The Mathematical Symbol "Clockwise Contour Integral (∲)"
The ∲ Symbol in Mathematics: Clockwise Contour Integral
Among the myriad of symbols in advanced mathematics, the ∲ or "Clockwise Contour Integral" symbol holds a distinctive place, especially in the field of complex analysis. This article aims to illuminate its usage and provide examples for enhanced comprehension.
Usage
The ∲ symbol signifies a contour integral taken over a path that encircles a region in a clockwise direction. It is mainly used in the context of complex analysis, where functions are integrated over curves or contours in the complex plane.
Examples
- Example 1: Basic contour integration:
Evaluate the integral of a function \( f(z) \) over a closed curve \( C \) that circles a region in the clockwise direction:
\( \int_C f(z) \, dz \)
This integration moves in the direction opposite to the conventional counterclockwise orientation. - Example 2: Relation with counterclockwise contour integral:
If you have a counterclockwise contour integral \( \oint f(z) \, dz \) over a curve \( C \), then the clockwise contour integral over the same curve is its negative:
\( ∲_C f(z) \, dz = -\oint_C f(z) \, dz \)
In summary, the ∲ symbol is fundamental in the domain of complex analysis. By understanding the implications of direction in contour integrals, mathematicians can better interpret and analyze functions in the complex plane.
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Codes for the ∲ Symbol
| The Symbol | ∲ | |
| Alt Code | Alt 8754 | |
| HTML Code | ∲ | |
| HTML Entity | ∲ | |
| CSS Code | \2232 | |
| Hex Code | ∲ | |
| Unicode | U+2232 |
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 8754. 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: "\2232";} </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: ∲ |
| 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+2232. 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 |
|---|---|
| 2232 [Hold down Alt] [Press x] | ∲ (The 2232 turns into ∲. Note that you can omit any leading zeros.) |
| JavaScript Text | Output |
|---|---|
| let str = "\u2232" document.write("My symbol: " + str) | My symbol: ∲ |