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.

## Closed Surface Examples

Examples of closed surfaces:

- Cube,
- Polyhedron,
- Sphere,
- Torus (an inflated inner tube).
- Gaussian surface: any closed surface through which an electric field passes [1].

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 [2].

## Do Closed Surfaces have a Boundary?

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 [2]. Sometimes closed surfaces are described as having*no*boundary [4], 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 [5].

## 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

[1] Chapter 24 Lecture. Retrieved April 22, 2021 from: http://www.physics.gsu.edu/dhamala/Phys2212/chap24.pdf

[2] 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.

[3] Lntcnnmc, CC BY-SA 3.0

[4] V9. Surface Integrals. Retrieved April 22, 2021 from: http://math.mit.edu/~jorloff/suppnotes/suppnotes02/v9.pdf

[5] Classification of Surfaces. Retrieved April 22, 2021 from: https://people.math.osu.edu/fiedorowicz.1/math655/classification.html