Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. Differential geometry is a vast subject. A comprehensive introduction would require prerequisites in several related subjects, and would take at least two or three semesters of courses. In this elementary introductory course we develop much of the language and many of the basic concepts of differential geometry in the simpler context of curves and surfaces in ordinary 3 dimensional Euclidean space. Our aim is to build both a solid mathematical understanding of the fundamental notions of differential geometry and sufficient visual and geometric intuition of the subject. We hope that this course is of interest to students from a variety of math, science and engineering backgrounds, and that after completing this course, the students will be in a position to (i) apply their knowledge and skills in this course to their related subjects, (ii) be ready to study more advanced topics such as global properties of curves and surfaces, geometry of abstract manifolds, tensor analysis, and general relativity.
The officially listed prerequisite is 01:640:311. But equally essential prerequisites from prior courses are Multivariable Calculus and Linear Algebra. Most notions of differential geometry are formulated with the help of Multivariable Calculus and Linear Algebra.
01:640:311, which itself requires Multivariable Calculus and Linear Algebra as prerequisites, is an important prerequisite because it helps students build mathematical maturity and gain the ability to understand, formulate and present precise mathematical concepts and proofs.
Manfredo DoCarmo, Differential Geometry of Curves and Surfaces (first edition), Prentice-Hall, 1976 (503 pp.); (ISBN# 0-13-212589-7; ISBN13: 9780132125895).
Syllabus, Spring 2008 . (for the DoCarmo text). Now obsolete.
This course is taught in the Spring semester.
Sections Taught This Semester:
- Spring 2009. Prof. Nussbaum
- Spring 2008. Prof. Han