Module-1 (14 hours): Definition of Space Curve, Arc length, Tangent, Normal, Binormal, Tangent and Osculating Plane, Principal Normal and Binormal, Curvature and Torsion, Serret-Frenet formulae, Curves on a surface, Contact between curves and Surfaces, Osculating Circle and Osculating Sphere, Tangent Surfaces, Involutes and Evolutes
Module-2 (10 hours): Intrinsic equation of space curves, Fundamental existence theorems for space curves, Surface representation, Regular and Singular points, Change, Tangent plane and normal, Surfaces of Revolution
Module-3 (12 hours): Metric on a Surface-The first fundamental form, Invariance of the metric, Direction coefficients on a Surface, Double family of curves, Isometric correspondence, Intrinsic properties, Geodesics and their Differential Equations, Canonical Geodesic equations, Geodesics on a surface of revolution, Gauss-Bonnet theorem, Second fundamental form, Gauss and Weingarten equations.

The objective of the course is to make the students go a step further from multivariable calculus.

The students can venture into the course on Differentiable Manifolds, a more abstract approcah to this study.

A. N. Pressely, Elementary Differential Geometry, Springer, 2012.

J.A. Thorpe, Elementary Topics in Differential Geometry, Springer, 2004.