Enkelpunktet

Enkelpunktet is a term used in Norwegian mathematics to refer to a point on a geometric object where the local structure is non-degenerate and behaves smoothly. In many contexts it serves as the counterpart to a singular point, indicating a point where the geometry or analysis is well-behaved.

In algebraic geometry, a plane curve defined by F(x, y) = 0 has an enkelpunktet at p if the gradient ∇F(p) is not zero. Equivalently, the partial derivatives do not vanish simultaneously at p. At such points the curve is locally a smooth one-dimensional manifold and possesses a well-defined tangent line. This non-singular behavior allows standard geometric and analytic tools to apply.

In differential geometry and the theory of manifolds, an enkelpunktet (often called a regular point in English) is a point where the differential (or Jacobian) has full rank. At a regular point the tangent space has the expected dimension, and local coordinates give a smooth structure. Singular points, where the differential loses rank, are the points where the local structure fails to be smooth.

Examples help illustrate the concept. The circle defined by x^2 + y^2 = 1 has enkelpunkter at every point, since the gradient is nonzero everywhere on the curve. By contrast, the cusp of the curve y^2 = x^3 has a singular point at the origin, which is not an enkelpunkt.

Usage varies by subfield, and modern texts often prefer terms like regular point or non-singular point. Enkeltpunktet remains a recognized Norwegian formulation encountered in historical or context-specific discussions of geometry and analysis.