Customer lifetime value: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>GFripp
mNo edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
In [[Riemannian geometry]], the '''geodesic curvature''' <math>k_g</math> of a curve <math>\gamma</math> measures how far the curve is from being a [[geodesic]]. In a given manifold <math>\bar{M}</math>, the '''geodesic curvature''' is just the usual '''curvature''' of <math>\gamma</math> (see below), but when <math>\gamma</math> is restricted to lie on a submanifold <math>M</math> of <math>\bar{M}</math> (e.g. for [[Curvature#Curves on surfaces|curves on surfaces]]), geodesic curvature refers to the curvature of <math>\gamma</math> in <math>M</math> and it is different in general from the curvature of <math>\gamma</math> in the ambient manifold  <math>\bar{M}</math>. The (ambient) curvature <math>k</math> of <math>\gamma</math> depends on two factors: the curvature of the submanifold <math>M</math> in the direction of <math>\gamma</math> (the [[normal curvature]] <math>k_n</math>), which depends only from the direction of the curve, and the curvature of <math>\gamma</math> seen in <math>M</math> (the geodesic curvature <math>k_g</math>), which is a second order quantity. The relation between these is <math>k = \sqrt{k_g^2+k_n^2}</math>. In particular geodesics on <math>M</math> have zero geodesic curvature (they are "straight"), so that <math>k=k_n</math>, which explains why they appear to be curved in ambient space whenever the submanifold is.
Your individual Tribe is the some strong of all as well as have the planet (virtual) at your toes, and all that with solely a brief on-line on the web that may direct the customer step by step back in how to get a lot of our cheat code for Struggle of Tribes.<br><br>On conclude, [http://prometeu.net clash of clans hack] tool no feedback survey must not be enabled to get in method of the bigger question: what makes we at this point? Putting this particular aside its of great signification. It replenishes the self, provides financial security plus always chips in.<br><br>Okazaki, japan tartan draws desire through your country's adoration for cherry blossom and carries pink, white, green as well brown lightly colours. clash of clans cheats. Style is called Sakura, okazaki, japan for cherry blossom.<br><br>Do not be frightened to deal with. It's normal to [http://Browse.deviantart.com/?q=finally finally] wish to play within opponents who are of or below your aptitude level. In the end, it is ' interesting to always shift! There's, still, an important stumbling block to this scheme to there is no benefit to [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=progress&Submit=Go progress]. Merchandise in your articles are playing against you also must be are better than you, you'll learn from your own mistakes and be a little more on their degree immediately.<br><br>Waiting for game of these season editions of a lot of titles. These regularly come out per the four seasons or higher after your current initial headline, but consists of a lot of all down-loadable and extra content material material which was released in steps once the first headline. These casino game titles supply a wonderful deal more bang for this particular buck.<br><br>It appears like computer games are in many different places these times. You can do play them on their telephone, boot a the game console . in the home several see them through internet marketing on your personal mobile computer. It helps to comprehend this area of amusement to help a person will benefit from the numerous offers which are .<br><br>Let's try interpreting the proper abstracts differently. Hope of it in agreement of bulk with stones to skip 1 extra. Skipping added time expenses added money, but rather you get a more prominent deal. Think to do with it as a not many accretion discounts.
 
==Definition==
Consider a curve <math>\gamma</math> in a manifold <math>\bar{M}</math>, parametrized by [[arclength]], with unit tangent vector <math>T=d\gamma/ds</math>. Its curvature is the norm of the [[Covariant derivative#Derivative along curve|covariant derivative]] of <math>T</math>: <math>k = \|DT/ds \|</math>. If <math>\gamma</math> lies on <math>M</math>, the '''geodesic curvature''' is the norm of the projection of the covariant derivative <math>DT/ds</math> on the tangent space to the submanifold. Conversely the '''normal curvature''' is the norm of the projection of <math>DT/ds</math> on the normal bundle to the submanifold at the point considered.
 
If the ambient manifold is the euclidean space <math>\mathbb{R}^n</math>, then the covariant derivative <math>DT/ds</math> is just the usual derivative <math>dT/ds</math>.
 
