Math 423/673 has two main aims: to understand the general version of the Fundamental Theorem of Calculus using differential forms and to understand Gauss curvature.
In Math 251 you meet four flavours of the Fundamental Theorem of Calculus: the FTC for curves in space, Green's Theorem, Stokes' Theorem, and the Divergence Theorem. In Math 423 we will learn how these four theorems are special cases of a more general result, which is easier to state than any one of them. (One you know what a differential form is!)
In the part of the course that focuses on curvature, we will study two remarkable properties of Gauss curvature. First, if you look at a surface such as a sphere or a torus (donut) in three dimensional space you can see how curved it is at each point. The Gauss curvature quantifies this concept by calculating how much the unit normal vector to the surface is changing direction at each point. In 1828, Gauss made the remarkable discovery that you can compute curvature simply by making measurements on the surface: You don't need to know how the surface is embedded in space, you just need to be able to compute distances and angles on the surface! The second remarkable property of Gauss curvature is that the total Gauss curvature, which is the integral of the curvature over the surface, is a topological invariant: If you deform the surface, but don't tear it, the total Gauss curvature doesn't change. This result is called the Gauss-Bonnet Theorem.
The major emphases of the course will be to solidify your understanding of Vector Calculus, to teach you to work with differential forms, to learn to compute curvature analytically, and to understand the big theorems discussed above. If you liked the sorts of calculations you do in Multivariable calculus, you'll love the Differential Geometry.
In addition to being an area of active research in its own right, Differential Geometry also has applications in the fields of biomedical imaging, computer graphics, computer vision, geometric design, scientific visualization, physics (eg general relativity), control theory, and optimization.
The text for the course will be "Elementary Differential Geometry" by Barrett O'Neill.
The prerequisites are Math 221 (Introduction to Linear Algebra) and Math 251 (Multivariable Calculus). In addition for general "mathematical maturity" it would be helpful, though not essential, if you have taken at least one 300 level Math course.
Questions? Contact me at zweck@umbc.edu
Last modified Oct 14th, 2010