Dynamic Visualization of Calculus

Mathematics Department Colloquium

September 18, 1997

Key Question:
How can we visualize and understand calculus of n variables?
 
Calculus of n - Variables Courses
# of Controlled Variables/ 
# of Controlling Variables 
Traditional 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 and image		 Adv. Calc. 
Vector Calc.
2 Calculus III 
Level Curves 
Graph of Function		 Linear Algebra 
Vector Calc. 
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.
7 Music G
 
Visualizations of the Derivative Concept
# of Controlled Variables/ 
# of Controlling Variables 
Traditional Method  
of Visualization		 1 2 3 ... 6
1 Velocity 
Slope of Tangents		 Velocity Vector 
Tangent Vector		 Velocity Vector 
Tangent Vector
2 The Gradient  
Tangent Plane		 The Jacobian (?) The Tangent Plane
3 The tangent plane to a level surface
?
...
6 ?
 

Familiar experiences with many variables: