Math 281: Multivariable Calculus Animations and Graphics

Below is a compilation of Animations and Computer Graphics.  They have been pulled together from various resources and course materials found on the Internet.  Many of the graphics are in Quicktime format.  ENJOY!

• Graph Points, Line Segments and Position vectors in the 3-D rectangular coordinate system.

Graphing applet written by Professor Leathrum at Jacksonville State University

• Point (x, y, z)
  A point (x, y, z) in three space.  The x-component is in red, y-component in green, z-component in pink

13.2   Vectors

• Graph Points, Line Segments and Position vectors in the 3-D rectangular coordinate system.

Graphing applet written by Professor Leathrum at Jacksonville State University

• Position vector
  The blue vector represents a position in space

13.3   The Dot Product

13.4   The Cross Product

• A Java Applet tutorial on the vector Cross Product.
  (courtesy of the Syracuse University Physics Department)

13.5        Equations of Lines and Planes

• Position vector
  The blue vector represents a position in space
• Line
  Line swept out as the values of a vector-valued function
• Normal line to a plane
  A plane with its normal line

13.6   Quadratic Surfaces

• The hyperboloid of one sheet and its horizontal traces
  (Quicktime version)
  courtesy of Professor John Putz, Alma College
• The hyperboloid of one sheet and its vertical traces
  (Quicktime version)
  courtesy of Professor John Putz, Alma College
• Circular hyperboloid of one sheet
  all z-slices are circles
• The hyperboloid of two sheets and its horizontal traces
  (Quicktime version)
  courtesy of Professor John Putz, Alma College
• The hyperboloid of two sheets and its vertical traces
  (Quicktime version)
  courtesy of Professor John Putz, Alma College
• The hyperbolic paraboloid and its horizontal traces
  (Quicktime version)
  courtesy of Professor John Putz, Alma College
• a surface with its traces, and the resulting level curves
  (courtesy of Professor John Putz, Alma College)
• same surface as above
  the same surface, but you can rotate it
• Hyperboliods of two sheets, a cone, and Hyperboloids of one sheet.   
  (courtesy of Professor John Putz, Alma College)

13.7   Cylindrical and Spherical Coordinates

• Cylindrical Coordinates
• Spherical Coordinates

• Circular helix
  Circular helix given by a vector-valued function.  Created by
• Trefoil knot
  Knotted curve given by a vector-valued function
• Graphing applet for 3D parameterized curves
  Graphing applet written by Professor Leathrum at Jacksonville State University

14.2   Derivatives and Integrals of Vector Functions

• 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
• Tangent vectors from the derivative
  The derivative of the elliptical helix as another vector-valued function.

14.3   Arc Length and Curvature

• Arc length animation (Quicktime format)
  Approximating arc length on a helix-like curve.

14.4   Motion in Space: Velocity and Acceleration

• Animation of a particle moving along a curve  r(t ) = <(sin 3t ) (cos t ), (sin 3t ) (sin t )>.

The velocity vector is shown in RED, the acceleration vector in GREEN.

• Notice that the velocity vector is always tangent to the path and points in the direction of motion, lengthening as the particle gains speed and shortening as it slows down.
• The acceleration vector acts to some degree in a direction orthogonal to the velocity vector to move the particle off its course.  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.
• Graphing Applet which graphs a parametric curve, velocity vector, acceleration vector, along with the T-N-B frame.

Graphing applet written by Professor Leathrum at Jacksonville State University

• Direction of the Velocity and Acceleration Vectors.  Uses a matchbox car analogy to describe the vectors direction.

Website created by Tom Henderson of Glenbrook South High School.

• Graphs functions of the form z = f (x,y) using 3-dimensional rectangular coordinates.

Graphing applet written by Professor Leathrum at Jacksonville State University

• a surface with its traces, and the resulting level curves
  (courtesy of Professor John Putz, Alma College)
• same surface as above
  the same surface, but you can rotate it
• Flagstaff USGS Shaded Relief Map -- We usually think of starting with a surface and obtaining the level curves (contours) from it. This site gives examples of shaded relief maps which start with a contour map (a map of the level curves) and mathematically reconstruct the surface from the level curves!

