Dynamic Visualization of Calculus:
More Concepts to Drive Technology
A paper presented at
The Eleventh Annual International Conference on
Technology in Collegiate Mathematics
by

# Martin Flashman |Department of Mathematics Humboldt State University Arcata, CA 95521

e-mail: flashman@axe.humboldt.edu
http://flashman.neocities.org

Introductory Comments: Traditional approaches to the concepts of calculus of several variables and vectors have emphasized graphs in static visualizations. Using the Geometer's Sketchpad, I will present some of these concepts with dynamic visualizations focused on

• mapping (transformation) figures [2 var, 1-2, 2-1,2-2,3-3]
• scalar fields, and
• vector fields
• that can make these concepts more accessible.

A Little Background Propoganda for more power enabling software: In a previous paper given at the ICTCM 5 in 1992 entitled "Concepts to Drive Technology," I examined different levels of technology and how sufficiently powerful technology can enable its user to explore and illustrate concepts that otherwise would be impracticable to develop or explain effectively. In other words:
Concepts drive the applications of a sufficiently powerful technology, and not vice versa.

Though there are many concepts that have a distinctly dynamic character to their understanding, most current mathematical software, computer algebra systems, and graphing calculators remain static, not providing adequate power for a fully dynamic treatment of concepts.

Major exceptions to these static limitations have been in the work done with geometry either using software like the Geometers' Sketchpad, Cabri on the TI 92 calculator, and several special applications (especially on the world wide web) such as those built using JAVA.

Key Question:
How can we visualize and understand the concepts vector calculus and the calculus of several variables?

 # of Controlled Variables/  # of Controlling Variables  Trad`l (static) Method of Visualization 1 2 3 ... 6 1 Precalculus  Calculus I&II  Graphs of functions Calc. III  Graphs of Parametric and Vector Curves in the plane- Trace and image Calc. III  "Graphs" of Parametric and Vector Curves in space -Trace (image) Adv. Calc.  Vector Calc. 2 Calculus III  Level Curves  Graph of Function Linear Algebra  Vector Calc.  Vector Fields Transformation (mapping) Figure Calc. III  Vector Calc.  Graph of parametric Surfaces in space Adv. Calc. 3 Calculus III  Level Surfaces Linear Algebra  Transformation (mapping) Figure Linear Algebra  Transformation (mapping) Figure Adv. Calc. ... Adv. Calc.  Vector Calc. Adv. Calc. Adv. Calc. 8 Music (Guitar)

• Transformation figures
• Derivatives:
• Linearity: Tangent planes and linear estimating functions.
• Visualizing partial derivatives.
• The Chain Rule 1-2-1
• The Gradient and the Differential.