To step up one dimension is to consider the level surfaces of a real-valued function of three real variables. We can try to imagine these surfaces as stacked, somehow, in four dimensions--a challenge for us all. I use the example

    whose level surface, f (x, y, z) = k, is a hyperboloid of two sheets when k < 0, a hyperboloid of one sheet when k > 0, and an elliptic cone when k = 0. The animation that I use shows these surfaces "stacked" in time. (See Animation 8.) This is a very nice analogue to the previous example of a hyperbolic paraboloid. There, the level curves are hyperbolas with branches in one direction when k < 0, hyperbolas with branches in the other direction when k > 0, and two intersecting lines when k = 0.

    Velocity and acceleration vectors

    To demonstrate the vector concepts of velocity and acceleration, I created an animation of a particle moving along a curve ( r(t ) = [(sin 3t ) (cos t ), (sin 3t ) (sin t )] ) at varying speed. At the point in the course where I use this, the students know that, given a vector-valued function of a real variable which is to be viewed as a position function, the velocity and acceleration vectors are the first and second derivatives, respectively, and that the speed is the magnitude of the velocity vector. At the chalkboard, I explain that the velocity vector, always tangent to the path, points in the direction of motion, lengthening as the particle gains speed and shortening as it slows down. This concept has not been particularly difficult to teach using the chalkboard. Harder to communicate, though, is the dual role of the acceleration vector: that it acts to some degree in a direction orthogonal to the velocity vector to move the particle off its course; and that it acts to some degree in the direction of the velocity vector when the particle is gaining speed and in a direction opposite to the velocity vector when the particle is slowing down. Animation 9 illustrates this behavior very well; the velocity vector shown in red, the acceleration vector in green.

    Directional derivatives and gradient vectors

    The most difficult idea to convey in the entire course, for me, is that of the directional derivative and the gradient vector. The animation that I've created to help me, uses the function

    The animation shows:

  • the surface
  • a unit vector rotating about the point (1, 1, 0)
  • a rotating plane parallel to the unit vector
  • the traces of the planes in the surface
  • the tangent lines to the traces at (1, 1, f (1, 1))
  • the gradient vector (shown in green)
  • (See Animation 10.) As the traces and tangents rotate along with the planes and unit vectors, the effect of a changing unit vector is evident. Moreover, the animation makes very clear what is so difficult to show by other means: that the maximum slope away from a point occurs when the unit vector has the direction of the gradient vector.


    These demonstrations represent a significant leap in my ability to communicate these ideas. They tie the abstract to the concrete, bind a concept to an image that we can call up in our mind's eye.

    When I was writing Animation 10, I couldn't quite make it work before it was time that I teach the topic, so I taught directional derivatives and gradient vectors the old way, using a chalkboard and prepared transparency. That night, I found a way to get what I was after (by foiling the hidden-line algorithm a little) and used the demonstration on the following day. After class, two people thanked me for writing it. "We didn't understand before," they said. "But we do now."