A surface defined by a single chart (or coordinate map). Specifically, a parameterized surface X⊆ℝ³ is a function r:U → ℝ³ defined on an open set U⊆ℝ² such that

• • r is a continuous (see continuous function) bijection from U onto X and r⁻¹ is continuous; that is, r is a homeomorphism between U and X;

• • ∂r/∂x, ∂r/∂y exist and are continuous;

• • (smoothness condition) ∂r/∂x and ∂r/∂y are linearly independent.

The last condition means that the tangent vectors ∂r/∂x and ∂r/∂y span a 2-dimensional vector space called the tangent space. The smooth condition is equivalent to EG–F² ≠ 0, where E,2F,G are the coefficients of the first fundamental form. See also atlas, transition map.