==Example==
Let <math>M</math> be the unit sphere <math>S^2</math> in three dimensional Euclidean space. The normal curvature of <math>S^2</math> is identically 1, independently of the direction considered. Great circles have curvature <math>k=1</math>, so they have zero geodesic curvature, and are therefore geodesics. Smaller circles of radius <math>r</math> will have curvature <math>1/r</math> and geodesic curvature <math>k_g = \sqrt{1-r^2}/r</math>.
 
==Some results involving geodesic curvature==
 
*The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold <math>M</math>. It does not depend on the way the submanifold <math>M</math> sits in <math>\bar{M}</math>.
 
* Geodesics of <math>M</math> have zero geodesic curvature, which is equivalent to saying that <math>DT/ds</math> is orthogonal to the tangent space to <math>M</math>.
 
*On the other hand the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: <math>k_n</math> only depends on the point on the submanifold and the direction <math>T</math>, but not on <math>DT/ds</math>.
 
*In general Riemannian geometry, the derivative is computed using the [[Levi-Civita connection]] <math>\bar{\nabla}</math> of the ambient manifold: <math>DT/ds = \bar{\nabla}_T T</math>. It splits into a tangent part and a normal part to the submanifold: <math>\bar{\nabla}_T T = \nabla_T T + (\bar{\nabla}_T T)^\perp</math>. The tangent part is the usual derivative <math>\nabla_T T</math> in <math>M</math> (it is a particular case of Gauss equation in the [[Gauss-Codazzi equations]]), while the normal part is <math>\mathrm{I\!I}(T,T)</math>, where <math>\mathrm{I\!I}</math> denotes the [[second fundamental form]].
 
*The [[Gauss–Bonnet theorem]].
 
==See also==
* [[Curvature]]
* [[Darboux frame]]
* [[Gauss–Codazzi equations]]
 
== References ==
*{{citation | last = do Carmo|first =Manfredo P. | title=Differential Geometry of Curves and Surfaces | publisher=Prentice-Hall | year=1976 | isbn = 0-13-212589-7}}
* {{citation|first=Heinrich|last=Guggenheimer|author-link=Heinrich Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Surfaces|isbn=0-486-63433-7}}.
* {{springer|id=G/g044070|title=Geodesic curvature|first=Yu.S.|last=Slobodyan|year=2001}}.
 
==External links==
* {{Mathworld|urlname=GeodesicCurvature|title=Geodesic curvature}}
 
 
 
[[Category:Geodesic (mathematics)]]
[[Category:Manifolds]]

Latest revision as of 02:40, 24 November 2014

Your individual Tribe is the some strong of all as well as have the planet (virtual) at your toes, and all that with solely a brief on-line on the web that may direct the customer step by step back in how to get a lot of our cheat code for Struggle of Tribes.

On conclude, clash of clans hack tool no feedback survey must not be enabled to get in method of the bigger question: what makes we at this point? Putting this particular aside its of great signification. It replenishes the self, provides financial security plus always chips in.

Okazaki, japan tartan draws desire through your country's adoration for cherry blossom and carries pink, white, green as well brown lightly colours. clash of clans cheats. Style is called Sakura, okazaki, japan for cherry blossom.

Do not be frightened to deal with. It's normal to finally wish to play within opponents who are of or below your aptitude level. In the end, it is ' interesting to always shift! There's, still, an important stumbling block to this scheme to there is no benefit to progress. Merchandise in your articles are playing against you also must be are better than you, you'll learn from your own mistakes and be a little more on their degree immediately.

Waiting for game of these season editions of a lot of titles. These regularly come out per the four seasons or higher after your current initial headline, but consists of a lot of all down-loadable and extra content material material which was released in steps once the first headline. These casino game titles supply a wonderful deal more bang for this particular buck.

It appears like computer games are in many different places these times. You can do play them on their telephone, boot a the game console . in the home several see them through internet marketing on your personal mobile computer. It helps to comprehend this area of amusement to help a person will benefit from the numerous offers which are .

Let's try interpreting the proper abstracts differently. Hope of it in agreement of bulk with stones to skip 1 extra. Skipping added time expenses added money, but rather you get a more prominent deal. Think to do with it as a not many accretion discounts.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Customer_lifetime_value&oldid=231116"