Surface -- from Wolfram MathWorld

The word "surface" is an important term in mathematics and is used in many ways. The most common and straightforward use of the word is to denote a two-dimensional submanifold of three-dimensional Euclidean space. Surfaces can range from the very complicated (e.g., fractals such as the Mandelbrot set) to the very simple (such as the plane). More generally, the word "surface" can be used to denote an -dimensional submanifold of an -dimensional manifold, or in general, any codimension-1 subobject in an object (like a Banach space or an infinite-dimensional manifold).

Even simple surfaces can display surprisingly counterintuitive properties. For example, the surface of revolution of around the x-axis for (called Gabriel's horn) has finite volume but infinite surface area.

See also

Algebraic Surface, Compact Surface, Complete Surface, Developable Surface, Enneper's Minimal Surface, Hypersurface, Manifold, Minimal Surface, Orientable Surface, Orthogonal Surfaces, Riemann Surface, Smooth Surface, Solid, Surface with Boundary Explore this topic in the MathWorld classroom

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References

Andrews, P. "The Classification of Surfaces." Amer. Math. Monthly 95, 861-868, 1988.Endraß, S. "Home Page of S. Endraß." http://www.mathematik.uni-mainz.de/~endrass/Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, 1986.Francis, G. K. A Topological Picturebook. New York: Springer-Verlag, 1987.Gallier, J. H. Curves and Surfaces for Geometric Design: Theory and Algorithms. New York: Academic Press, 1999.Geometry Center. "The Topological Zoo." http://www.geom.umn.edu/zoo/.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Hunt, B. "Algebraic Surfaces." http://www.mathematik.uni-kl.de/~hunt/drawings.html.Javaview. "Classic Surfaces from Differential Geometry." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface.html.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 135, 1999.Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369-376, 1996.Morris, R. "A Parade of Surfaces: A Pictorial Introduction to Singularity Theory." http://www.amsta.leeds.ac.uk/~rjm/parade/.Nordstrand, T. "Gallery." http://jalape.no/math/mathgal.Nordstrand, T. "Surfaces." http://jalape.no/math/surfaces.von Seggern, D. CRC Standard Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 2007.Wagon, S. "Surfaces." Ch. 3 in Mathematica in Action. New York: W. H. Freeman, pp. 67-91, 1991.Wilkinson, S. "Intersections of Surfaces." Mathematica in Educ. Res. 8, 5-10, 1999.Yamaguchi, F. Curves and Surfaces in Computer Aided Geometric Design. New York: Springer-Verlag, 1988.

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Surface

Cite this as:

Weisstein, Eric W. "Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Surface.html

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