Uniform tiling - Revision history

en>Hyacinth: /* Self-dual tilings */ --> '''self-dual'''

2014-11-05T23:03:05Z

Self-dual tilings: --> '''self-dual'''

← Older revision		Revision as of 01:03, 6 November 2014
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	~~In [[non-standard analysis]], a field of mathematics, the '''increment theorem''' states~~ the ~~following: Suppose a [[Function (mathematics)\|function]] ''y'' = ''f''(''x'')~~ is ~~[[differentiable]] at ''x''~~ and that ~~Δ''x'' is [[infinitesimal]]~~. ~~Then~~		The individual who wrote the post is known as Jayson Hirano and he totally digs that title. For a whilst I've been in Alaska but I will have to transfer in a yr or two. Office supervising is what she does for a residing. To climb is something I really appreciate doing.<br><br>My blog post ... online psychic chat ([http://www.010-5260-5333.com/index.php?document_srl=1880&mid=board_ALMP66 Suggested Webpage])
	~~:<math>\Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x\,</math>~~

	~~for some infinitesimal ε, where~~
	~~: <math>\Delta y=f(x+\Delta x)-f(x).\,</math>~~

	~~If <math>\scriptstyle\Delta x\not=0</math> then we may write~~
	~~: <math>\frac{\Delta y}{\Delta x} = f'(x)+\varepsilon,</math>~~

	~~which implies that <math>\scriptstyle\frac{\Delta y}{\Delta x}\approx f~~'~~(x)</math>, or~~ in ~~other words that <math>\scriptstyle \frac{\Delta y}{\Delta x}</math> is infinitely close~~ to ~~<math>\scriptstyle f'(x)\,</math>,~~ or ~~<math>\scriptstyle f'(x)\,</math>~~ is ~~the [[standard part function\|standard part]] of~~ <~~math~~>~~\scriptstyle \frac{\Delta y}{\Delta x}~~<~~/math~~>.

	~~== See also ==~~
	*[[Non-standard calculus]]
	*''[[Elementary Calculus: An Infinitesimal Approach]]''
	*[[Abraham Robinson]]

	~~== References ==~~
	* [[Howard Jerome Keisler]]: ''[[Elementary Calculus: An Infinitesimal Approach]]''. First edition 1976; 2nd edition 1986. ~~This book is now out of print~~. ~~The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at~~ http://www.~~math~~.~~wisc.edu~~/~~~keisler/calc~~.~~html~~
	*{{cite book\|last=~~Robinson\|first=Abraham\| title=Non-standard analysis\|year=1996\| edition=Revised edition \| publisher=Princeton University Press\| isbn~~ = ~~0-691-04490-2}}~~


	~~{{Infinitesimals}}~~

	~~[[Category:Calculus]]~~
	~~[[Category:Non-standard analysis]~~]

en>Tomruen: /* Uniform tilings of the hyperbolic plane */

2013-10-21T07:43:26Z

Uniform tilings of the hyperbolic plane

← Older revision		Revision as of 09:43, 21 October 2013
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	~~Hi there~~, ~~I am Andrew Berryhill. Invoicing~~ is ~~what I do~~. ~~To climb is~~ some ~~thing I truly appreciate doing~~. ~~Her family members life~~ in ~~Ohio.~~<br><br>~~Also visit my website~~ - ~~tarot card readings (~~[http://~~hknews~~.~~classicmall~~.~~com~~.hk/~~groups~~/~~some~~-~~simple~~-~~tips~~-~~for~~-~~personal~~-~~development-progress/ click here~~])		In [[non-standard analysis]], a field of mathematics, the '''increment theorem''' states the following: Suppose a [[Function (mathematics)\|function]] ''y'' = ''f''(''x'') is [[differentiable]] at ''x'' and that Δ''x'' is [[infinitesimal]]. Then
			:<math>\Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x\,</math>

			for some infinitesimal ε, where
			: <math>\Delta y=f(x+\Delta x)-f(x).\,</math>

			If <math>\scriptstyle\Delta x\not=0</math> then we may write
			: <math>\frac{\Delta y}{\Delta x} = f'(x)+\varepsilon,</math>

			which implies that <math>\scriptstyle\frac{\Delta y}{\Delta x}\approx f'(x)</math>, or in other words that <math>\scriptstyle \frac{\Delta y}{\Delta x}</math> is infinitely close to <math>\scriptstyle f'(x)\,</math>, or <math>\scriptstyle f'(x)\,</math> is the [[standard part function\|standard part]] of <math>\scriptstyle \frac{\Delta y}{\Delta x}</math>.

			== See also ==
			*[[Non-standard calculus]]
			*''[[Elementary Calculus: An Infinitesimal Approach]]''
			*[[Abraham Robinson]]

			== References ==
			* [[Howard Jerome Keisler]]: ''[[Elementary Calculus: An Infinitesimal Approach]]''. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
			*{{cite book\|last=Robinson\|first=Abraham\| title=Non-standard analysis\|year=1996\| edition=Revised edition \| publisher=Princeton University Press\| isbn = 0-691-04490-2}}


			{{Infinitesimals}}

			[[Category:Calculus]]
			[[Category:Non-standard analysis]]

173.240.25.249: /* External links */

2012-08-07T03:22:37Z

External links

New page

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