# Hypersurface

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

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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 **R**^{n+1} is called a **hypersphere**. Hypersurfaces occur frequently in multivariable calculus as level sets.

In **R**^{n}, every closed hypersurface is orientable.^{[1]} Every connected compact hypersurface is a level set,^{[2]} and separates **R**^{n} 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*(*x*_{1}, *x*_{2}, ..., *x*_{n}) = 0, a homogeneous polynomial in the homogeneous coordinates.

Thus, it generalizes those algebraic curves *f*(*x*_{1}, *x*_{2}) = 0 (dimension one), and those algebraic surfaces *f*(*x*_{1}, *x*_{2}, *x*_{3}) = 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

## References

- ↑ Hans Samelson (1969) Orientability of hypersurfaces in
**R**^{n}, Proceedings of the American Mathematical Society 22(1): 301,2 - 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.

- Hazewinkel, Michiel, ed. (2001), "Hypersurface",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Shoshichi Kobayashi and Katsumi Nomizu (1969), Foundations of Differential Geometry Vol II, Wiley Interscience
- P.A. Simionescu & D. Beal (2004) Visualization of hypersurfaces and multivariable (objective ) functions by partial globalization,
*The Visual Computer*20(10):665–81.