powierchnie

Powierchnie is not a standard term in Polish; it is typically a misspelling of powierzchnie, the plural of powierzchnia. The ordinary meaning of powierzchnia is surface—the outer boundary of a three‑dimensional object or, more generally, a two‑dimensional manifold in mathematics and related fields.

In mathematics, a surface is a two‑dimensional object that can be embedded in three‑dimensional space. Common examples include the plane, the sphere, and the torus. A surface can be described locally by a parameterization r(u, v) in which each point is given by coordinates that depend on two parameters. A basic measurement associated with a surface is its area. For a parametric surface r(u, v), the area is computed as the double integral over the parameter domain of the magnitude of the cross product of the partial derivatives, Area = ∬ |r_u × r_v| du dv. Surfaces may also be described implicitly by equations f(x, y, z) = 0.

Surfaces have wide applications. In computer graphics, surfaces are approximated by meshes to render 3D objects. In engineering and architecture, surface geometry informs design and analysis. In geography and earth sciences, terrain is represented as surface models. In physics, interfaces between materials exhibit surface phenomena such as surface tension.

When encountered in Polish texts, powierchnie is usually a typographical error for powierzchnie and should be corrected to reflect the standard term.