punchcros.blogg.se - De carmo differential geometry

The ?rst stream contains the standard theoretical material on differential ge- etry of curves and surfaces. The material is given in two parallel streams. Who is who: - Riemann, - Serret, - Frenet, - Jacobi, - Gauss, - Christoffel, - Weingarten, - Caratheodory, - Meusnier, - Clairaut, - Bonnet.This concise guide to the differential geometry of curves and surfaces can be recommended to ?rst-year graduate students, strong senior students, and students specializing in geometry. Isometric transformation of catenoid to helicoid -(by Albert Chern).Minimal surfaces - (3-min slide-show by Hojoo Lee).Tractrix - Wikipedia page (scroll down for animations!).Serret-Frenet frame - Wikipedia page (scroll down!),.Soap film and Minimal surfaces (15 min video by Juan Bragado).Soap films (2 min video by Henry Segerman).Involutes of different curves (including the involute of the circle).Catenary: arhes and bridges in architecture, anchoring of marine objects.Some applications of differential geometry: Detailed Solutions for questions from Problems Class 8 (scheduled for ).Questions to discuss in revision lectures.Questions discussed in the Problems Class.There will be weekly sets of exercises stared questions to hand in on Thursdays, weeks 3,5,7,9.If you have any questions you are very welcome to ask (during the lectures, after a lecture, during office hours, in any other convenient time or via e-mail) !!! Week 19: Local and global Gauss-Bonnet theorem.

Meusnier's theorem, asymptotic curves lines of curvature.

Week 16: Curves on surfaces normal and geodesic curvatures.

Week 14: Global theorems about curvature, Theorema Egregium Christoffel symbols.

Week 12: The Weingarten map, the second fundamental form Gauss curvature and mean curvature.

Week 11: Revision of Michaelmas term isometries and conformal maps.

Week 10: Smooth maps between surfaces the Gauss map.

Week 9:Area of subsets of surfaces, calculation in terms of the coefficients of the first fundamental form, examples families of curves on surfaces.

Week 8:First fundamental form, coefficients of the first fundamental form arc length of a curve on a surface, metric, coordinate curves and angles.

Week 7: Special surfaces: surfaces of revolution, canal surfaces, ruled surfaces tangent vectors and tangent plane.

Week 6: Parametrized surfaces in Euclidean space, regular surfaces as level sets, examples change of parameters.

Open sets in R^n, smooth maps R^n -> R^m, Jacobi matrix, differential implicit function theorem.

Week 5: Local canonical form of space curves.

Fundamental theorem of local theory of space curves. Curvature and torsion for non unit speed space curves geometric meaning of curvature and torsion.

Week 4: Space curves, principal normal vector, binormal vector, curvature, torsion, Serret-Frenet equations, curvature and torsion for unit speed space curves.

Radius and center of curvature, evolute and involute of a plane curve.

Week 3: Vertices and inflection points of plane curves, four vertex theorem, fundamental theorem of local theory of plane curves.

Tangent and normal vectors, curvature of a plane curve. Week 2: Length of a curve, arc length, existence of arc length parametrization, examples.Week 1: Introduction and overview of the course, idea of curvature of a curve and surface, definition of a curve, trace, tangent vector, regular curve.Plane and space curves, arc length, tangent and normal vectors, curvature, local and global properties embedded surfaces, tangent planes, curves on surfaces intrinsic geometry of a surface, metric, length, area, first fundamental form maps between surfaces, Gauss map isometries and conformal maps, the Weingarten map, the second fundamental form, Gauss curvature and mean curvature, minimal surfaces, Theorema Egregium, Christoffel symbols, normal and geodesic curvatures, Meusnier's theorem, asymptotic curves, lines of curvature, geodesics, Clairaut's relations, global and local Gauss-Bonnet theorems. Preliminary course content (subject to change): Bolton, A First Course in Differential Geometry, in the library Do Carmo, Differential Geometry of Curves and Surfaces, in the library Will not follow any of these strictly, the material can be found in them. The lectures are based on the following books. Office hours: Tuesday 11:30 - 12:30 and by appointment Michaelmas 2019 - Epiphany 2020 Time and place:Į-mail: anna dot felikson at durham dot ac dot uk