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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · x^2 - y^2

Algebra · Expressions

Analyse x^2 - y^2

x^{2} - y^{2}

Àwọn ìṣàmúlò-ètò

x^{2} - y^{2}
An expression in x, y. Here is what it does.
\left(x - y\right) \left(x + y\right)
Factored form.

Fi àwọn àgbèwọlé hàn

x^{2} - y^{2}

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Bii o ṣe le: Surfaces and Gaussian curvature

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Differential Geometry

Curves, arc length and curvature