DIFFERENTIAL GEOMETRY-II & TENSOR ANALYSIS

Paper Code: 
MAT 224
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

Paper Code

Paper Title

MAT224

Differential Geometry-II & Tensor Analysis (Theory)

The students will be able to –

 

CO45: To get introduced to geodesics on a surface and their characterization. Discuss the fundamental Theorem for regular surfaces.

CO46: To understand geodesics as distance minimizing curves on surfaces and find geodesics on various surfaces.

CO47: To be introduced to Christoffel symbols and their expression in terms of metric coefficients and their derivatives.

CO48: To Discuss Gauss Bonnet theorem and its implication for a geodesic

CO49: Understand concepts of tensor variables and difference from scalar or vector variables.

CO50: Understand the reason why the tensor analysis is used and explain usefulness of the tensor analysis.

CO51: Derive base vectors, metric tensors and strain tensors in an arbitrary coordinate system.

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 surface  , Single differential equation of a geodesic, Geodesic on a surface of revolution, Geodesic on a conoidal surface, Geodesics on conicoids  (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.
  • Clifford Henry Taube’s, Differential Geometry, Oxford University Press, 2011.
  • Dirk J. Struik, Lectures on Classical Differential Geometry, Addison Wesley Publishing Company, London, 1961.
  • H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha sahitya Prakashan, 2012.
  • Erwin Kreyszig, Differential Geometry, Dover Publishing,1991.
  • 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.
  • Prasun Kumar Nayak, Tensor Calculus and Differential Geometry, PHI Learning  Private Limited, 2012
Academic Year: 
2020-2021
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