Curves and surfaces
in 3-space, tangent vectors, normal vectors, curvature,
introduction to topology, manifolds, tangent spaces, multilinear
algebra and tensors. Normally offered in alternate years.
Differential geometry uses the methods of multivariable calculus and
linear algebra to study curves, surfaces, and higher-dimensional
"manifolds". The subject was initiated by Gauss, further developed by
Riemann, and has seen important advances in this century due to Cartan
and Lie. The concept of a manifold and the geometric structures
associated to it play central roles in such fields as dynamical
systems, topology, harmonic analysis and differential equations. Many
basic ideas are involved in differential geometry: how can space be
curved, what is the relationship between local and global information,
why don't you always end up where you started when going around a loop?
Differential geometry is now a central tool in theoretical physics;
this century has seen the "geometrization of physics" with significant
applications to the fields of general relativity , classical mechanics
and elementary particle theory.
The first part of the course will study surfaces, both in three- and
higher-dimensional spaces, using as its main tool vector fields on these
surfaces. We will treat geodesics, parallel transport and curvature. We
then discuss differential forms and a general form of Stokes's Theorem.
This will lead to a significant and beautiful result, the Gauss-Bonnet
Theorem, relating local geometric to global topological information. It
is a prototype of many important modern results. The later part of the
course will be an introduction to differentiable manifolds.
The course will be accessible to students with a background in
vector calculus and linear algebra. Some knowledge of differential
equations will also be useful. In fact, the course may be seen as
putting the finishing touches on these courses, unifying them and
showing their significant applications.
The grade will be based on term tests and assignments (70%) and a final
project (30%). The text has not been chosen.
Prerequisite:AS/SC/AK/MATH 3010 3.0;
AS/SC/MATH 2022 3.0 or AS/SC/AK/MATH 2222 3.0;
or permission of the course coordinator.
Exclusion:AS/SC/MATH 3450 3.0.
Coordinator: S. Scull