Welcome to Prof. Jason Osborne’s Mathematica Demo Central...In this Slide Show you will find many demos related to Differential Geometry that you can explore. The companion contact sheet (Mathematica Demo Central Contact Sheet (.pdf)) shows a list of all the demos that are available to you.

Directions: You can view this Slide Show in full screen modeby following Palettes → Slide Show → Start Presentation (see below screen shot)

(1) You can tab through the Slides using the navigation bar in the slide show navigator .

(2) After selecting your demo, select the cell right below the com-ment (*main code hiding below*) which looks and evaluate it using

"#(3) You will then be free to explore any interactive graphics that areavailable to you.

Welcome to Prof. Jason Osborne’s Mathematica Demo Central...In this Slide Show you will find many demos related to Differential Geometry that you can explore. The companion contact sheet (Mathematica Demo Central Contact Sheet (.pdf)) shows a list of all the demos that are available to you.

Directions: You can view this Slide Show in full screen modeby following Palettes → Slide Show → Start Presentation (see below screen shot)

(1) You can tab through the Slides using the navigation bar in the slide show navigator .

(2) After selecting your demo, select the cell right below the com-ment (*main code hiding below*) which looks and evaluate it using

"#(3) You will then be free to explore any interactive graphics that areavailable to you.

2 DifferentialGeometryDemos.nb

Slide 1 of 7 (Welcome to Mathematica)Description: In this demo, some quick facts and examples are given to introduce you to Mathematica.

Quick Facts:* Site license for student and faculty personal machines:(http://support.appstate.edu/software/mathematica-wolfram)

* Hands-On Start to Mathematica video tutorials:(http://www.wolfram.com/broadcast/screencasts/handsonstart)

* Mathematica Demo Central written tutorials: (http://mathsci2.appstate.edu/~osbornejm/CourseLinks.html → MathematicaDemoCentral)

Quick Examples (screen shot):

DifferentialGeometryDemos.nb 3

Slide 2 of 7 (Differential Geometry) Space Curve with Velocity and AccelerationDescription: For a defined vector-valued function in one variable,explore the graphical representation of the “velocity” and “acceleration”as vectors at a point (or arrows).

4 DifferentialGeometryDemos.nb

Slide 3 of 7 (Differential Geometry) Space Curve with unit Tangent and NormalDescription : For a defined vector - valued function in one variable,explore the graphical representation of the unit "Tangent" and "Normal"as vectors at a point (or arrows) and their relationship to “Curvature”

DifferentialGeometryDemos.nb 5

Slide 4 of 7

(Differential Geometry) Curvature of Curve on 2D Surface: (Euler, circa 1760) and (Meusnier, 1776) Description : For an explicit function in two variables, exploreEuler’s result relating the curvature of a curve on a surfaceto the curvatures in the “principal” directions.

6 DifferentialGeometryDemos.nb

Slide 5 of 7 (Differential Geometry) Coordinate Lines and Directional Derivative of f :ℝ2 → ℝDescription: For a defined function of 2 variables, explore the concept of coordinate curves,curve in a defined direction, and the directional derivative.

DifferentialGeometryDemos.nb 7

Slide 6 of 7 (Differential Geometry) Directional Derivative of f :ℝ3 → ℝ

8 DifferentialGeometryDemos.nb

Slide 7 of 7 (Differential Geometry) Chain Rule and Jacobian Dfwhere f :ℝ2 → ℝ3

Description: For a defined parametric surface and coordinate curve, explore the Jacobian as a crucial component of the Chain Rule.

where f :ℝ2 → ℝ3

DifferentialGeometryDemos.nb 9 10 DifferentialGeometryDemos.nb