Welcome to
Multivariate Calculus
Paul Blanchard

Quicktime animations

• Animation of parameterized line
  Illustrates sum of position and scalar multiples of direction
• Tangent vector as a limit
  Animation to illustrate the limiting process involved in the definition of the derivative: control h using the slider
• Arc length animation
  Control of delta t using the slider
• Chain rule animation
• Constrained max/min
  Animation of constrained max/min for f(x,y) = 2x²+4y² subject to the constraint x²+y²=1
• Riemann sums converging to the integral
• an animation of many y-slices
• Line integral animation

Martin Kraus - Author of LiveGraphics3D

LiveGraphics3D

• Position and direction vectors for a line
  Blue position vector and red direction vector
• The tangent vector at the point
  The derivative vector
• Elliptic cylinder
  an example of a generalized cylinder
• Surface of revolution
• Limit at (0,0) does not exist
  graph of 2xy/(x²+y²)
• a graph with a local maximum
  graph of 9-x²-y² with a critical point at (0,0)
• a graph of a saddle
  graph of y²-x² with a critical point at (0,0)
• a graph of a monkey saddle
  graph of (1/3)x³-xy² with a critical point at (0,0)
  this is an example of a degenerate critical point at (0,0)
• a graph of a function with two critical points
  graph of y-(1/12)y³-(1/4)x²+(1/2)
  this example has two critical points---one local max and one saddle
• Integration over a general region
• Small parallelogram tangent to the surface
• A "cube" in space determined by spherical coordinates
• 40 different spherical "cubes"

Animated LiveGraphics3D

• Line
  Line swept out as the values of a vector-valued function
• Circular helix
  Circular helix given by a vector-valued function
• Trefoil knot
  Knotted curve given by a vector-valued function
• Tangent vectors from the derivative
  The derivative of the elliptical helix as a vector-valued function.
• Derivative of a curve that sweeps out a trefoil knot
  Animation to illustrate the speed at which a point traverses a curve
• Constrained max/min
  Animation of constrained max/min for f(x,y) = 2x²+4y² subject to the constraint x²+y²=1

Animations from other sites

• Cross product
  (courtesy of the Syracuse University Physics Department)
• Area in polar coordinates (Quicktime format)
  courtesy of Professor Lou Talman, Metropolitan State College of Denver
• Another surface of revolution. (Quicktime format)
  courtesy of Professor Lou Talman, Metropolitan State College of Denver
• Directional derivatives and the gradient (Quicktime format)
  (courtesy of Professor John Putz, Alma College)
• Applet to illustrate area conversion
  Area conversion applet written by Professor Tom Leathrum, Jacksonville State University

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