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.


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.


  • 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.

Mathematical symbol 'Clockwise Contour Integral'

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Codes for the ∲ Symbol

The Symbol
Alt CodeAlt 8754
HTML Code∲
HTML Entity∲
CSS Code\2232
Hex Code∲

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 TextOutput
<b>My symbol: &#8754;</b>My symbol: ∲

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

HTML TextOutput
<b>My symbol: &cwconint;</b>My symbol: ∲

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

CSS and HTML TextOutput
span:after {
content: "\2232";}
<span>My symbol:</span>
My symbol: ∲

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

HTML TextOutput
<b>My symbol: &#x2232;</b>My symbol: ∲
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2232 to place the ∲ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);

(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:
[Hold down Alt]
[Press x]

(The 2232 turns into ∲. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2232. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2232"
document.write("My symbol: " + str)
My symbol: ∲