Geometry of Curves & Surfaces | Department of Mathematics

Geometry of Curves & Surfaces

A Major Elective for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 102 Calculus II or MAT 103 Mathematical Methods I. And MAT 160 Linear Algebra I.

Overview: This course combines the traditional approach to learn the basic concepts of curves and surfaces with the symbolic manipulative abilities of Mathematica. Students will learn and study the classical curves/surfaces as well as more interesting curves/surfaces using computer methods. For example, to see the effect of change of parameter, the student will explore and observe with the help of Mathematica and then the mathematical proof of the observation will be developed in the class.

Detailed Syllabus:

1- Curves in the plane: Length of a curve, Vector fields along curves. Famous plane curves: cycloids, lemniscates of Bernoulli, cardioids, catenary, cissoid of Diocles, tractrix, clothoids, pursuit curves.

2- Regular curve, curvature of a curve in a plane, curvature and torsion of a curve in R3. Determining a plane curve from given curvature.

3- Global properties of plane curves: Four vertex theorem, Isoperimetric inequality.

4- Curves on Sphere. Loxodromes on spheres, animation of curves on a sphere.

5- Review of calculus in Euclidean space.

6- Surfaces in Euclidean spaces: Patches in R3, local Gauss map, Regular surface, Tangent vectors.

7- Example of surfaces: Graphs of a function of two variables, ellipsoid, stereographic ellipsoid, tori, paraboloid, seashells.

8- Orientable and Non-orientable surfaces. Mobius strip, Klein Bottle.

9- The shape operator, normal curvature, Gaussian and mean curvature, fundamental forms.

10- Surfaces of revolution

 

References:

  1. Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition by Elsa Abbena, Simon Salamon, Alfred Gray.
  2. Elementary Differential Geometry by A.N. Pressley, Springer Undergraduate Mathematics Series.
  3. Differential Geometry of Curves and Surfaces by Manfredo DoCarmo.

Past Instructors: Pradip Kumar

Course Code: 
MAT332
Course Credits: 
4.00
Department: 
Course Level: