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| | My name is Theresa (29 years old) and my hobbies are Games Club - Dungeons and Dragons, Monopoly, Etc. and Radio-Controlled Car Racing.<br><br>Feel free to surf to my web-site: [http://www.ozelkervanasml.k12.tr/ziyaretci-defteri Fifa 15 Coin Generator] |
| 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.
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| Let ''ƒ'' and ''g'' be functions. Now consider:
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| :<math>\int (af(x)+bg(x))\, dx.</math>
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| By the [[sum rule in integration]], this is
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| :<math>\int af(x)\, dx+\int bg(x)\, dx.</math>
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| By the [[constant factor rule in integration]], this reduces to
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| :<math>a\int f(x)\, dx+b\int g(x)\, dx.</math>
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| Hence we have
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| :<math>\int (af(x)+bg(x))\, dx=a\int f(x)\, dx+b\int g(x)\, dx.</math>
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| ==Operator notation==
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| 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.
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| 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''.
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| That is, we treat the ''arbitrary constant of integration'' as a notation for a [[coset]] ''f'' + ''C''; and all is well with the argument.
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| {{DEFAULTSORT:Linearity Of Integration}}
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| [[Category:Integral calculus]]
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