DIFFERENTIAL GEOMETRY-I
- Acquaint with the fundamentals of differential geometry primarily by focussing on the theory of curves and surfaces in three spaces.
- 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.
- Learn about tangent spaces, Surfaces, Gauss map, Geodesics on surfaces and curvature of plane curves.
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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: CLO16- 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 CLO17-:The theory of surfaces introduces the fundamental quadratic Forms of a surface, intrinsic and extrinsic geometry of surfaces, and the Gauss-Bonnet theorem. CLO18-Analyse the equivalence of two curves by applying some theorems. CLO19- Understand Gauss map-Geodesics. Express definition and parameterization of surfaces. CLO20- Integrate differential forms on surfaces.
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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 Periodic Intervals. Written assignment(s) Semester End Examination
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Theory of curves: Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and binormal, Curvature, Torsion, Serret-Frenet's formulae.
Osculating circle and osculating sphere, Existence and uniqueness theorems for space curves, Bertrand curves, Involutes, Evolutes.
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.
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.
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.
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- P.P.Gupta & 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.
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- Erwin Kreyszig, Differential Geometry, Dover Publishing,1991.
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- 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.
