Hadamard three-lines theorem - Revision history

2601:2:4D00:27B:593E:D535:A4E9:4370: /* Proof */ This is already "displayed" so "displaystyle" is redundant.

2015-01-02T21:53:01Z

Proof: This is already "displayed" so "displaystyle" is redundant.

← Older revision		Revision as of 23:53, 2 January 2015
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	~~{{Unreferenced\|date~~=~~January 2011}}~~		Landon is what my [http://search.Huffingtonpost.com/search?q=wife+loves&s_it=header_form_v1 wife loves] to get in touch with me but you can simply call me anything you like. I've often cherished living in Texas. I am now a generation and distribution officer but I approach on altering it. What I truly appreciate executing is to fish and now I am striving to [http://www.dict.cc/englisch-deutsch/gain+funds.html gain funds] with it. I am not excellent at webdesign but you may well want to verify my website: http://www.couleursafrique.org/modules/profile/userinfo.php?uid=767180<br><br>
	~~In [[differential calculus~~]~~], the~~ '~~''domain-straightening theorem''' states that, given~~ a [~~[vector field~~]~~] <math>X<~~/~~math> on a [[manifold]], there exist local coordinates <math>y_1, \dots, y_n<~~/~~math> such that <math>X = \partial~~ / ~~\partial y_1<~~/~~math> in a neighborhood of a point where~~ <~~math~~>X<~~/math~~> ~~is nonzero. The theorem is also known as '''straightening out of a vector field'''.~~

	~~The~~ [~~[Frobenius theorem (differential topology)\|Frobenius theorem]] in differential geometry can be considered as a higher dimensional generalization of this theorem.~~		Look into my web-site :: donde comprar zapatillas asics; [http://www.couleursafrique.org/modules/profile/userinfo.php?uid=767180 http://www.couleursafrique.org/modules/profile/userinfo.php?uid=767180],

	~~== Proof ==~~
	~~It is clear that we only have to find such coordinates at 0 in <math>\mathbb{R}^n</math>. First we write <math>X = \sum_j f_j(x) {\partial \over \partial x_j}</math> where <math>x<~~/~~math> is some coordinate system at <math>0</math>. Let <math>f = (f_1, \dots, f_n)<~~/~~math>~~. ~~By linear change of coordinates, we can assume <math>f(0) = (1, 0, \dots, 0)~~.</~~math> Let <math>\Phi(t, p)<~~/~~math> be the solution of the initial value problem <math>\dot x = f(x), x(0) = p<~~/~~math> and let~~
	~~:<math>\psi(x_1, \dots, x_n)~~ = ~~\Phi(x_1, (0, x_2, \dots, x_n)).</math>~~
	~~<math>\Phi</math> (and thus <math>\psi</math>) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that~~
	:~~<math>{\partial \over \partial x_1} \psi(x) = f(\psi(x))</math>,~~
	~~and, since <math>\psi(0, x_2, \dots, x_n) = \Phi(0, (0, x_2, \dots, x_n)) = (0, x_2, \dots, x_n)<~~/~~math>, the differential <math>d\psi<~~/~~math> is the identity at <math>0</math>~~. ~~Thus, <math>y = \psi^{-1}(x)</math> is a coordinate system at <math>0</math>~~. ~~Finally, since <math>x = \psi(y)<~~/~~math>, we have: <math>{\partial x_j \over \partial y_1} = f_j(\psi(y)) = f_j(x)<~~/~~math> and so <math>{\partial \over \partial y_1} = X<~~/~~math>~~
	~~as required~~.

	~~[[Category:Differential calculus]~~]

