### September 18, 1997

Key Question:
How can we visualize and understand calculus of n variables?

 # 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

 # 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:

• Populations of animal species in geographic region control levels of different kinds of vegetation.
• A class with 30 students  taking 10 quizzes and 3 exams in a course and thereby controlling the class average on each of these tests.
• Piano- 10 fingers and 2 feet control 88 keys
• Guitar 4 +3 fingers control 6 strings.

## A Personal History:

• Basic and APL Programs (IBM 5100)
• Graphing with tabs and format
• Apple IIe ARBPLOT (R. B. Brown)
• Templates for graphs, tangent lines, tangent fields, etc.
• Mastergrapher
• X(PLORE) on 8088. (David Meredith)
• Interactive graphing.
• Tangent and Vector Fields
• Transformation (Mapping) Figures (Demo)
• Darts
• Casio 7000, HP 28S, HP48S
• Graphing Functions
• X(PLORE) for MAC
• Dynamic Geometry
• Locus and Trace
• Animation
• Limited connection to coordinate geometry
• Programming through scripts
• HP48G
• Transformation figures
• Interactive features
• Animation-
• built with 3 dimensional visualization
• Programmed/ interactive control
•  Mathwrite Library: Interactive Calculus (from www)
• Templates with dynamic interactive controls (Demo)
• Wingeom (Andover Freeware)
• Animation
• Coordinate geometry
• Coordinate geometry (Demo?)
• Extended Tools (Demo?)

#### Calculus Concepts and Visualization

• 2 Variables: Functions and Relations
• Tables
• Graphs
• Transformation Figures (Demo GS)
• Derivatives: Tangents and Rates (Velocity)
• 3 (or 4) Variables:
• 1 controlling 2 (or 3)
• Tables
• Parametrics
• Vectors
• Graphs: images and traces
• Transformation figures
• Line to plane (space) (Demo GS)
• Line to 2 (or 3) lines (Demo GS)
• Derivatives:
• Visualized as vectors (Demo GS)
• Visualizing components
• 2 or 3 controlling 1
• Tables
• Vectors
• Graphs
• Transformation figures (Demos GS)
• Plane to line
• 3 lines to single line.
• Derivatives:
• Linearity: Tangent planes and linear functions.
• Visualizing partial derivatives. (Demo GS)
• The Gradient and the Differential.
• The Derivative Matrix.
• 3 variables controlling 3 variables.
• (Final Demo)