Math 540 syllabus
Course description: Local and global theory of curves and surfaces in three-dimensional space. Topics covered are from the following: Local and global theory of curves and surfaces in three-dimensional space: curvature and torsion of curves and the Frenet frame, local change of coordinates on surfaces and differentiability of functions on surfaces, orientability of surfaces, covariant differentiation, Gauss map, first and second fundamental forms, principle curvatures, Gauss and Mean curvatures, geodesics, the exponential map, the Jacobi equation, Gauss’ Theorem Egregium, the Gauss-Bonnet Theorem, the Bonnet-Myers Theorem, brief introduction to the intrinsic and higher dimensional view-point.
Prerequisites: C or better in MA 227, C or better in MA 238 and C or better in MA 237
Suggested Textbook: Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo, Second Edition, Dover Publications, Inc.
Learning outcomes: Upon the successful completion of the course a student will:
- perform basic calculations in local coordinates,
- understand coordinate-invariance,
- recall and describe emblematic examples of surfaces with simple curvature properties,
- understand the role of geodesics in generalizing Euclidean geometry,
- have some rudimentary understanding of how curvature constricts global topology.