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Introduction to Differential Geometry 1

Walter Seaman
Organization: University of Iowa
Department: Mathematics
URL: http://www.math.uiowa.edu/~seaman/
Education level


In this course we will study geometric quantities defined for curves and surfaces both in Euclidean space and also from the "intrinsic" viewpoint. We will study the curvature and torsion functions for curves. We study the Gauss map, Gauss, mean and principal curvatures for surfaces in space, and Gauss curvature for abstractly defined surfaces. We also study "extremal" objects, such as distance and energy minimizing curves on surfaces, and area minimizing surfaces. The course web page includes an example of such an area-minimizing surface (the "catenoid"). The course will conclude with various forms of the Gauss-Bonnet theorem. These ideas and many techniques from Differential Geometry have applications in Physics, Chemistry,

Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., by Alfred Gray

The modus operandi for the Spring 2000 course was that students downloaded Mathematica notebooks containing information and exercises about geometric topics from the course homepage. They modified those notebooks by typing in answers to exercises, and then uploaded the modified files back to the homepage for grading and comments. Each student also wrote two team written term paper projects (one on curves and one on surfaces). For these, students were allowed to examine, download and copy template Mathematica term papers which we had written. There were explicit instructions about which parts of the templates they could use for their own term papers (mostly Mathematica programs), and how significant modifications had to be made to other parts (mostly our geometric observations about the curves or surfaces under investigation).

We used Mathematica for most numerical, algebraic and multivariable Calculus based computations, numerical differential equation solving, graphing with colorings and animations, and other programming, to investigate geometric topics. The course covered traditional material with greater student involvement because computing made difficult examples and computations accessible, but use of computing also allowed us to easily investigate topics which previously would have been impossible to study without extensive and highly time-consuming hand computations.

We constructed thirteen Mathematica notebooks, using Alfred Gray's Mathematica Differential Geometry programming as the starting point. These programs form a cornerstone of Gray's book Modern Differential Geometry of Curves and Surfaces using Mathematica, Second Edition, which was the course text. Each Mathematica notebook focused on one geometric topic. The Mathematica programs needed for the investigation of that topic were all included in that notebook. Each notebook contains explanatory background notes on the topics under investigation, the Mathematica programming used in that investigation, explanations of how the programming is used, references to Gray's text, and exercises. The exercises have students i. write up Mathematical-analytical observations, proofs and speculations, ii. modify Mathematica programming to investigate examples not covered in the notebook, iii. explain what various Mathematica based Differential Geometry programs do and how these are used in exploring Differential Geometric topics.

In the two team written term papers students investigated geometric and analytic attributes first (after the first month) for a family of curves and then second (at the end of the course), the corresponding attributes for surfaces. Mathematical-analytical observations, explanations, and Mathematica programming were used to investigate these attributes.

We assumed that the students in the course had not used Mathematica before. The first homework assignment in the Spring 2000 course was a Mathematica notebook which led students through a series of examples showing how Mathematica does numerical computation, algebraic manipulation, Calculus and differential equations symbolic and numerical solving, and one, two and three dimensional graphics (including colorings and animations).

The spring 2000 course also included a one hour lecture three times per week for the fifteen week semester. In these lectures we often presented a traditional covering of the course topics on a blackboard. But we also included in roughly half of the classes, a five to fifteen minute presentation on computer computations and graphics used in studying particular geometric topics. The computer presentations (using a ceiling-mounted computer projection device) were taken from the Mathematica notebooks which students had to study and work on for homework problems.

In the Spring 2000 Introduction to Differential Geometry 1 course, we used Mathematica and web materials to cover the traditional topics in beginning Differential Geometry: the study of the geometric attributes of curves in the plane, curves in space, and surfaces. However, the presentation and delivery of the materials, the work students did with those materials, was very different from traditional approaches. The technological component of the course also allowed students to investigate geometric topics which were genuinely inaccessible in traditional formats of such a course. Students' grades in this course were based entirely on their work on computer files. We weighted the grades as follows: 60% answering homework questions in Mathematica notebooks, 20% for each of their two Mathematica generated team-written term papers, which included programming used to investigate geometric attributes of curves and surfaces.

All the course materials were posted on the course homepage. In particular, the course materials were and still are already accessible in remote or 'distance' format. Anyone who has a web browser and a copy of Mathematica can do all the course work for the course as we ran it in spring 2000. We invite reviewers to examine these web materials. Simply log in to the course homepage, click on the 'login' button at the bottom of this page, and using the username 'g0uest' and password 'guest' one goes to the main course page. Once on the main course page, the Mathematica notebook assignments can been seen in the document brought up by clicking on the "Assignments" button.

Our experience with this course was very positive. It was, for us, the most interesting version of Differential Geometry which we had ever taught. The student reaction to this course was very positive, too. From undergraduates to graduate students in Mathematics and Engineering, they seemed to have little difficulty using the internet or Mathematica components of the course. The team projects went smoothly, too. There were some really nice results (copies on request). Students seemed to genuinely appreciate the insights provided by Mathematica's graphical and computational capabilities. Examples of some of the graphics used in teaching Differential Geometric topics in the course with Mathematica are assembled on the "Differential Geometry Images" page which is linked to our homepage http://www.math.uiowa.edu/~seaman/.

  • Plane Curves
  • Mathematica Introduction
  • Plane Curves with Mathematica
  • Plane Curve Constructions
  • Curves in 3D
  • Space Curve Constructions
  • Multi Variable Calculus
  • Surfaces
  • Examples
  • Metrics (first fundamental form, induced metric)
  • Surfaces in 3D (Gauss map, shape operator, curvatures)
  • Surfaces with Mathematica
  • Surfaces of revolution, surfaces of constant curvature
  • Intrinsic Geometry
  • Geodesics
  • Gauss-Bonnet

*Mathematics > Calculus and Analysis > Differential Geometry