109.9.15.148: /* Applications */ punctuation

2013-07-24T14:21:53Z

Applications: punctuation

← Older revision		Revision as of 16:21, 24 July 2013
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	~~For a wonderful several households in~~ the ~~USA nowadays you will find out~~ that ~~they have~~ a ~~Coleman Ice Chest. Empty all products out of the chest freezer and retailer them in an additional freezer or the refrigerator to avoid thawing. You can also store them in~~ a ~~cooler filled with ice or~~, ~~in a worst case situation~~, ~~sandwiched in between layers of blankets to enable slow the thawing method. Turn the chest freezer down to its lowest setting-switch it all the way off~~, ~~if achievable-and unplug it. Break any large chunks of accumulated ice and frost out of the freezer by hand.~~<br><br>~~Dab the cloth~~ in ~~the bowl~~ of ~~hot water and use the hot cloth to take away ice stuck to the inside~~ of ~~the freezer. Wait 15 to 20 minutes and then take away any ice nevertheless stuck to the inside~~. The ~~steam from the bowl should really get rid~~ of ~~the remaining ice~~. ~~Superior ice~~ / ~~cold retention is obtained by utilizing premium polyurethane insulation injected below stress within all wall cavities and the lid~~. ~~An ice chest~~ is ~~most likely what people are familiar with~~.<br><br>~~This tea is just lemons~~, ~~limes, and oranges cut up and squeezed and mixed with instant~~ (~~YES~~, ~~I Mentioned Immediate) tea~~, ~~sugar~~, and lots and lots of fantastic ice. Then I pour in lots of ice and the fruit and fresh juice and fill it up with water. Pour tea mix and sugar into the beverage container or ice chest and add water to dissolve both. Add ice (to fill about halfway) ~~and pour fruit and fruit juice in. Stir extremely properly~~.<br><br>~~The last component~~ of the ~~equation is that I cover the ice chest with a thick towel that I hold wet with river~~ (~~you can use ocean~~) ~~water. Wintersmiths~~, founded by American brothers Chris Little and Pat Tiny, identified a tremendous quantity of good results in 2013 with the Ice Baller. The Ice Chest makes use of straightforward tap water frozen overnight to produce four completely round slow-melting ice spheres. Obtain an ice chest that will most effective suit your wants.<br><br>~~I looked extremely seriously at the high-priced coolers a handful of years ago~~, ~~before we created a lengthy~~ (~~2-month) tent camping trip~~, ~~over half of it in 90+ temps. I finally decided to start out out with a Coleman Xtreme 70-quart cooler that expense much less than $50~~ (~~shown in the photo~~) ~~and see if it would do what we needed. The bigger the cooler, the much more effective it will be — if you use that extra space for a lot more ice~~.<br><br>The low-priced blue and red ones from the hardware store for 30 bucks will preserve ice and beer for days now compaired to their old ones that could do it for a single day. If you liked this posting and you would like to obtain far more facts pertaining to [http:/~~/www.bestcoolerreviewshq.com/best-cooler-for-the-money/ Bestcoolerreviewshq.Com] kindly go to the page. I got an older metal coleman cooler~~ and ~~am incredibly happy with how it keeps items cool compared to all my plastic coolers and I have various of the plastic ones. The [http:~~/~~/www.sharkbayte.com/keyword/ice+lasted ice lasted] substantially longer in the Igloo than it did~~ in ~~the Coleman~~.<br><br>~~Spot a towel on a table~~ and ~~place the cooler on this towel. Simply because cool air sinks~~, ~~it's greatest to have the cooler of ice elevated—otherwise~~, ~~the cool air will simply linger around the floor. Open the cooler lid or~~, ~~if achievable~~, ~~take away the lid entirely. For most effective results~~, the ~~ice really should be level with or slightly above~~ the ~~top rated rim of the cooler~~. ~~As the fan blows air over the ice~~, ~~the quickly surrounding air will be much cooler. I just applied this cooler on~~ a ~~recent camping trip~~. ~~Really like my 94 qt ice chest!!!~~<br><br>So right here I am at your disposal a complete pectoral exercise that will aid you gain additional muscle in the chest in a short time to incorporate in such training exercises. To assistance avoid the food from going undesirable, ~~you can wrap the food in newspaper~~ and ~~store it in a cool bag, also identified~~ as ~~an insulated bag, or spot it in seal-tight freezer bags and retain it in a cooler filled with ice. A cooler bag covers almost everything from hikes to parties~~.		{{Unreferenced\|date=January 2011}}
			In [[differential calculus]], the '''domain-straightening theorem''' states that, given a [[vector field]] <math>X</math> on a [[manifold]], there exist local coordinates <math>y_1, \dots, y_n</math> such that <math>X = \partial / \partial y_1</math> in a neighborhood of a point where <math>X</math> is nonzero. The theorem is also known as '''straightening out of a vector field'''.

			The [[Frobenius theorem (differential topology)\|Frobenius theorem]] in differential geometry can be considered as a higher dimensional generalization of this theorem.