15.2   Limits and Continuity

• a graph where the limit at (0,0) exists
  graph of sin(x²+y²)/(x²+y²)
• a graph where the limit at (0,0) does not exist
  graph of 2xy/(x²+y²)

15.3   Partial Derivatives

• A function whose mixed partials are unequal -- A function must satisfy certain conditions in order for the mixed partials to be equal. This site discusses these conditions and provides an example of a function which does not meet them. Thanks to Ali Eftekhari

15.4   Tangent Planes and Differentials

• A non-differentiable function with partial derivatives everywhere.
• A function without a limit, although limits exist along all lines
• A function with a single critical point, which is a local minimum but not a global minimum
• A non-differentiable function with all directional derivatives
• A function whose mixed partials are unequal

15.5   The Chain Rule

15.6   Directional Derivatives and the Gradient Vector

• Temperature Gradients -- See the use of gradients in a temperature map. Test your knowledge!
• Animation of the Directional Derivatives of      
  (courtesy of Professor John Putz, Alma College)

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)

15.7   Maximum and Minimum Values

• 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

15.8   Lagrange Multipliers

• Constrained max/min
  Animation of constrained max/min for f(x,y) = 2x²+4y² subject to the constraint x²+y²=1
• Quicktime version of the above animation (in Quicktime format)

• Volumes and Riemann sums converging to the integral

16.2   Iterated Integrals

16.3   Double Integrals over General Regions

16.4   Double Integrals in Polar Coordinates

• Area in polar coordinates
  courtesy of Professor Lou Talman, Metropolitan State College of Denver
• Area between two polar curves
  once again, courtesy of Professor Lou Talman

16.5   Applications of Double Integrals

16.6   Surface Area

16.7   Triple Integrals

• Beams, Bending, and Boundary Conditions: Centroids -- This site contains many questions to deepen your understanding of the centroid of an object.

16.8   Triple Integrals in Cylindrical and Spherical Coordinates

• A "cube" in space determined by spherical coordinates
• 40 different spherical "cubes"

16.9   Change of Variables in Multiple Integrals

• Applet to illustrate area conversion
  Area conversion applet written by Professor Leathrum at Jacksonville State University

• Applet that graphs a vector field in the plane and its flow curves.

Graphs the vector field in the plane given by the vector-valued function F (x,y)=<f (x,y),g(x,y)> and flow curves given parametrically as (x(t),y(t)) from starting point (x₀,y₀) associated with the value t=0.

Graphing applet written by Professor Leathrum at Jacksonville State University

• Vector fields (pdf format)  (examples)
• Java Applet to visualize vector fields in the plane 

(courtesy of Shannon Holland, Dr. Matthias Kawski of Arizona State university)

• Flows of Vector Fields

You can choose a vector field and observe how a region in the plane is carried around by the flow.

17.2   Line Integrals

• Surface whose surface area can be calculated by a path integral
• Line integral animation (in Quicktime format)
• constant vector field around the positively-oriented unit circle
• constant vector field along a piecewise linear curve
• nonconstant vector field along the top-half of the positively-oriented unit circle

17.3   The Fundamental Theorem for Line Integrals

17.4   Green's Theorem

17.5   Curl and Divergence

• Graphing Applet which demonstrates the effects of the divergence and curl of the vector field.
  • If divergence is positive, then the vertices move apart, resulting in a quadrilateral with area larger than the original square; if divergence is negative, the vertices move together.
  • If curl is positive, then the vertices rotate counterclockwise; if curl is negative, the rotation is clockwise.

Graphing applet written by Professor Leathrum at Jacksonville State University

17.6   Parametric Surfaces and Their Areas

17.7   Surface Integrals

17.8   Stokes' Theorem

17.9   The Divergence Theorem