For differential geometry usage, see glossary of differential geometry and topology.

Ackley's function of three variables, with time the 3rd variable.

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n 1 dimensions is a hypersurface. Equivalently, the codimension of a hypersurface is one. For example, the n-sphere in Rn+1 is called a hypersphere. Hypersurfaces occur frequently in multivariable calculus as level sets.

In Rn, every closed hypersurface is orientable.[1] Every connected compact hypersurface is a level set,[2] and separates Rn in two connected components,[2] which is related to the Jordan–Brouwer separation theorem.

In algebraic geometry, a hypersurface in projective space of dimension n is an algebraic set (algebraic variety) that is purely of dimension n 1. It is then defined by a single equation f(x1, x2, ..., xn) = 0, a homogeneous polynomial in the homogeneous coordinates.

Thus, it generalizes those algebraic curves f(x1, x2) = 0 (dimension one), and those algebraic surfaces f(x1, x2, x3) = 0 (dimension two), when they are defined by homogeneous polynomials.

A hypersurface may have singularities, and hence is not necessarily a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.

See also


  1. Hans Samelson (1969) Orientability of hypersurfaces in Rn, Proceedings of the American Mathematical Society 22(1): 301,2
  2. 1 2 Elon L. Lima, "The Jordan-Brouwer separation theorem for smooth hypersurfaces", The American Mathematical Monthly, Vol. 95, No. 1 (Jan., 1988), pp. 39–42.
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