Information AboutSurface |
| CATEGORIES ABOUT SURFACE | |
| surfaces | |
| geometric topology | |
| mathematical theorems | |
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In Mathematics , specifically in Topology , a surface is a Two-dimensional Manifold . The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean Space , '''E'''3. On the other hand, there are also more exotic surfaces, that are so "contorted" that they cannot be Embedded in three-dimensional space at all. 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 in general it is not possible to extend any one coordinate patch to the entire surface; surfaces, like manifolds of all dimensions, are usually constructed by patching together multiple coordinate systems. 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. DEFINITIONS AND FIRST EXAMPLES A ''(topological) surface with boundary'' is a Hausdorff Topological Space in which every point has an open Neighbourhood Homeomorphic to some Open Subset of the closed half space of E2 (Euclidean 2-space). The neighborhood, along with the homeomorphism to Euclidean space, is called a ''(coordinate) chart''. The set of points that have an open neighbourhood homeomorphic to E2 is called the ''interior'' of the surface; it is always non- Empty . The Complement of the interior is called the ''boundary''; it is a one-manifold, or union of closed curves. The simplest example of a surface with boundary is the closed Disk in E2; its boundary is a circle. A surface with an empty boundary is called ''boundaryless''. (Sometimes the word surface, used alone, refers only to boundaryless surfaces.) A ''closed'' surface is one that is boundaryless and Compact . The two-dimensional sphere, the two-dimensional Torus , and the Real Projective Plane are examples of closed surfaces. The Möbius Strip is a surface with only one "side". In general, a surface is said to be ''orientable'' if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because deleting a point or disk from the real projective plane produces the Möbius strip). More generally, it is common in Differential and Algebraic Geometry to study surfaces with Singularities , such as self-intersections, cusps, etc. EXTRINSICALLY DEFINED SURFACES AND EMBEDDINGS | ||
| Image:ProjectivePlaneAsSquaresvg | "http://wwwinformationdelightinfo/information/entry/real_projective_plane" class="copylinks">Real Projective Plane |
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| Image:KleinBottleAsSquaresvg | "http://wwwinformationdelightinfo/information/entry/Klein_bottle" class="copylinks">Klein Bottle |
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