Hypersurface

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For differential geometry usage, see glossary of differential geometry and topology.
Ackley3.gif

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, so not a submanifold in the strict sense. "Primal" is an old term for an irreducible hypersurface.

See also[edit]

References[edit]

  1. ^ Hans Samelson, "Orientability of hypersurfaces in Rn", Proceedings of the American Mathematical Society, Vol. 22, No. 1 (Jul., 1969), pp. 301-302.
  2. ^ a b Elon L. Lima, "The Jordan-Brouwer separation theorem for smooth hypersurfaces", The American Mathematical Monthly, Vol. 95, No. 1 (Jan., 1988), pp. 39-42.