User:Philc 0780/Sandbox: Difference between revisions - Wikipedia

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=Random=

=Partial Differentiation=

==Usage==

If you have a function of two variables such that <math>p = f(q,r)\,</math>

then the following equation is true in general

:<math> \frac{dp}{dx}= \frac{\partial p}{\partial q} \frac{dq}{dx} + \frac{\partial p}{\partial r} \frac{dr}{dx}</math>

this allows you to calculate the rate of change of <math>p</math> given the rate of change of <math>q</math> and <math>r</math>. For example, in the case of a cone

:<math>V = \frac{\pi r^2 h}{3}</math>

:<math>\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3}</math>

:<math>\frac{ \partial V}{\partial h} = \frac{ 2 \pi r^2}{3}</math>

this means that the following is true

:<math> \frac{dV}{dt}= \frac{ 2 \pi r }{3} \bigg ( h \frac{dr}{dt} + r \frac{dh}{dt} \bigg )</math>

=Ref Desk Q's=

== electrons ==

[[Image:Electrons-RefDeskQ.jpg|thumb|left|350px]]

I asked a similar question this yesterday, though this one seems much more complicated to me. I will go through my calucaltions so far, so if there are any errors, you can help me rectify them.

If you have n electrons distributed evenly around a circle radius r, distance x from another electron, what is the force of repulsion experienced by the electron not on the circle from the circle of electrons. Electrostatic repulsion is a inverse square law, and the coefficent of electrostatic force can be considered as k.

*r = radius (OA, OC)

*x = CY

*y = OB

*a = AY

*α = AOY

*β = BAY

using cosine rule;

<math>a = \sqrt{x^2 + (x+r)^2 - 2r(r+x)\cos\alpha}</math>

so the force, being an inverse square is;

<math>F = \frac{k}{x^2 + (x+r)^2 - 2r(r+x)\cos\alpha}</math>

now, seeing as always in the circle the components in y-direction will cancel, we can consider only those in the x-direction in order to caluclate a resultant force we need to consider only the x-directions components, to find this consider the triangle ABY, the angle β and distance OB are neccesary.

using sine rule.

<math>y=-r\cos(\alpha)\,</math>

therefore using sine rule,

<math>\sin\beta = \frac{r-r\cos(\alpha)+x}{\sqrt{x^2 + (x+r)^2 - 2r(r+x)\cos\alpha}} </math>

so the x direction component of F, or Fsinβ is;

<math>\mathit{F_x}=\frac{k(r-r\cos(\alpha)+x)}{\big[x^2 + (x+r)^2 - 2r(r+x)\cos\alpha\big]^\frac{3}{2}}</math>

So to consider all the electrons; then <math>\alpha = \frac{2\pi\mathit{k}}{N}</math> where N number of electrons. and k varies from 1 to N. Therfore a sum of all the forces would be.

<math>\mathit{F_{total}}=\frac{k}{N}\sum_{k=1}^N\bigg[\frac{r-r\cos(\frac{2\pi k}{N})+x}{\big[x^2 + (x+r)^2 - 2r(r+x)\cos(\frac{2\pi k}{N})\big]^\frac{3}{2}}\bigg]</math>

Now this is it, I really need some help simplifying this massive mess, and like yesterday I really need the summation series out of the equation if that's possible. Thank you.

==data handling==

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==Russian Launch Vehicles Template==

<div style="background:#B0C4DE; text-align: center; border: 1px solid #667766" class="NavHead"><div style="float:left;margin-left:2px;">{{~~Tnavbar-~~navbar|mini=1|Cold War}}</div>'''Russian Launch Vehicles'''

</div>

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|[[1998]] ''[[CAF Champions League 1998|Details]]''

|'''[[ASEC Abidjan]]''' [[~~Image~~File:Flag of ~~Cote~~Côte d'Ivoire.svg|20px]] ([[Côte d'Ivoire]])

|1st leg: '''4-2''' 2nd leg: '''0-0'''

|[[Dynamos]] [[Image:Flag of Zimbabwe.svg|20px]] ([[Zimbabwe]])

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|[[Espérance]] [[Image:Flag of Tunisia.svg|20px]] ([[Tunisia]])

|[[Al-Ahly]] [[Image:Flag of Egypt.svg|20px]] ([[Egypt]])

|[[ASEC Abidjan]] [[~~Image~~File:Flag of ~~Cote~~Côte d'Ivoire.svg|20px]] ([[Côte d'Ivoire]])

|- align=center bgcolor=#F5FAFF

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|[[Raja Casablanca]] [[Image:Flag of Morocco.svg|20px]] ([[Morocco]])

|[[TP Mazembe]] [[Image:Flag of the Democratic Republic of the Congo.svg|20px]] ([[DRC]])

|[[ASEC Abidjan]] [[~~Image~~File:Flag of ~~Cote~~Côte d'Ivoire.svg|20px]] ([[Côte d'Ivoire]])

|- align=center bgcolor=#D0E7FF

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The European Nations Cup changed its name to the UEFA European Football Championship for 1968 and also a new format was introduced. Eight groups of seeded teams played each other twice and the top side of each group proceeded to two-legged quarter finals. The semi-finals and final were played in the host country of [[Italy national football team|Italy]] who won the competition after a 2-0 replay of the final, having drawn 1-1 in the first against Yugoslavia.

In the 1972 tournament, the same structure was retained, with [[Belgium national football team|Belgium]] being the host of the finals. [[~~West~~ Germany national football team|~~West~~ Germany]] won the competition, beating the Soviet Union 2-0 in the final. The 1976 final round was hosted by [[Yugoslavia]]. In the final, [[Czechoslovakia national football team|Czechoslovakia]] squandered a two-goal lead before penalties were needed. When [[Uli Hoeness]] missed, it allowed [[Antonín Panenka]] to chip into the space vacated by [[Sepp Maier]]'s anticipatory dive for a Czechoslovakian victory.

===Another Change===

In 1980 UEFA introduced a new format, which saw eight teams go to the finals instead of four, and then play each other in two groups of four teams; the winner of each group then proceeded to the final. ~~West~~ Germany faced [[Belgium national football team|Belgium]] in the final, which they won 2-0 after two goals from [[Horst Hrubesch]]. However the newer format was not retained in 1984, and a different format was used, again employing group stages, but this time in place of only the quarter-finals. The top two teams from each of the two groups progressed to the semi-finals. The competition was held in France for the second time. The hosts won 2-0 in Paris against Spain with goals from [[Michel Platini]] and [[Bruno Bellone]].

The [[1988 European Football Championship|1988 competition]] was held in [[~~West~~ Germany]], while the format from the [[1984 European Football Championship|1984 competition]] was retained. [[Marco Van Basten]] led the [[Netherlands national football team|Netherlands]], scoring what was later voted the best goal ever in the competition <ref name="UEFAEuro2004Final_MatchDayPublication">{{cite news|title=UEFA Euro 2004 Final Match Day Publication|publisher=UEFA|date=July 4th 2006|pages=59}}</ref>.

===Great Political Change===

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from:22/06/1966 till:15/07/1966 shift:(20,-6) text:"[[Brazil national football team|——————— Brazil]]" color:Brazil

from:15/07/1966 till:25/07/1966 shift:(20,-3) text:"[[USSR national football team|———————————— USSR]]" color:USSR

from:25/07/1966 till:30/07/1966 shift:(20,+15) text:"[[~~West~~ Germany national football team|~~West~~ Germany]]" color:West_Germany