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
• 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
• Surface of revolution

Partial Derivatives

• Limit at (0,0) does not exist
  graph of 2xy/(x²+y²)
• Chain rule animation (Quicktime format)
• 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
• A "cube" in space determined by spherical coordinates
• 40 different spherical "cubes"

Differential Equations

• Exercise involving sketching solution curve from component graphs (Quicktime format)
• Eigenvalues and oscillators (Quicktime format)