Differential Geometry-II & Tensor Analysis

Paper Code: 
MAT224
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Understand the role of tensors in differential geometry.
  2. Understand the interpretation of the curvature tensor, Geodesic curvature, Gauss and Weingarten formulae.
  3. Learn and apply problem-solving with differential geometry to diverse situations.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

MAT224

Differential Geometry-II & Tensor Analysis (Theory)

The students will be able to –

 

CO54: Know the Interpretation of the curvature tensor, Geodesic curvature, Gauss and Weingarten formulae.
CO55: Apply problem-solving with differential geometry to diverse situations in physics, engineering and in other mathematical contexts.
CO56: Understand the role of Gauss’s Theorem Egregious and its consequences.

CO57: Explain the basic concepts of tensors

CO58: Know the role of tensors in differential geometry.

CO59: Learn various properties of curves including Frenet Serret formulae and their applications.

 

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
Asymptotic lines: Definition, Differential equation of a asymptotic lines, Theorems on asymptotic lines, Curvature and torsion of an asymptotic line, Gauss's formulae, Gauss's characteristic equation, Weingarten equations, Mainardi-Codazzi equations, Fundamental existence theorem for surfaces.
 
 
 
Unit II: 
II
15.00
Parallel surfaces, Gaussian and mean curvature for a parallel surface, Bonnet's theorem on parallel surfaces, Geodesics: Definition, General differential equation of a geodesic on a surfacer r'=r'(u,v), Single differential equation of a geodesic, Geodesic on a surface of revolution, Geodesic on a conoidal surface, Geodesics on conoids (Joachimsthal theorem).
 
 
 
Unit III: 
III
15.00
Geodesic curvature, Geodesic curvature in form of Gauss coefficient, Bonnet’s formula for Geodesic curvature and torsion of a Geodesic, Normal angle, Geodesic torsion, Gauss-Bonnet Theorem (Joachimsthal theorem).
 
Unit IV: 
IV
15.00
Tensor Analysis: Definition, Kronecker delta, Symmetric tensor, Skew Symmetric tensor, Quotient law of tensor, Relative tensor, Metric tensor, Indicator, Permutation symbols and Permutation tensor, Christoffel symbols and their properties, Covariant differentiation of tensor, Ricci's theorem.
 
Unit V: 
V
15.00
Intrinsic derivative, Differential equation of geodesic of a metric, Geodesic coordinates, Reimann-Christoffel tensor and its properties, Covariant curvature tensor, Einstein space, Bianchi's identity, Einstein tensor, Flat space, Isotropic point, Schur's theorem.
 
Essential Readings: 
  • J.L. Bansal, Differential Geometry, Jaipur Publishing House, Jaipur, 2014.
  • J.L. Bansal, Tensor Analysis, Jaipur Publishing House, Jaipur, 2012.
  • P.P. Gupta and G.S. Malik, Differential Geometry, Pragati Prakashan, 2012.
  • Raj Bali, Tensor Analysis, Navkar Publication, Ajmer, 2012.
References: 
  • Clifford Henry Taube’s, Differential Geometry, Oxford University Press, 2011.
  • Dirk J. Struik, Lectures on Classical Differential Geometry, Addison Wesley Publishing Company, London, 2003.
  • H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha sahitya Prakashan, 2012.
  • Erwin Kreyszig, Differential Geometry, Dover Publishing, 2003.
  • B.D. Neill, Elementary Differential Geometry, Academic Press, London, 2006.
  • 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.
  • Prasun K. Nayak, Tensor Calculus and Differential Geometry, PHI Learning Private Limited, 2012.
 
Academic Year: 
2022-2023
  • Printer-friendly version
  • PDF version