Animations for
Multivariable Calculus
and Differential Equations
Paul Blanchard

Geometry

• Line with 2 points
  Line in space determined by two points
  LiveGraphics3D applet courtesy of Martin Kraus
• Animation of parameterized line (Quicktime format)
  Illustrates sum of position and scalar multiples of direction
• Cross product
  (courtesy of the Syracuse University Physics Department)
• Trefoil knot
  Knotted curve given by a vector-valued function
• Tangent vector as a limit
  Animation to illustrate the limiting process involved in the definition of the derivative
• Tangent vector as a limit (Quicktime format)
  Animation to illustrate the limiting process involved in the definition of the derivative: control h using the slider
• The tangent vector at the point
  The derivative vector that results from the above two animations
• Tangent vectors from the derivative
  The derivative of the elliptical helix as another vector-valued function.
• Arc length animation (Quicktime format)
  Control of delta t using the slider
• Derivative of a curve that sweeps out a trefoil knot
  Animation to illustrate the speed at which a point traverses a curve
• Area in polar coordinates (Quicktime format)
  courtesy of Professor Lou Talman, Metropolitan State College of Denver
• Surface of revolution
• Another surface of revolution (BIG, i.e., over 4MB) (Quicktime format)
  courtesy of Professor Lou Talman, Metropolitan State College of Denver
• Matching question on first exam: Surface A and Surface B

Partial Derivatives

• Limit at (0,0) does not exist
  graph of 2xy/(x²+y²)
• Chain rule animation (Quicktime format)
• Directional derivatives and the gradient (Quicktime format)
  (courtesy of Professor John Putz, Alma College)
• Constrained max/min (Quicktime format)
  Animation of constrained max/min for f(x,y) = 2x²+4y² subject to the constraint x²+y²=1

Multiple Integrals

• Riemann sums converging to the integral (Quicktime format)
• one x-slice
• an animation of many y-slices (Quicktime format)
• Integration over a general region
• Animation of iterated integration with x-slices
• Applet to illustrate area conversion
  Area conversion applet written by Professor Tom Leathrum, Jacksonville State University
• A "cube" in space determined by spherical coordinates
• 40 different spherical "cubes"

Differential Equations

• DETools
  The following examples are on the CD that comes with Blanchard, Devaney, Hall, Differential Equations, 2nd edition, published by Brooks/Cole Publishing Co.
  • MassSpring
  • HMSGlider
  • Pendulums
• Phase plane versus component graphs of predator-prey equations (Quicktime format)
• 3D graphs of solutions
• Exercise involving sketching solution curve from component graphs (Quicktime format)
  During the talk, I illustrated the answer, but I'm not going to post the answer here.
• Eigenvalues and oscillators (Quicktime format)

"How to" Examples

• LiveGraphics3D
• Arc length animation (Quicktime format)

Chautauqua Short Course, Boston University, July 19-21, 2004

Contact Information