DIFFERENTIAL GEOMETRY - I

Paper Code: 
MAT124
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Unit I: 
I
15.00
Theory of curves- Space curves, Tangent, Contact of curve and surface, Osculating plane, Principal normal and Binormal, Curvature, Torsion, Serret-Frenet's formulae.
 
Unit II: 
II
15.00
Osculating circle and Osculating sphere, Existence and Uniquenss theorems, Bertrand curves, Involute, Evolutes. 
 
Unit III: 
III
15.00
Ruled surface, Developable surface, Tangent plane to a ruled surface. Necessary and sufficient condition that a surface ℑ=f () should represent a developable surface. 
 
Unit IV: 
IV
15.00
Metric of a surface, First, second and third fundamental forms. Fundamental magnitudes of some important surfaces, Orthogonal trajectories. normal curvature, Meunier's theorem,.
 
Unit V: 
V
15.00
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. Asymptotic lines. 
 
Essential Readings: 
  1. Thorpe, ‘Elementary Topics in Differential Geometry’, Second Edition, Springer
  2. Verlag, New York, (1985).
  3. J.L.Bansal & P.R.Sharma, Differential Geometry,JPH,Jaipur
  4. Raj Bali, Differential Geometry, Navkar Publication, Ajmer
 
References: 
  1. T.J.Willmore, ‘An Introduction to Differential Geometry’, Oxford University Press, London, (1972).
  2. Dirk J. Struik, ‘Lectures on Classical Differential Geometry’, Second Edition, Addison Wesley Publishing Company, London, (1961).
 
Academic Year: 
2015-2016
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