Sectional density - Revision history

2602:306:C410:1A80:C685:8FF:FE86:7AEF at 08:35, 24 December 2014

2014-12-24T08:35:37Z

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	In the [[mathematics\|mathematical]] theory of [[partial differential equation]]s (PDE), the '''Monge cone''' is a geometrical object associated with a first-order equation. It is named for [[Gaspard Monge]]. In two dimensions, let

	~~:<math>F(x,y,u,u_x,u_y) = 0\qquad\qquad (1)</math>~~

	be a ~~PDE~~ for ~~an unknown real~~-~~valued function ''u'' in two variables ''x'' and ''y''~~. ~~Assume that~~ this ~~PDE is non-degenerate in the sense that <math>F_{u_x}<~~/~~math> and <math>F_{u_y}<~~/~~math> are not both zero in the domain of definition~~. ~~Fix a point (''x''<sub>0<~~/~~sub>, ''y''<sub>0</sub>, ''z''<sub>0<~~/~~sub>) and consider solution functions ''u'' which have~~		Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Good luck on continue learning how to make a wordpress website. In the event you adored this short article along with you wish to obtain more details with regards to [http://conniecolin.com/xe/?document_srl=242317 wordpress backup] generously check out our own website. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. There are number of web services that offer Word press development across the world. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>But before choosing any one of these, let's compare between the two. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. Provide the best and updated information to the web searchers and make use of these wonderful free themes and create beautiful websites. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>A built-in widget which allows you to embed quickly video from popular websites. I didn't straight consider near it solon than one distance, I got the Popup Ascendancy plugin and it's up and lengthways, likely you make seen it today when you visited our blog, and I yet customize it to fit our Thesis Wound which gives it a rattling uncomparable visage and search than any different popup you know seen before on any added journal, I hump arrogated asset of one of it's quatern themes to make our own. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. Where from they are coming, which types of posts are getting top traffic and many more. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. An ease of use which pertains to both internet site back-end and front-end users alike. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Page speed is an important factor in ranking, especially with Google. Get started today so that people searching for your type of business will be directed to you.

	~~:<math>z_0~~ = ~~u(x_0, y_0)~~.~~\qquad\qquad (2)</math>~~

	~~Each solution~~ to ~~(1) satisfying (2) determines~~ the ~~[[tangent plane]]~~ to ~~the graph~~

	:<~~math~~>~~z = u(x,y)\,~~<~~/math~~>

	~~through the point (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>)~~. ~~As the pair (''u''<sub>''x''</sub>~~, ~~''u''<sub>''y''</sub>) solving (1) varies~~, the ~~tangent planes [[Envelope (mathematics)\|envelope]] a cone in '''R'''<sup>3</sup> with vertex at (''x''<sub>0</sub>~~,~~''y''<sub>0</sub>,''z''<sub>0</sub>), called the '''Monge cone'''. When ''F''~~ is ~~[[Differential equations#Types~~ of ~~differential equations\|quasilinear]], the Monge cone degenerates~~ to ~~a single line called~~ the ~~'''Monge axis'''~~. ~~Otherwise,~~ the ~~Monge cone is a proper cone since a nontrivial and non-coaxial one-parameter family~~ of ~~planes through~~ a ~~fixed point envelopes a cone~~. ~~Explicitly~~, ~~the original partial differential equation gives rise to a scalar-valued function on the [[cotangent bundle]]~~ of ~~''R''~~<~~sup~~>3<~~/sup~~>, ~~defined at a point (''x'',~~'~~'y'',''z'') by~~
	~~:<math>adx+bdy+cdz \mapsto F(x,y,z,~~-~~a/c~~,~~-b/c)~~.~~</math>~~
	~~Vanishing of ''F'' determines~~ a ~~curve~~ in the ~~[[projective plane]] with [[homogeneous coordinates]] (''a'':''b'':''c'')~~. ~~The [[dual curve]] is a curve in~~ the ~~projective [[tangent space]] at~~ the ~~point,~~ and ~~the affine cone over this curve is the Monge cone~~. ~~The cone may have multiple branches~~, ~~each one an affine cone over a simple closed curve~~ in the ~~projective tangent space~~.


	~~As the base point (''x''~~<~~sub~~>0<~~/sub~~>,'~~'y''<sub>0</sub>~~,~~''z''<sub>0</sub>) varies~~, ~~the cone also varies. Thus the Monge cone is~~ a ~~cone field~~ on '~~''R'''<sup>3</sup>~~. ~~Finding solutions of (1) can thus be interpreted as finding~~ a ~~surface which is everywhere tangent to~~ the ~~Monge cone at the point. This~~ is ~~the [[method~~ of ~~characteristics]]~~.

	The ~~technique generalizes to scalar first~~-~~order partial differential equations~~ in ~~''n'' spatial variables; namely~~,

	~~:<math>F\left(x_1~~,~~\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right) = 0~~.~~</math>~~

	~~Through each point <math>(x_1^0~~,~~\dots,x_n^0, z^0)</math>, the Monge cone (or axis in the quasilinear case) is the envelope~~ of ~~solutions of the PDE with <math>u(x_1^0,\dots,x_n^0) = z^0</math>~~.

	~~==Examples==~~
	~~;Eikonal equation~~
	~~The simplest fully nonlinear equation is the [[eikonal equation]]. This has the form~~
	:<~~math~~>~~\|\nabla u\|^2 = 1,~~<~~/math~~>
	~~so that the function ''F''~~ is ~~given by~~
	~~:<math>F(x,y,u,u_x,u_y) = u_x^2+u_y^2-1~~.~~</math>~~
	~~The dual cone consists~~ of 1-~~forms ''adx''+''bdy''+''cdz'' satisfying~~
	~~:<math>a^2+b^2~~-~~c^2=0~~.~~</math>~~
	~~Taken projectively, this defines~~ a ~~circle. The dual curve is also a circle~~, ~~and so the Monge cone at each point~~ is ~~a proper cone.~~

	~~==See also==~~
	* [[Tangent cone]]

	~~==References==~~
	* {{cite book\|title=Methods of ~~mathematical physics~~, ~~Volume 2\|author=[[David Hilbert]] and [[Richard Courant]]\|publisher=Wiley Interscience\|year=1989}}~~
	* {{springer\|id=m/m064630\|title=Monge cone\|author=Ivanov, A.~~B.}}~~
	* {{cite book\|author=Monge, G.~~\|title=Application de l'analyse à la géométrie\|publisher=Bachelier\|year=1850}} {{fr}}~~

	~~[[Category:Partial differential equations]]~~

en>Francis Flinch: Undid revision 590033359 by 164.151.129.211 (talk)rm deviations from references

2014-01-10T09:21:02Z

Undid revision 590033359 by 164.151.129.211 (talk)rm deviations from references

← Older revision		Revision as of 11:21, 10 January 2014
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	~~I am Eddy from Gilbert~~. ~~I am learning~~ to ~~play~~ the ~~Pedal Steel Guitar~~. ~~Other hobbies are Racquetball~~.<br><br>~~My website~~; [~~http~~://~~conniecolin~~.~~com~~/xe/~~?document_srl~~=~~242317 wordpress backup~~]		In the [[mathematics\|mathematical]] theory of [[partial differential equation]]s (PDE), the '''Monge cone''' is a geometrical object associated with a first-order equation. It is named for [[Gaspard Monge]]. In two dimensions, let

			:<math>F(x,y,u,u_x,u_y) = 0\qquad\qquad (1)</math>

			be a PDE for an unknown real-valued function ''u'' in two variables ''x'' and ''y''. Assume that this PDE is non-degenerate in the sense that <math>F_{u_x}</math> and <math>F_{u_y}</math> are not both zero in the domain of definition. Fix a point (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) and consider solution functions ''u'' which have

			:<math>z_0 = u(x_0, y_0).\qquad\qquad (2)</math>

			Each solution to (1) satisfying (2) determines the [[tangent plane]] to the graph

			:<math>z = u(x,y)\,</math>

			through the point (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>). As the pair (''u''<sub>''x''</sub>, ''u''<sub>''y''</sub>) solving (1) varies, the tangent planes [[Envelope (mathematics)\|envelope]] a cone in '''R'''<sup>3</sup> with vertex at (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>), called the '''Monge cone'''. When ''F'' is [[Differential equations#Types of differential equations\|quasilinear]], the Monge cone degenerates to a single line called the '''Monge axis'''. Otherwise, the Monge cone is a proper cone since a nontrivial and non-coaxial one-parameter family of planes through a fixed point envelopes a cone. Explicitly, the original partial differential equation gives rise to a scalar-valued function on the [[cotangent bundle]] of ''R''<sup>3</sup>, defined at a point (''x'',''y'',''z'') by
			:<math>adx+bdy+cdz \mapsto F(x,y,z,-a/c,-b/c).</math>
			Vanishing of ''F'' determines a curve in the [[projective plane]] with [[homogeneous coordinates]] (''a'':''b'':''c''). The [[dual curve]] is a curve in the projective [[tangent space]] at the point, and the affine cone over this curve is the Monge cone. The cone may have multiple branches, each one an affine cone over a simple closed curve in the projective tangent space.


			As the base point (''x''<sub>0</sub>,''y''<sub>0</sub>,''z''<sub>0</sub>) varies, the cone also varies. Thus the Monge cone is a cone field on '''R'''<sup>3</sup>. Finding solutions of (1) can thus be interpreted as finding a surface which is everywhere tangent to the Monge cone at the point. This is the [[method of characteristics]].

			The technique generalizes to scalar first-order partial differential equations in ''n'' spatial variables; namely,

			:<math>F\left(x_1,\dots,x_n,u,\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_n}\right) = 0.</math>

			Through each point <math>(x_1^0,\dots,x_n^0, z^0)</math>, the Monge cone (or axis in the quasilinear case) is the envelope of solutions of the PDE with <math>u(x_1^0,\dots,x_n^0) = z^0</math>.

			==Examples==
			;Eikonal equation
			The simplest fully nonlinear equation is the [[eikonal equation]]. This has the form
			:<math>\|\nabla u\|^2 = 1,</math>
			so that the function ''F'' is given by
			:<math>F(x,y,u,u_x,u_y) = u_x^2+u_y^2-1.</math>
			The dual cone consists of 1-forms ''adx''+''bdy''+''cdz'' satisfying
			:<math>a^2+b^2-c^2=0.</math>
			Taken projectively, this defines a circle. The dual curve is also a circle, and so the Monge cone at each point is a proper cone.

			==See also==
			* [[Tangent cone]]

			==References==
			* {{cite book\|title=Methods of mathematical physics, Volume 2\|author=[[David Hilbert]] and [[Richard Courant]]\|publisher=Wiley Interscience\|year=1989}}
			* {{springer\|id=m/m064630\|title=Monge cone\|author=Ivanov, A.B.}}
			* {{cite book\|author=Monge, G.\|title=Application de l'analyse à la géométrie\|publisher=Bachelier\|year=1850}} {{fr}}

			[[Category:Partial differential equations]]

en>Francis Flinch: The form factor or drag coefficient of the bullet/ballistic coeficient mainly rules the aerodymaic efficiency.

2012-07-13T17:03:40Z

The form factor or drag coefficient of the bullet/ballistic coeficient mainly rules the aerodymaic efficiency.

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