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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)

Lecture 18 - Existence of Geodesics, Geodesics on Surfaces of Revolution

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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