Scalar potential - Revision history

en>Fgnievinski: /* Altitude as gravitational potential energy */

2014-08-17T02:28:07Z

Altitude as gravitational potential energy

← Older revision		Revision as of 04:28, 17 August 2014
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	~~In [[mathematics]]~~, a ~~[[function (mathematics)\|function]] on a [[normed vector space]] is said to~~ '~~''vanish at infinity''' if~~		Hello there, I am Adrianne and I totally delve that name. Vermont has always been my current home and I really like every day living this. Gardening is what I do regular. I am a people manager but soon I'll be without any help. You can find my blog site here: http://prometeu.net<br><br>Here is my webpage :: [http://prometeu.net how to hack clash of clans with cydia]
	~~:<math>f(x)\to 0</math> as <math>\\|x\\|\to \infty~~.~~</math>~~

	~~For example, the function~~
	:~~<math>f(x)=\frac{1}{x^2+1}</math>~~
	~~defined on the [[real line]] vanishes at infinity.~~

	More generally, a function <math>f</math> on a [[locally compact space]] (which may not have a norm) vanishes at infinity if, given any [[positive number]] <math>\epsilon</math>, there is a [[compact space\|compact]] subset <math>K</math> such that
	:~~<math>\\|f(x)\\| < \epsilon<~~/~~math>~~
	~~whenever the point <math>x</math> lies outside of <math>K<~~/~~math>~~. ~~For a given [[locally compact]] space~~ <~~math~~>~~\Omega~~<~~/math~~>~~, the [[set (mathematics)\|set]] of such functions~~
	:~~<math>f~~:~~\Omega\rightarrow\mathbb{F}</math>~~
	~~(where <math>\mathbb{F}</math> is either the~~ [~~[field (mathematics)\|field]] <math>\mathbb{R}</math> of [[real number]]s or the field <math>\mathbb{C}<~~/~~math> of [[complex number]]s) forms an <math>\mathbb{F}<~~/~~math>-vector space with respect to [[pointwise]] [[scalar multiplication]] and [[addition]], often denoted <math>C_{0}(\Omega)</math>~~.

	~~Both of these notions correspond to the intuitive notion of adding a point "at infinity" and requiring the values of the function to get arbitrarily close~~ to ~~zero as we approach it. This "definition" can be formalized in many cases by adding a [[locally compact#The point at infinity\|point at infinity]].~~

	~~==Rapidly decreasing==~~
	~~Refining the concept, one can look more closely at the ''rate of vanishing''~~ of ~~functions at infinity. One of the basic intuitions of [[mathematical analysis]] is that the [[Fourier transform]] interchanges [[smooth function\|smoothness]] conditions~~ with ~~rate conditions on vanishing at infinity. The '''rapidly decreasing''' test functions of [[tempered distribution]] theory are [[smooth function]]s that are~~

	~~:o(\|''x''\|<sup>−''N''</sup>)~~

	for all ''N'', as \|''x''\| → ∞, and such that all their [[partial derivative]]s satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding [[Distribution (mathematics)\|distribution theory]] of ''tempered distributions'' will have the same good property.

	~~==See also==~~
	* [[Schwartz space]]
	~~==References==~~
	~~{{Refimprove\|date=January 2008}}~~
	* {{cite book\|author=Hewitt, E and Stromberg, K\|year=1963\|title=Real and abstract analysis\|publisher=Springer-Verlag}}

	~~[[Category:Mathematical analysis]~~]

en>ArmbrustBot: /* References */re-categorisation per CFDS, replaced: Category:Potential → Category:Potentials using AWB

2013-10-09T14:15:03Z

References: re-categorisation per CFDS, replaced: Category:Potential → Category:Potentials using AWB

← Older revision		Revision as of 16:15, 9 October 2013
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	~~Roberto~~ is ~~the name You love~~ to ~~be which is called with although it can not the name attached to my birth certificate. My colleagues say it~~'~~s not good for me but specifically what I love doing is considered~~ to be to ~~drive but I do have been taking~~ on ~~innovative new things lately~~. ~~South Carolina is where personal home~~ is~~. I used to be unemployed but I am~~ a [~~http~~://~~www.dict.cc~~/~~englisch-deutsch/cashier~~.~~html cashier~~] ~~but~~ the ~~promotion for no reason comes. I've been working on my website for some enough time now. Check it competeing here~~: ~~http~~://~~circuspartypanama.com~~<br><br>~~Feel free~~ to ~~surf~~ to ~~my webpage~~ ... [~~http~~:/~~/circuspartypanama~~.~~com clash~~ of ~~clans hack ipad~~]		In [[mathematics]], a [[function (mathematics)\|function]] on a [[normed vector space]] is said to '''vanish at infinity''' if
			:<math>f(x)\to 0</math> as <math>\\|x\\|\to \infty.</math>

			For example, the function
			:<math>f(x)=\frac{1}{x^2+1}</math>
			defined on the [[real line]] vanishes at infinity.

			More generally, a function <math>f</math> on a [[locally compact space]] (which may not have a norm) vanishes at infinity if, given any [[positive number]] <math>\epsilon</math>, there is a [[compact space\|compact]] subset <math>K</math> such that
			:<math>\\|f(x)\\| < \epsilon</math>
			whenever the point <math>x</math> lies outside of <math>K</math>. For a given [[locally compact]] space <math>\Omega</math>, the [[set (mathematics)\|set]] of such functions
			:<math>f:\Omega\rightarrow\mathbb{F}</math>
			(where <math>\mathbb{F}</math> is either the [[field (mathematics)\|field]] <math>\mathbb{R}</math> of [[real number]]s or the field <math>\mathbb{C}</math> of [[complex number]]s) forms an <math>\mathbb{F}</math>-vector space with respect to [[pointwise]] [[scalar multiplication]] and [[addition]], often denoted <math>C_{0}(\Omega)</math>.

			Both of these notions correspond to the intuitive notion of adding a point "at infinity" and requiring the values of the function to get arbitrarily close to zero as we approach it. This "definition" can be formalized in many cases by adding a [[locally compact#The point at infinity\|point at infinity]].

			==Rapidly decreasing==
			Refining the concept, one can look more closely at the ''rate of vanishing'' of functions at infinity. One of the basic intuitions of [[mathematical analysis]] is that the [[Fourier transform]] interchanges [[smooth function\|smoothness]] conditions with rate conditions on vanishing at infinity. The '''rapidly decreasing''' test functions of [[tempered distribution]] theory are [[smooth function]]s that are

			:o(\|''x''\|<sup>−''N''</sup>)

			for all ''N'', as \|''x''\| → ∞, and such that all their [[partial derivative]]s satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding [[Distribution (mathematics)\|distribution theory]] of ''tempered distributions'' will have the same good property.

			==See also==
			* [[Schwartz space]]
			==References==
			{{Refimprove\|date=January 2008}}
			* {{cite book\|author=Hewitt, E and Stromberg, K\|year=1963\|title=Real and abstract analysis\|publisher=Springer-Verlag}}

			[[Category:Mathematical analysis]]

69.112.163.78: /* See also */

2012-06-30T01:58:45Z