MATH 436 (Differential Geometry of Curves and Surfaces I)
| DESCRIPTION |
Starting with multi-variable calculus, this course will
develop the
theme of invariants attached to the intrinsic and extrinsic geometry of
curves and surfaces. Using local coordinates, invariants will be
defined, which will later turn out to be independent of the choice of
coordinates.
The contrasts between intrinsic and extrinsic concepts will be
emphasized.
The notion of a smooth submanifold will be explored in detail, as will
various notions of curvature. The various notions of curvature of
surfaces are related to curvature and torsion of curves. The
contrast
between local and global phenomena is also emphasized. In the
past
the course has dealt with surfaces of revolution, ruled surfaces,
minimal
surfaces, special curves on surfaces, Gauss's "Theorema Egregium" and
the
Gauss-Bonnet theorem. |
| PREREQUISITES |
MATH 241; and either MATH 240 or MATH 461 |
| TOPICS |
Curves in the plane and Euclidean space
Curvature and torsion, moving frames
Smooth surfaces in Euclidean space
Tangent spaces and normal vector fields
Orientability
First and second fundamental form
Normal curvature
Intrinsic geometry of surfaces
|
| TEXT |
Text(s)
typically used in this course. |
|
