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

Lecture 04 - Frenet-Serret Formula

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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