maths.free › Differential Geometry › Surfaces and Gaussian curvature
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.
例題 · x^2 - y^2
ステップバイステップ
- x^{2} - y^{2}
An expression in x, y. Here is what it does.
- \left(x - y\right) \left(x + y\right)
Factored form.
答えを表示
今度はあなたの番 問題を選択し、タイプしたり描いたりしてください。 すべてのステップは絵で、答えはあなたが尋ねるまで隠されています。
ここで使用された記号
どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。
どうやって: Surfaces and Gaussian curvature
- An expression in x, y. Here is what it does.
- Factored form.
よくある質問
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.