Shahada ya kwanza (juu)Geometry & TopologyImepangwa
Differential geometry of curves and surfaces
Curvature and torsion of space curves, the first and second fundamental forms of surfaces, Gaussian and mean curvature, Gauss's Theorema Egregium, geodesics and the Gauss–Bonnet theorem.
math.geom.curves-surfaces
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 37 (hadi chini kabisa)
Utajifunza
- Compute curvature and torsion using the Frenet–Serret frame.
- Compute the first and second fundamental forms and the Gaussian curvature of standard surfaces.
- State the Theorema Egregium and the Gauss–Bonnet theorem and check them on spheres and tori.
Fasili
- The curvature of a unit-speed curve γ is κ = |γ″|.
- The Gaussian curvature K of a surface at a point is the product of its principal curvatures.
- A geodesic on a surface is a curve whose acceleration is everywhere normal to the surface.
Nadharia
- Theorema Egregium: Gaussian curvature depends only on the first fundamental form, so it is intrinsic.
- Gauss–Bonnet: for a compact oriented surface S without boundary, ∫_S K dA = 2πχ(S).
- No map from an open region of a sphere to the plane preserves all distances, because their Gaussian curvatures differ.
Mawazo potofu ya kawaida
- Thinking a cylinder is curved in the Gaussian sense; its Gaussian curvature is zero.
- Confusing the curvature of a curve with the Gaussian curvature of a surface.
Ugunduzi wa hiari
- Smooth manifolds The Theorema Egregium motivates intrinsic geometry, which is studied on abstract manifolds with no surrounding space.
- ← Euclidean geometry and proof
Njia za kujifunza
Usomaji zaidi
- Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces (1976)