- Textbook:
- Elementary Differential Geometry
(second edition), Barrett O'Neill.
- Prerequisites:
-
The prerequisite is one or more 200-level calculus course, such
as 242, 243, 280, or 285.
- Homework:
- 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.
- Exams:
- 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.
- Outline:
-
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.
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Last edited 30aug02
gfrancis@uiuc.edu