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Enabling longer space missions

The 50th anniversary of the Apollo 11 moon landing has reignited interest in space travel. However, almost any mission beyond the moon, whether manned or unmanned, will require the spacecraft to remain fully operational for ...

Scientists find way to cut nanoparticle toxicity levels

Bioengineers and biophysicists from the National Research Nuclear University MEPhI, the Sechenov First Moscow State Medical University, the Universite de Reims Champagne-Ardenne in France, and the University of Tubingen in ...

Folded paper creates portable lab for field laboratory tests

Monitoring and tracking biological threats or epidemics require the ability to carry out medical and laboratory tests in the field during a disaster or other austere situations. Expensive laboratory equipment is often unavailable ...

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball or bagel. On the other hand, there are surfaces which cannot be embedded in three-dimensional Euclidean space without introducing singularities or intersecting itself — these are the unorientable surfaces.

To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide coordinates on it — except at the International Date Line and the poles, where longitude is undefined. This example illustrates that not all surfaces admits a single coordinate patch. In general, multiple coordinate patches are needed to cover a surface.

Surfaces find application in physics, engineering, computer graphics, and many other disciplines, primarily when they represent the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface.

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