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Math 323, Differential Geometry, Fall 2002

in 154 Henry, MWF at 10am (sect C1).
George Francis, 344 Illini Hall,
333-4794; mailbox in 250 Altgeld
Office hours:
Elementary Differential Geometry (second edition), Barrett O'Neill.
The prerequisite is one or more 200-level calculus course, such as 242, 243, 280, or 285.
There will be frequent homework assignments, due on Wednesdays, often with problems assigned from the textbook. Students are encouraged to keep a journal and complete a project. The homework, project, and general class participation counts for 50% of the course grade.
There will be an test in September, in class, and a take-home test in November. The material covered and dates will be announced later. These tests count for 20% of your grade. The final exam covers the entire course, and counts for 30% of the course grade. The final exam will be at the time assigned by the timetable.
This course is an introduction to the geometry of differentiable curves and surfaces in ordinary 3D-space. We look for properties of their shape, such as their curvature, which are independent of how the object is parametrized by explicit functions, or where it is placed in space. For this purpose we present a thorough treatment of 3D-vector calculus, implicit function theorems, the Frenet-Serret theory of moving frames, sectional, mean and Gaussian curvatures for surfaces. Students are encouraged to explore applied topics of special interest to them further. Possible topics include camera paths, splines for curves and surfaces, fairness and other curvature related attributes of industrial interest.

The high point of the course is the Gauss-Bonnet theorem. The Gauss-curvature is intrinsic to the surface, in the sense that bending a surface without stretching (roll a sheet of paper into a cylinder) does not change it. Moreover, the integral of the Gauss curvature over the entire surface is invariant even if we stretch and distort the surface topologically.