المرحلة الجامعية المتقدمةGeometry & Topologyمُخطَّط له
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
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يعتمد على
عدد المتطلبات السابقة الإلزامية (حتى الأساس): 37
ستتعلّم
- 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.
تعريف
- 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.
مبرهنة
- 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.
مفاهيم خاطئة شائعة
- 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.
اكتشافات اختيارية
- Smooth manifolds The Theorema Egregium motivates intrinsic geometry, which is studied on abstract manifolds with no surrounding space.
- ← Euclidean geometry and proof
مسارات التعلّم
قراءات إضافية
- Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces (1976)