Schedule of Classes Course Description

Medgar Evers College
Course Prefix: MTH                                     Course Number: 324                                    
Course Title: Introduction to Differential Geometry
Subject: Mathematics
Minimum Credits: 3.0                                     Maximum Credits: 3.0                                     Hours per week: 3.0


This course is designed to provide students in the Mathematical Sciences Program with an introduction to the classical (local) differential geometry of curves and surfaces in R3 using vector methods. The concepts of arc length, curvature, torsion along with the fundamental systems of basic unit vectors and the associated lines and planes will be discussed. The Serret-Frenet formulas and their application and the moving trihedron will be investigated in detail. The representation problem in terms of the natural parameter (arc lengths) and the general theory of smooth space (twisted or gauche) curves will be emphasized, as will the representation problem and elementary theory of smooth surfaces embedded in Euclidean space. The First and Second Fundamental Forms will be presented and the various curves on embedded surfaces (such as lines of curvature, asymptotic lines, and directions) will be discussed, as will Meusnier's theorem, Euler's theorem and the Dupin indicatrix. Elementary principles and methods of the tensor calculus will be introduced as a means of investigating the Fundamental Theorem of Surface Theory, the Gauss-Weingarten equations, and the Mainardi-Codazzi equations. The Theorema Egrigium of Gauss will be discussed, as will the concepts of geodesics and geodesic coordinates. The course will conclude with an analysis of the classical Gauss-Bonnet Theorem and its implications.

Prerequisite: MTH 204 & MTH 207 or MTH 320 or permission from chairperson of the department of mathematics.
Start Date:    09/27/2006                         End Date: