Differential Geometry-II & Tensor Analysis
This course will enable the students to –
- Understand the role of tensors in differential geometry.
- Understand the interpretation of the curvature tensor, Geodesic curvature, Gauss and Weingarten formulae.
- Learn and apply problem-solving with differential geometry to diverse situations.
Course Outcomes (COs):
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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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. 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.
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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 |
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, 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 r ⃗=r ⃗(u,v), Single differential equation of a geodesic, Geodesic on a surface of revolution.
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).
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. 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.
- 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, 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.
