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:''For differential geometry usage, see glossary of differential geometry and topology.'' 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.〔Hans Samelson, "Orientability of hypersurfaces in Rn", Proceedings of the American Mathematical Society, Vol. 22, No. 1 (Jul., 1969), pp. 301-302.〕 Every connected compact hypersurface is a level set,〔Elon L. Lima, "The Jordan-Brouwer separation theorem for smooth hypersurfaces", The American Mathematical Monthly, Vol. 95, No. 1 (Jan., 1988), pp. 39-42.〕 and separates Rn in two connected components,〔 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== * Affine sphere * Coble hypersurface * Polar hypersurface * Null hypersurface * Dwork family 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「hypersurface」の詳細全文を読む スポンサード リンク
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