## The Mathematical Symbol "Surface Integral (∯)"

The "Surface Integral" Symbol (∯)

The ∯ symbol represents a double integral, more specifically known as the surface integral. It looks similar to two overlapping integral signs (∯) and is used in multivariable calculus to compute integrals over surfaces.

## Visual Representation

The symbol visually appears as two integral signs placed next to and overlapping each other, looking somewhat like this: ∯.

## Usage in Mathematics

The surface integral finds its application in various fields of mathematics, particularly when dealing with vector fields. The surface integral of a scalar function over a surface \( S \) in three-dimensional space is represented as:

\[ \iint_S f(x,y,z) \, dS \]Where \( dS \) is a differential element on the surface \( S \).

## Application in Physics

One of the most known applications of the surface integral in physics is in the computation of flux through a closed surface. The flux of a vector field \( \vec{F} \) through surface \( S \) is given by:

\[ \iint_S \vec{F} \cdot \hat{n} \, dS \]Where \( \hat{n} \) is the outward unit normal vector to the surface at each point and \( dS \) is a differential element on the surface.

## Representation in Other Contexts

In LaTeX, a popular typesetting system used for mathematical and scientific documents, the surface integral is represented by the command `\oiint` (provided certain packages are in use).

## Conclusion

The surface integral is a fundamental concept in multivariable calculus and has various applications, especially in the field of physics, where it's used to calculate quantities like electric and magnetic flux. The ∯ HTML entity provides a way to represent this symbol in web documents.

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## Codes for the ∯ Symbol

The Symbol | ∯ | |

Alt Code | Alt 8751 | |

HTML Code | ∯ | |

HTML Entity | ∯ | |

CSS Code | \222F | |

Hex Code | ∯ | |

Unicode | U+222F |

## How To Insert the ∯ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ∯ symbol is to copy and paste it into your document.Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8751. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)(Method 3) Use the HTML Decimal Code (for webpages).

HTML Text | Output |
---|---|

<b>My symbol: ∯</b> | My symbol: ∯ |

(Method 4) Use the HTML Entity Code (for webpages).

HTML Text | Output |
---|---|

<b>My symbol: ∯</b> | My symbol: ∯ |

(Method 5) Use the CSS Code (for webpages).

CSS and HTML Text | Output |
---|---|

<style> span:after { content: "\222F";} </style> <span>My symbol:</span> | My symbol: ∯ |

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML Text | Output |
---|---|

<b>My symbol: ∯</b> | My symbol: ∯ |

**0x222F**to place the ∯ symbol on your canvas. For example:

JavaScript Text |
---|

const x = "0x"+"E9" ctx.fillText(String.fromCodePoint(x), 5, 5); |

Output |

∯ |

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ∯ is**U+222F**. The important part is the hexadecimal number after the

**U+**, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:

Type | Output |
---|---|

222F [Hold down Alt] [Press x] | ∯ (The 222F turns into ∯. Note that you can omit any leading zeros.) |

JavaScript Text | Output |
---|---|

let str = "\u222F" document.write("My symbol: " + str) | My symbol: ∯ |