Differential Geometry-I
This course will enable the students to –
- Acquaint with the fundamentals of differential geometry primarily by focusing 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 curve.
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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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24MAT 124 |
Differential Geometry-I (Theory) |
CO19: Compute quantities of geometric interest such as curvature and torsion and comprehend their concepts for curves and surfaces in space. CO20: Develop arguments in the geometric description of curves and surfaces to establish basic properties of the osculating circle and osculating sphere. CO21: Determine and calculate envelopes and the edge of regression of curves. Analyze the metrics of a surface. CO22: Explain the fundamental magnitudes of some important surfaces and analyze mean and Gaussian curvature. CO23: Explore the radius of curvature and curvature of normal sections and develop the relation between fundamental forms. CO24: Contribute effectively in course-specific interaction. |
Approach in teaching: Interactive Lectures, Discussion, Informative videos
Learning activities for the students: Self learning assignments, Effective questions, Topic presentation, Assigned tasks |
Quiz, Class Test, Individual projects, Open Book Test, Continuous Assessment, 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, Gaussian curvature, Umbilics.
Radius of curvature of any normal section at an umbilic on , Radius of curvature of a given section through any point on Lines of curvature, Principal radii, Relation between fundamental forms, Curvature of the normal section.
- J.L. Bansal and P.R. Sharma, Differential Geometry, Jaipur Publishing House Jaipur, 2013.
- P.P. Gupta and 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.
SUGGESTED READING
- T. J. Willmore, An Introduction to Differential Geometry, Oxford University Press, London, 1997.
- Erwin Kreyszig, Differential Geometry, Dover Publishing, 2003.
- H.K. Pathak and J.P. Chauhan, Differential Geometry, Shiksha Sahitya Prakashan, 2012.
- Clifford Henry Taube’s, Differential Geometry, Oxford university press, 2011.
- B. D. Neill, Elementary Differential Geometry, Academic Press, London, 2006.
- Millan and G. D. Parker, Elements of Differential Geometry, PHI, 1977.
- D. Somasundaram, Differential Geometry, Narosa Publishing House, New Delhi, 2005.
e- RESOURCES
- http://etananyag.ttk.elte.hu/FiLeS/downloads/_01_Csikos_Differential_geometry.pdf
- https://nptel.ac.in/courses/111104095
- https://www.mathcity.org/msc/notes/differential-geometry-kaushef_salamat
- https://www.pdfdrive.com/elementary-differential-geometry-d185499035.html
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