DIFFERENTIAL GEOMETRY-I

Paper Code: 
MAT 124
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Acquaint with the fundamentals of differential geometry primarily by focusing on the theory of curves and surfaces in three spaces. 
  2. Compute quantities of geometric interest such as curvature, as well as develop a facility to compute in various specialized systems, such as semi geodesic coordinates or ones representing asymptotic lines or principal curvatures. 
  3. Learn about tangent spaces, Surfaces, Gauss map, Geodesics on surfaces and curvature of plane curve.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

MAT124

Differential Geometry-I (Theory)

The students will be able to –

 

CO16: The theory of curves studies global properties of curves such as the four vertex theorem. Study the concept of Curvature of plane curves and surface

CO17: The theory of surfaces introduces the fundamental quadratic Forms of a surface, intrinsic and extrinsic geometry of surfaces, and the Gauss-Bonnet theorem.

CO18: Analyse the equivalence of two curves by applying some theorems.

CO19: Understand Gauss map-Geodesics. Express definition and parameterization of surfaces.

CO20: Integrate differential forms on surfaces.

 

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching, PowerPoint presentations.

 

Learning activities for the students:

Self learning assignments, Effective questions, Seminar presentation, Posters and Charts preparation.

Class test, Semester end examinations, Quiz, Assignments, Presentation, Individual

 

 

Unit I: 
I
15.00

Theory of curves: Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and binormal, Curvature, Torsion, Serret-Frenet's formulae.

Unit II: 
II
15.00

Theory of curves: Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and binormal, Curvature, Torsion, Serret-Frenet's formulae.

Unit III: 
III
15.00
Envelops and edge of regression, Ruled surfaces, Developable surfaces, Tangent plane to a ruled surface, Necessary and sufficient condition that a surface ζ=f (ξ,η) should represent a developable surface, Metric of a surface: First, second and third fundamental forms.
 
Unit IV: 
IV
15.00
Fundamental  magnitudes of some important surfaces, Orthogonal  trajectories, normal curvature, Meunier's theorem, Principal directions and principal curvatures, First curvature, Mean curvature, Gaussion  curvature, Umbilics.
 
Unit V: 
V
15.00

Radius of curvature of any normal section at an umbilic on z = f (x,y), Radius of curvature of a given section through any point on z = f(x,y), Lines of curvature, Principal radii, Relation between fundamental forms, Curvature of the normal section.

Essential Readings: 
 
  • J.L. Bansal and P.R. Sharma, Differential Geometry, Jaipur Publishing House Jaipur, 2013.
  • P.P. Gupta and G.S. Malik, Differential Geometry, Pragati Prakashan, Meerut, 2012.
  • Prasun Kumar Nayak, Tensor Calculus and Differential Geometry, PHI Learning Pvt. Ltd., 2012.
  • Raj Bali, Differential Geometry, Navkar Publication, Ajmer, 2012.
  • T. J. Willmore,   An Introduction to Differential Geometry, Oxford University Press, London, 1972.
  • Dirk J. Struik, Lectures on Classical Differential Geometry, Addison Wesley Publishing Company, London, 1961.
  • Erwin Kreyszig, Differential Geometry, Dover Publishing, 1991.
  • H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha Sahitya Prakashan, 2012.
  • Clifford Henry Taube’s, Differential Geometry, Oxford university press, 2011.
  • B. D. Neill, Elementary Differential Geometry, Academic Press, London, 1996.
  • Nirmala Prakash, Differential Geometry, Tata McGraw Hill, 1981.
  • Millan and G. D. Parker, Elements of Differential Geometry, PHI, 1977.
  • D. Somasundaram, Differential Geometry, Narosa Publishing House, New Delhi, 2005.
References: 
 

 

Academic Year: 
2020-2021
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