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
that can make these concepts more accessible.mapping (transformation) figures [2 var, 12, 21,22,33] scalar fields, and vector fields
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) 