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Surfaces and Gaussian curvature
Principal curvatures, Gauss's Theorema Egregium, geodesics.
At each point a surface bends most and least in two perpendicular directions; their product is the Gaussian curvature: positive on a sphere, zero on a cylinder, negative on a saddle. Gauss proved it can be measured from inside the surface. Picture it: the saddle z = x² − y², curving up one way and down the other. Think it: the same curvature, in four dimensions, is what gravity is in general relativity.
مثال عملي · x^2 - y^2
خطوة خطوة
- x^{2} - y^{2}
An expression in x, y. Here is what it does.
- \left(x - y\right) \left(x + y\right)
Factored form.
كشفت الإجابة
الآن أنت اختار مشكلة، أو طبعها أو رسمها بنفسك، كل خطوة، صورة، الإجابة مخفية حتى تسأل.
الرموز المستخدمة هنا
لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.
كيف: Surfaces and Gaussian curvature
- An expression in x, y. Here is what it does.
- Factored form.
أسئلة يطرحها الناس
What is curvature?
For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.