Curves and Surfaces (Prof. Sudipta Dutta, IIT Kanpur) | Mathematics

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.