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, E³. 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.