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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.

Exemple de travail: x^2 - y^2

Algebra · Expressions

Analyse x^2 - y^2

x^{2} - y^{2}

Étape par étape

  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.

Révèle la réponse
x^{2} - y^{2}

Symbols used here

\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\kappa,\ K
curvature, Gaussian curvature
How fast a curve turns; the product of a surface's principal curvatures.

How to: Surfaces and Gaussian curvature

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

Questions people ask

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.

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