## The Mathematical Symbol "Partial Differential (∂)"

The "Partial Differential" Symbol (∂): A Key Notation in Calculus

In the world of mathematics, especially in calculus and mathematical physics, the ∂ symbol, known as the "Partial Differential", holds significant importance. It is used to denote partial derivatives, which are derivatives of a function with respect to one of its several variables, treating all others as constant. Let's dive into the intricacies of ∂ and its various applications.

## Understanding the ∂ Symbol

The concept of differentiation is central to calculus, and when dealing with functions of multiple variables, partial differentiation becomes crucial. The ∂ symbol aids in distinguishing partial derivatives from ordinary derivatives.

Example 1: A Simple Function

For a function \(f(x, y)\), the partial derivative with respect to \(x\) is represented as ∂f/∂x. This computation considers only the changes in \(f\) due to changes in \(x\), assuming \(y\) remains constant.

Example 2: Thermodynamics

In thermodynamics, properties like entropy \(S\) can be a function of volume \(V\) and temperature \(T\). The partial derivative of entropy with respect to volume at constant temperature is represented as (∂S/∂V)_{T}.

## Applications of the ∂ Symbol

The ∂ symbol finds applications in various branches of science and engineering:

**Mathematics:**In multivariable calculus for defining gradients, curl, and divergence.**Physics:**Especially in thermodynamics and quantum mechanics.**Engineering:**In fields like fluid dynamics and electrical engineering.**Economics:**For understanding rates of change in multivariate functions.

The notation offers a concise way to express how a function changes with respect to specific variables while keeping others constant. This allows for a granular understanding of dynamic systems with interdependent variables.

In summary, the ∂ symbol provides a powerful mathematical tool for understanding and analyzing functions of multiple variables. Its ubiquity across scientific disciplines underscores its fundamental role in shaping our comprehension of complex systems and phenomena.

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## Codes for the ∂ Symbol

The Symbol | ∂ | |

Alt Code | Alt 8706 | |

HTML Code | ∂ | |

HTML Entity | ∂ | |

CSS Code | \2202 | |

Hex Code | ∂ | |

Unicode | U+2202 |

## 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 8706. 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: "\2202";} </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: ∂ |

**0x2202**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+2202**. 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 |
---|---|

2202 [Hold down Alt] [Press x] | ∂ (The 2202 turns into ∂. Note that you can omit any leading zeros.) |

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

let str = "\u2202" document.write("My symbol: " + str) | My symbol: ∂ |