Syllabus
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Course Code: Elective 1 MMATH21 -306 Course Name: Differential Geometry |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 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. |
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| 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 equations of Gauss and of Codazzi: Gauss’s formulae for ࢘ଵଵ, ࢘ଵଶ, ࢘ଶଶ, Gauss characteristic
equation, Mainardi-Codazzi relations, alternative expression, Bonnet’s theorem, derivatives of
the angle ߱. 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. |
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