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In mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy.

1 Topology

In what follows, all surfaces are considered to be second-countable two dimensional manifolds.

There is a complete classification of closed (i.e compact without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of three infinite collections:

• Spheres with n handles attached (called n-tori). These are orientable surfaces with Euler characteristic 2-2n, also called surfaces of genus n.
• Projective planeSurfaces Geometric topology In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. It is often described intus with n handles attached. These are non-orientable surfaces with Euler characteristic 1-2n.
• Klein bottleIn mathematics, the Klein bottle is a certain non- orientable surface, i. a surface (a two-dimensional topological space), for which there is no distinction between the "inside" and the "outside" of the surface. The Klein bottle was first described in 188s with n handles attached. These are non-orientable surfaces with Euler characteristic -2n.

Therefore Euler characteristic and orientabilityIn geometry and topology, a surface in is called non-orientable if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed. Otherwise the surface is said to be orientable . Examples in low dimensions Surfaces w describe a compact surfaces up to homeomorphismThis word should not be confused with homomorphism. In topology, two geometrical objects (or "spaces") are called homeomorphic if, roughly speaking, the first can be deformed into the second by stretching and bending; cutting is also allowed, but only if (and if surfaces are smooth then up to diffeomorphismIn mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Here is definition Given two differentiable manifolds M and N a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N a).

Compact surfaces with boundary are just these with one or more removed diskA disk or disc is anything that resembles a flattened cylinder in shape. More specifically: In biology, an intervertebral disc is a cartilaginous joint between vertebrae in the spine of vertebrate animals. In mathematics, a disk is a geometrical object.s. A compact surface can be embedded in R³ if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theoremIn differential topology, the Whitney embedding theorem states that Any smooth second-countable -dimensional manifold can be embedded in Euclidean -space. The result is sharp, in particular the projective -space cannot be embedded into Euclidean -space A that any surface can be embedded in R⁴.

To make some models, attach the sides of these (and remove the corners to puncture):

This notion of a surface is distinct from the notion of an algebraic surface. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a manifold.

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Topics: Surface Surfaces Algebraic Compact Euler Characteristic...

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SurfaceIn mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the Surface waveIn physics, a surface wave is a wave that is guided along the interface between two different media for a mechanical wave, or by a refractive index gradient for an electromagnetic wave. An example is the waves at the surface of water, ocean surface waves. Surface Transportation ActThe Surface Transportation Act ( US) of 1987 was the first transportation bill of the post-interstate era. It nominally gave power to apportion money to the Secretary of Transportation. Most noticeably it allowed states to raise the speed limit to 65 mile Surface chemistrySurface chemistry is the study of chemical phenomena that occur at the interface of two phases, usually between a gas and a solid or between a liquid and a solid. One important field of surface chemical research is related to surface functionalization, wh Surface tensionIn physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. It is the effect that allows insects (such as the water strider) to walk on water, and causes capillary action, for example. Surface pressureSurface pressure is the atmospheric pressure at a point of Earth's surface. It is directly proportional to the mass of air over that point. For numerical reasons, atmospheric models such as general circulation models (GCMs) usually predict the nondimensio Surface normal Surface-to-air missile Surface caching Surface-mount technology Surface-water hydrology Surface of revolution Surface energy Surface acoustic wave Surface Marker Buoy