 # Closed Surface: Simple Definition, Examples

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A closed surface contains a volume of space, enclosed from all directions; It consists of one connected, hollow piece that has no holes and doesn’t intersect itself.

A closed surface has a natural positive direction, where a unit normal vector N points away from the interior (i.e. points toward the outside). A normal vector pointed outward indicates the “negative” direction. The two parts of this shape are oriented by two unit normal vectors.

## Closed Surface Examples Closed surfaces (L to R): Cube, Polyhedron, Sphere, Torus.

Examples of closed surfaces:

• Cube,
• Polyhedron,
• Sphere,
• Torus (an inflated inner tube).
• Gaussian surface: any closed surface through which an electric field passes .

A Few examples of surfaces that are not closed: a plane, a sphere with a point removed, a crumpled tin can with a cross-section that looks like a figure-8 (it intersects itself), an infinite cylinder .

## Do Closed Surfaces have a Boundary? A Möbius strip .

The interior of a closed surface is a completely enclosed, single finite region of space; The closed surface itself is the boundary of this region . Sometimes closed surfaces are described as having no boundary , but there are cases where closed surfaces do have boundary (for example, a cylinder or Möbius strip). To avoid this problem with classification, many discussions on closed surfaces concentrate on simple closed surfaces, like cubes and spheres .

## Calculating Integrals

When you have many surfaces (for example, a pyramid has four), calculating integrals becomes a challenge. The divergence theorem allows you to combine multiple surface integrals and use one triple (volume) integral instead.

## References

 Chapter 24 Lecture. Retrieved April 22, 2021 from: http://www.physics.gsu.edu/dhamala/Phys2212/chap24.pdf
 MIT. (2007). 18.02 Multivariable Calculus. Retrieved April 22, 2021 from: https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/readings/divergance_thm.pdf. CC Share Alike 4.0.
 Lntcnnmc, CC BY-SA 3.0 , via Wikimedia Commons
 V9. Surface Integrals. Retrieved April 22, 2021 from: http://math.mit.edu/~jorloff/suppnotes/suppnotes02/v9.pdf
 Classification of Surfaces. Retrieved April 22, 2021 from: https://people.math.osu.edu/fiedorowicz.1/math655/classification.html

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Stephanie Glen. "Closed Surface: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/closed-surface/
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