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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.
Kugwira ntchito chitsanzo · x^2 - y^2
Gawo ndi Gawo
- x^{2} - y^{2}
An expression in x, y. Here is what it does.
- \left(x - y\right) \left(x + y\right)
Factored form.
Kusonyeza yankho
Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.
M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.
Kulembetsa KulowaZithunzi
Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.
Momwe mungagwiritsire ntchito: Surfaces and Gaussian curvature
- An expression in x, y. Here is what it does.
- Factored form.
Mafunso omwe anthu amafunsa
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.