Constant of integration - Revision history

en>Mark viking: /* Origin of the constant */ Added wl

2014-09-19T00:47:49Z

Origin of the constant: Added wl

← Older revision		Revision as of 02:47, 19 September 2014
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	~~In [[calculus]], '''linearity'''~~ is ~~a fundamental property of the [[integral]] that follows from the [[sum rule in integration]]~~ and ~~the [[constant factor rule in integration]]. Linearity of integration is related to the linearity of [[summation]], since integrals~~ are ~~thought of as infinite sums.~~

	~~Let ''&fnof;''~~ and ~~''g'' be functions. Now consider:~~

	~~:<math>\int (af(x)+bg(x))\, dx.</math>~~

	~~By the [[sum rule in integration]], this is~~

	~~:<math>\int af(x)\~~, ~~dx+\int bg(x)\~~, dx.~~</math>~~

	~~By the [[constant factor rule in integration]], this reduces to~~

	~~:<math>a\int f(x)\, dx+b\int g(x)\, dx~~.<~~/math~~>

	~~Hence we have~~

	:<~~math~~>~~\int (af(x)+bg(x))\, dx=a\int f(x)\, dx+b\int g(x)\, dx.</math>~~

	~~==Operator notation==~~
	~~The [[differential operator]] is linear — if we use the Heaviside '''D''' notation~~ to ~~denote this, we may extend '''D'''<sup>−1</sup>~~ to ~~mean the first~~ [~~[integral]]. To say that '''D'''<sup>−1<~~/~~sup> is ''therefore'' linear requires a moment to discuss the [[arbitrary constant of integration]]; '''D'''<sup>−1<~~/~~sup> would be straightforward to show linear if the arbitrary constant of integration could be set to zero~~.

	~~Abstractly, we can say that '''D''' is a [[linear transformation]] from some vector space ''V'' to another one, ''W''. We know that '''D'''(''c'') = 0 for any constant function ''c''~~. ~~We can by general theory ([[mean value theorem]])identify the subspace ''C'' of ''V'', consisting of all constant functions as the whole kernel of '''D'''~~. ~~Then by [[linear algebra]] we can establish that '''D'''<sup>−1<~~/~~sup> is a well~~-~~defined linear transformation that is bijective on Im '''D''' and takes values in ''V''/''C''.~~

	~~That is, we treat the ''arbitrary constant of integration'' as a notation for a [[coset]] ''f'' + ''C''; and all is well with the argument.~~

	~~{{DEFAULTSORT:Linearity Of Integration}}~~
	~~[[Category:Integral calculus]~~]

en>Addbot: Bot: Migrating 9 interwiki links, now provided by Wikidata on d:q1355804 (Report Errors)

2013-02-25T22:45:53Z

Bot: Migrating 9 interwiki links, now provided by Wikidata on d:q1355804 (Report Errors)

← Older revision		Revision as of 00:45, 26 February 2013
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	~~Nothing~~ to ~~tell about myself at all~~.<br>~~I enjoy of finally being a part of~~ this ~~community~~.<br>~~I just wish Im useful~~ in ~~some way~~ .<br><br>~~Also visit my webpage~~ ... [~~http:~~//~~hemorrhoidtreatmentfix~~.~~com/prolapsed-hemorrhoid-treatment prolapsed hemorrhoid treatment~~]		{{Unreferenced\|date=December 2009}}
			In [[calculus]], '''linearity''' is a fundamental property of the [[integral]] that follows from the [[sum rule in integration]] and the [[constant factor rule in integration]]. Linearity of integration is related to the linearity of [[summation]], since integrals are thought of as infinite sums.

			Let ''&fnof;'' and ''g'' be functions. Now consider:

			:<math>\int (af(x)+bg(x))\, dx.</math>

			By the [[sum rule in integration]], this is

			:<math>\int af(x)\, dx+\int bg(x)\, dx.</math>

			By the [[constant factor rule in integration]], this reduces to

			:<math>a\int f(x)\, dx+b\int g(x)\, dx.</math>

			Hence we have

			:<math>\int (af(x)+bg(x))\, dx=a\int f(x)\, dx+b\int g(x)\, dx.</math>

			==Operator notation==
			The [[differential operator]] is linear — if we use the Heaviside '''D''' notation to denote this, we may extend '''D'''<sup>−1</sup> to mean the first [[integral]]. To say that '''D'''<sup>−1</sup> is ''therefore'' linear requires a moment to discuss the [[arbitrary constant of integration]]; '''D'''<sup>−1</sup> would be straightforward to show linear if the arbitrary constant of integration could be set to zero.

			Abstractly, we can say that '''D''' is a [[linear transformation]] from some vector space ''V'' to another one, ''W''. We know that '''D'''(''c'') = 0 for any constant function ''c''. We can by general theory ([[mean value theorem]])identify the subspace ''C'' of ''V'', consisting of all constant functions as the whole kernel of '''D'''. Then by [[linear algebra]] we can establish that '''D'''<sup>−1</sup> is a well-defined linear transformation that is bijective on Im '''D''' and takes values in ''V''/''C''.

			That is, we treat the ''arbitrary constant of integration'' as a notation for a [[coset]] ''f'' + ''C''; and all is well with the argument.

			{{DEFAULTSORT:Linearity Of Integration}}
			[[Category:Integral calculus]]

en>Nbarth: /* Reason for a constant difference between antiderivatives */ 1/x

2012-04-14T16:43:58Z

Reason for a constant difference between antiderivatives: 1/x

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