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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.

Mfano wenye matokeo · x^2 - y^2

Algebra · Expressions

Analyse x^2 - y^2

x^{2} - y^{2}

Hatua kwa hatua

  1. x^{2} - y^{2}

    An expression in x, y. Here is what it does.

  2. \left(x - y\right) \left(x + y\right)

    Factored form.

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x^{2} - y^{2}

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Jinsi ya Kufaulu: Surfaces and Gaussian curvature

  1. An expression in x, y. Here is what it does.
  2. Factored form.

Maswali ambayo watu huuliza

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.

Mengi zaidi katika Differential Geometry

Curves, arc length and curvature