Syllabus

Course Code: MMATH21-306    Course Name: Elective - I ) Differential Geometry

MODULE NO / UNIT COURSE SYLLABUS CONTENTS OF MODULE NOTES
1 Curves: Tangent, principal normal, curvature, binormal, torsion, Serret-Frenet formulae, locus of center of curvature, spherical curvature, locus of centre of spherical curvature, curve determined by its intrinsic equations, helices, spherical indicatrix of tangent, etc., involutes, evolutes, Bertrand curves.
2 Envelopes and Developable Surface : Surfaces, tangent plane, normal. One parameter family of surfaces; Envelope, characteristics, edge of regression, developable surfaces. Developables associated with a curve; Osculating developable, polar developable, rectifying developable. Two parameter family of surfaces; Envelope, characteristic points and examples.
Curvilinear Coordinates, First order magnitudes, directions on a surface, the normal, second order magnitudes, derivatives of n, curvature of normal section, Meunier’s theorem.
(Relevant portions from the books ‘Differential Geometry of Three Dimensions’ by C.E. Weatherburn)
3 Curves on a surface : Principal directions and curvatures, first and second curvatures, Euler’s theorem, Dupin’s indicatrix, the surface z = f(x, y), surface of revolution. Conjugate systems; conjugate directions, conjugate systems. Asymptotic lines, curvature and torsion. Isometric lines; isometric parameters. Null lines, minimal curves.
4 The equation of gauss and if codazzi: Gauss's formula for r11, r12, r22, Gauss characteristics equation, mainardi-Cadazzi realation, alternative expression, Bonnet's theorem, derivatives of the angel w.
Geodesics: Geodesic property, equations of geodesics, surface of revolution, torsion of a geodesic. Curves in relation to Geodesics; Bonnet’s theorem, Joachimsthal’s theorems, vector curvature, geodesic curvature, Bonnet’s formula.
(Relevant portions from the books ‘Differential Geometry of Three Dimensions’ by C.E. Weatherburn)
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