DIFFERENTIAL GEOMETRY-II & TENSOR ANALYSIS
- 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.
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Learning Outcomes |
Learning and teaching strategies |
Assessment |
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After the completion of the course the students will be able to: CLO42- To get introduced to geodesics on a surface and their characterization. Discuss the fundamental Theorem for regular surfaces. CLO43- To understand geodesics as distance minimizing curves on surfaces and find geodesics on various surfaces. CLO44-To be introduced to Christoffel symbols and their expression in terms of metric coefficients and their derivatives. CLO45- To Discuss Gauss Bonnet theorem and its implication for a geodesic CLO46- Understand concepts of tensor variables and difference from scalar or vector variables. CLO47- Understand the reason why the tensor analysis is used and explain usefulness of the tensor analysis. CLO48- 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 Learning activities for the students: Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical
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Presentations by Individual Student Class Tests at end of each unit. Written assignment(s) Semester End Examination |
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 on a conoidal surface, Geodesics on conicoids (Joachimsthal theorem).
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, Einstein tensor, Flat space, Isotropic point, Schur's theorem.
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- Clifford Henry Taube’s,Differential Geometry, Oxford University Press,2011.
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- Erwin Kreyszig, Differential Geometry, Dover Publishing,1991.
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- Nirmala Prakash, Differential Geometry, Tata McGraw Hill,1981.
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- D.Somasundaram, Differential Geometry, Narosa PublishingHouse, New Delhi, 2005.
- Prasun Kumar Nayak, Tensor Calculus and Differential Geometry, PHI Learning Private Limited, 2012.
