Clockwise Integral

The Mathematical Symbol "Clockwise Integral (∱)"

The ∱ Symbol in Mathematics: Clockwise Integral

In the diverse lexicon of mathematical notations, the ∱ or "Clockwise Integral" symbol finds its niche primarily within calculus, particularly in the context of contour integrals in complex analysis.

Usage

The ∱ symbol represents a contour integral taken in the clockwise direction around a closed curve in the complex plane.

Examples

• Example 1: Basic contour integral:
  If \( f(z) \) is a complex function and \( C \) is a closed curve oriented in the clockwise direction, the integral of \( f \) over \( C \) is denoted by ∱_C \( f(z) \) dz.
• Example 2: Evaluating a clockwise contour integral:
  For a function \( f(z) = z^2 \) and a circle \( C \) of radius 1 centered at the origin, taken in the clockwise direction, the value of ∱_C \( z^2 \) dz can be evaluated using techniques of complex analysis.

In summary, the ∱ symbol offers a succinct way to express contour integrals in the clockwise direction, which can have distinct results from their counterclockwise counterparts. A good grasp of this notation is essential for those studying or working with complex analysis.

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Codes for the ∱ Symbol

The Symbol	∱	copy
Alt Code	Alt 8753	copy
HTML Code	∱	copy
HTML Entity	∱	copy
CSS Code	\2231	copy
Hex Code	∱	copy
Unicode	U+2231	copy

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 8753. 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: &#8753;</b>	My symbol: ∱

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

HTML Text	Output
<b>My symbol: &cwint;</b>	My symbol: ∱

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

CSS and HTML Text	Output
<style> span:after { content: "\2231";} </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: &#x2231;</b>	My symbol: ∱

On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2231 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+2231. 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
2231 [Hold down Alt] [Press x]	∱ (The 2231 turns into ∱. Note that you can omit any leading zeros.)

In JavaScript, the syntax is \uXXXX. So, our example would be \u2231. (Note that the format is 4 hexadecimal characters.)

JavaScript Text	Output
let str = "\u2231" document.write("My symbol: " + str)	My symbol: ∱

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