			== Proof ==
			It is clear that we only have to find such coordinates at 0 in <math>\mathbb{R}^n</math>. First we write <math>X = \sum_j f_j(x) {\partial \over \partial x_j}</math> where <math>x</math> is some coordinate system at <math>0</math>. Let <math>f = (f_1, \dots, f_n)</math>. By linear change of coordinates, we can assume <math>f(0) = (1, 0, \dots, 0).</math> Let <math>\Phi(t, p)</math> be the solution of the initial value problem <math>\dot x = f(x), x(0) = p</math> and let
			:<math>\psi(x_1, \dots, x_n) = \Phi(x_1, (0, x_2, \dots, x_n)).</math>
			<math>\Phi</math> (and thus <math>\psi</math>) is smooth by smooth dependence on initial conditions in ordinary differential equations. It follows that
			:<math>{\partial \over \partial x_1} \psi(x) = f(\psi(x))</math>,
			and, since <math>\psi(0, x_2, \dots, x_n) = \Phi(0, (0, x_2, \dots, x_n)) = (0, x_2, \dots, x_n)</math>, the differential <math>d\psi</math> is the identity at <math>0</math>. Thus, <math>y = \psi^{-1}(x)</math> is a coordinate system at <math>0</math>. Finally, since <math>x = \psi(y)</math>, we have: <math>{\partial x_j \over \partial y_1} = f_j(\psi(y)) = f_j(x)</math> and so <math>{\partial \over \partial y_1} = X</math>
			as required.

			[[Category:Differential calculus]]

en>GrouchoBot: r2.7.2) (Robot: Adding fr:Théorème des trois droites de Hadamard

2012-06-02T19:11:32Z

r2.7.2) (Robot: Adding fr:Théorème des trois droites de Hadamard

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For a wonderful several households in the USA nowadays you will find out that they have a Coleman Ice Chest. Empty all products out of the chest freezer and retailer them in an additional freezer or the refrigerator to avoid thawing. You can also store them in a cooler filled with ice or, in a worst case situation, sandwiched in between layers of blankets to enable slow the thawing method. Turn the chest freezer down to its lowest setting-switch it all the way off, if achievable-and unplug it. Break any large chunks of accumulated ice and frost out of the freezer by hand. Dab the cloth in the bowl of hot water and use the hot cloth to take away ice stuck to the inside of the freezer. Wait 15 to 20 minutes and then take away any ice nevertheless stuck to the inside. The steam from the bowl should really get rid of the remaining ice. Superior ice / cold retention is obtained by utilizing premium polyurethane insulation injected below stress within all wall cavities and the lid. An ice chest is most likely what people are familiar with. This tea is just lemons, limes, and oranges cut up and squeezed and mixed with instant (YES, I Mentioned Immediate) tea, sugar, and lots and lots of fantastic ice. Then I pour in lots of ice and the fruit and fresh juice and fill it up with water. Pour tea mix and sugar into the beverage container or ice chest and add water to dissolve both. Add ice (to fill about halfway) and pour fruit and fruit juice in. Stir extremely properly. The last component of the equation is that I cover the ice chest with a thick towel that I hold wet with river (you can use ocean) water. Wintersmiths, founded by American brothers Chris Little and Pat Tiny, identified a tremendous quantity of good results in 2013 with the Ice Baller. The Ice Chest makes use of straightforward tap water frozen overnight to produce four completely round slow-melting ice spheres. Obtain an ice chest that will most effective suit your wants. I looked extremely seriously at the high-priced coolers a handful of years ago, before we created a lengthy (2-month) tent camping trip, over half of it in 90+ temps. I finally decided to start out out with a Coleman Xtreme 70-quart cooler that expense much less than $50 (shown in the photo) and see if it would do what we needed. The bigger the cooler, the much more effective it will be — if you use that extra space for a lot more ice. The low-priced blue and red ones from the hardware store for 30 bucks will preserve ice and beer for days now compaired to their old ones that could do it for a single day. If you liked this posting and you would like to obtain far more facts pertaining to [http://www.bestcoolerreviewshq.com/best-cooler-for-the-money/ Bestcoolerreviewshq.Com] kindly go to the page. I got an older metal coleman cooler and am incredibly happy with how it keeps items cool compared to all my plastic coolers and I have various of the plastic ones. The [http://www.sharkbayte.com/keyword/ice+lasted ice lasted] substantially longer in the Igloo than it did in the Coleman. Spot a towel on a table and place the cooler on this towel. Simply because cool air sinks, it's greatest to have the cooler of ice elevated—otherwise, the cool air will simply linger around the floor. Open the cooler lid or, if achievable, take away the lid entirely. For most effective results, the ice really should be level with or slightly above the top rated rim of the cooler. As the fan blows air over the ice, the quickly surrounding air will be much cooler. I just applied this cooler on a recent camping trip. Really like my 94 qt ice chest!!! So right here I am at your disposal a complete pectoral exercise that will aid you gain additional muscle in the chest in a short time to incorporate in such training exercises. To assistance avoid the food from going undesirable, you can wrap the food in newspaper and store it in a cool bag, also identified as an insulated bag, or spot it in seal-tight freezer bags and retain it in a cooler filled with ice. A cooler bag covers almost everything from hikes to parties.