Английская Википедия:Integral symbol
Шаблон:Short description Шаблон:Distinguish Шаблон:Infobox symbol
The integral symbol: Шаблон:Block indent is used to denote integrals and antiderivatives in mathematics, especially in calculus.
History
The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings;[1][2] it first appeared publicly in the article "Шаблон:Lang" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.[3][4] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.
Typography in Unicode and LaTeX
Fundamental symbol
The integral symbol is Шаблон:Unichar in Unicode[5] and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (named entity).
The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.
The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh").
Extensions of the symbol
Related symbols include:[5][6]
| Meaning | Unicode | LaTeX | ||
|---|---|---|---|---|
| Double integral | ∬ | U+222C | <math>\iint</math> | \iint
|
| Triple integral | ∭ | U+222D | <math>\iiint</math> | \iiint
|
| Quadruple integral | ⨌ | U+2A0C | <math>\iiiint</math> | \iiiint
|
| Contour integral | ∮ | U+222E | <math>\oint</math> | \oint
|
| Clockwise integral | ∱ | U+2231 | ||
| Counterclockwise integral | ⨑ | U+2A11 | ||
| Clockwise contour integral | ∲ | U+2232 | Шаблон:Intorient | \varointclockwise
|
| Counterclockwise contour integral | ∳ | U+2233 | Шаблон:Intorient | \ointctrclockwise
|
| Closed surface integral | ∯ | U+222F | Шаблон:Oiint | \oiint
|
| Closed volume integral | ∰ | U+2230 | Шаблон:Oiiint | \oiiint
|
Typography in other languages
In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.[7]
Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol:
<math display="block">
\int_0^5 f(t) \,\mathrm{d}t, \quad
\int_{g(t) = a}^{g(t) = b} f(t) \,\mathrm{d}t.
</math>
By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits:
<math display="block">
\int\limits_0^T f(t) \,\mathrm{d}t, \quad
\int\limits_{\!\!\!\!\!g(t) = a\!\!\!\!\!}^{\!\!\!\!\!g(t) = b\!\!\!\!\!} f(t) \,\mathrm{d}t.
</math>
See also
Notes
References
External links
- ↑ Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Шаблон:Webarchive ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Шаблон:Webarchive ("Methodi tangentium inversae exempla", November 11, 1675).
- ↑ Шаблон:Cite web
- ↑ Шаблон:Citation
- ↑ Шаблон:Cite book
- ↑ 5,0 5,1 Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
