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### Contact

Jenny Vanden Eynden
Mathematics Professor
Email: jenny.vandeneynden@gcccd.edu
Phone: 619.644.7294

# 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!

### 13.1        Three-Dimensional Coordinate Systems

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

Graphing applet written by Professor Leathrum at Jacksonville State University

### 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

• 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)

### 14.4   Motion in Space: Velocity and Acceleration

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

Website created by Tom Henderson of Glenbrook South High School.

### 15.1   Functions of Several Variables

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!

### 15.3   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

### 15.6   Directional Derivatives and the Gradient Vector

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)

### 17.1   Vector Fields

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

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

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

### 17.5   Curl and Divergence

Graphing applet written by Professor Leathrum at Jacksonville State University

### 17.9   The Divergence Theorem

Last Updated: 09/10/2014

### Contact

Jenny Vanden Eynden
Mathematics Professor
Email: jenny.vandeneynden@gcccd.edu
Phone: 619.644.7294

A Member of the Grossmont-Cuyamaca Community College District