The Mathematical Symbol "Double Integral (∬)"
Exploring the "Double Integral" Symbol (∬): A Dive into Advanced Calculus
Calculus, the mathematical study of change and motion, has many symbols that stand testament to its rich history and depth. One such symbol that stands out in multivariable calculus is the "Double Integral", denoted as ∬.
Unveiling ∬: The Basics
The double integral is employed to compute the volume under a surface in three-dimensional space. It's an extension of the concept of a single integral, which calculates the area under a curve. With a double integral, instead of integrating over a one-dimensional interval, we integrate over a two-dimensional region.
Example 1:
Consider a function \( f(x,y) \) that represents a surface over a region \( R \) in the xy-plane. The volume below \( f \) and above \( R \) is given by:
∬ ∬ R f(x,y) dA
where \( dA \) is a differential area element in the region \( R \).
Example 2:
Let's take a simple function, \( f(x,y) = x^2 + y^2 \), over the square region where \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq 1 \). The double integral, which gives the volume under the paraboloid and above the square region, can be represented as:
∬ ∬ [0,1]x[0,1] (x^2 + y^2) dxdy
The ∬ symbol is pivotal in advanced calculus, especially when navigating problems involving volumes and more complex spatial relationships in higher dimensions. As calculus continues to evolve and find new applications, the double integral remains a foundational concept, guiding our understanding of space and volume.
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Codes for the ∬ Symbol
| The Symbol | ∬ | |
| Alt Code | Alt 8748 | |
| HTML Code | ∬ | |
| HTML Entity | ∬ | |
| CSS Code | \222C | |
| Hex Code | ∬ | |
| Unicode | U+222C |
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 8748. 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: "\222C";} </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+222C. 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 |
|---|---|
| 222C [Hold down Alt] [Press x] | ∬ (The 222C turns into ∬. Note that you can omit any leading zeros.) |
| JavaScript Text | Output |
|---|---|
| let str = "\u222C" document.write("My symbol: " + str) | My symbol: ∬ |