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1 2 DifferentialGeometryDemos.nb Calculus III Demos Table of Contents (ToC) : by Prof. Jason Osborne * Mathematica Demo Central (Welcome) * Mathematica Demo Central (Quick Facts) * Check Velocity and Acceleration * Unit Tangent and Unit Normal Vectors T, N and Curvature, κ * Curve Curvature Results (Euler and Meusnier) * Directional Derivative f : R 2 R * Directional Derivative f : R 3 R * Chain Rule and Jacobian * Spherical and Cylindrical Frame Field * Spherical and Cylindrical CoFrame Field * Adapted Frame and CoFrame to Surface (!! Maple tensoraddons dependency) * Frame and CoFrame (!! Maple tensoraddons dependency) * Vectors and (1,0)-tensors * Parallel Transport Equations (!! Maple tensoraddons dependency) Welcome to the Mathematica Demo Central (Differential Geometry) Demos written by Prof. Jason Osborne Directions: You can view this Slide Show in full screen mode by 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 comment (*main code hiding below*) which looks and evaluate it using $% (3) You will then be free to explore any interactive graphics that are available to you.

2 DifferentialGeometryDemos.nb 3 4 DifferentialGeometryDemos.nb Quick Facts and Examples: * Site license for student and faculty personal machines: ( * (Video Tutorials) Hands-On Start to Mathematica: ( (Differential Geometry) Space Curve with Velocity and Acceleration Description: 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). * Sample Code and Basic Syntax (screen shot):

3 DifferentialGeometryDemos.nb 5 6 DifferentialGeometryDemos.nb (Differential Geometry) Space Curve with unit Tangent and Normal Description : 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 (Differential Geometry) Curvature of Curve on 2D Surface: (Euler, circa 1760) and (Meusnier, 1776) Description : For an explicit function in two variables, explore Euler s result relating the curvature of a curve on a surface to the curvatures in the principal directions.

4 DifferentialGeometryDemos.nb 7 8 DifferentialGeometryDemos.nb (Differential Geometry) Coordinate Lines and Directional Derivative of f : R 2 R Description: For a defined function of 2 variables, explore the concept of coordinate curves, curve in a defined direction, and the directional derivative. (Differential Geometry) Directional Derivative of f : R 3 R

5 DifferentialGeometryDemos.nb 9 10 DifferentialGeometryDemos.nb (Differential Geometry) Chain Rule and Jacobian Df where f : R 2 R 3 Description: For a defined parametric surface and coordinate curve, explore the Jacobian as a crucial component of the Chain Rule. (Differential Geometry) Spherical and Cylindrical Frame Field Description: Select from Spherical or Cylindrical coordinates to explore these frame fields.

6 DifferentialGeometryDemos.nb DifferentialGeometryDemos.nb (Differential Geometry) Spherical and Cylindrical Co-Frame Field Description: Select from Spherical or Cylindrical coordinates to explore these co-frame fields. (Differential Geometry) Adapted Frame and Co-Frame on Surface Note: There is a dependency in this demo on the Maple code found in the tensoraddons package. Description: After compiling Maple tensoraddons code for SetupSurfaceAdaptedFrame Example 2, explore the relationship of the surface adapted frame to its co-frame.

7 DifferentialGeometryDemos.nb DifferentialGeometryDemos.nb (Differential Geometry) Frame and Co-Frame Note: There is a dependency in this demo on the Maple code found in the tensoraddons package. Description: After compiling Maple tensoraddons code for SetupFrame Example 3, explore the relationship between the frame and its co-frame. (Differential Geometry) Vectors are (1,0)- Tensors Description: Show that a vector can be (a) viewed as a 1D storage device and (b) transforms appropriately. Since both (a) and (b) are satisfied then we can view vectors as a (1,0)- tensor.

8 DifferentialGeometryDemos.nb DifferentialGeometryDemos.nb (Differential Geometry) Acceleration Decomposition (ToDo...see Maple code tensoraddons for the time being) (Differential Geometry) Transport Equations Note: There is a dependence in this code on the Maple demo GeodesicAndTransportEquations of tensoraddons. Run the Maple demo first to generate the necessary starting data for this Mathematica demo which then handles all of the plotting. Description: For a given 2D or 3D coordinate curve and with 3 or 4 ICs, respectively, explore the Transport Equation solutions.

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