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Curves and Surfaces

Curves and Surfaces. Instructor: Prof. Sudipta Dutta, Department of Mathematics and Statistics, IIT Kanpur. This course is intended for undergraduate students in Indian Universities with a background in Differential Calculus of Several Variables. Such a course was broadcasted in March 2016 under MOOC (NPTEL- IV) and that background will be enough to follow that course. It is kind of a threshold level compilation of lectures to Differential Geometry on which there is hardly any standard course at under graduate level in most universities. (from nptel.ac.in)

Level Curves and Locus, Definition of Parametric Curves


Module I. Curves in R2 and R3
Lecture 01 - Level Curves and Locus, Definition of Parametric Curves, Tangent, Arc Length, Arc Length Parameterization
Lecture 02 - How Much a Curve Is Curved, Signed Unit Normal and Signed Curvature, Rigid Motions, Constant Curvature
Lecture 03 - Curves in R3, Principal Normal and Binormal, Torsion
Lecture 04 - Frenet-Serret Formula
Lecture 05 - Simple Closed Curve and Isoperimetric Inequality
Module II. Surfaces 1: Smooth Surfaces
Lecture 06 - Surfaces and Parametric Surfaces, Regular Surface and Non-example of Regular Surface, Transition Maps
Lecture 07 - Transition Maps Of Smooth Surfaces, Smooth Function Between Surfaces, Diffeomorphism
Lecture 08 - Reparameterization
Lecture 09 - Tangent, Normal
Lecture 10 - Orientable Surfaces, An Example of Non-orientable Surface
Module III. Surfaces 2: First Fundamental Form
Lecture 11 - Examples of Surfaces: Ruling Surfaces, Surfaces of Revolution
Lecture 12 - First Fundamental Form
Lecture 13 - Stereographic Projection, Conformal Mapping
Lecture 14 - Curvature of Surfaces
Lecture 15 - Euler's Theorem
Module IV. Surface 3: Curvature and Geodesics
Lecture 16 - Regular Surfaces Locally as Quadratic Surfaces, Gaussian Curvature, Mean Curvature
Lecture 17 - Geodesics, Geodesic Equations
Lecture 18 - Existence of Geodesics, Geodesics on Surfaces of Revolution
Lecture 19 - Geodesics on Surfaces of Revolution, Clairaut's Theorem
Lecture 20 - Pseudosphere, Geodesics on Pseudosphere
Lecture 21 - Classification of Quadratic Surface
Lecture 22 - Surface Area and Equiareal Map

References
Curves and Surfaces
Instructor: Prof. Sudipta Dutta, Department of Mathematics and Statistics, IIT Kanpur. This course is intended for students with a background in Differential Calculus of Several Variables.