formulasearchengine - User contributions [en]

Valve RF amplifier

2013-12-21T04:45:08Z

27.252.40.56: /* Bandwidth of valve vs solid state amplifiers */ Grammar (an valve-> a valve)

[[File:ATHLETE robot climbing a hill.jpg|thumb|300px|The [[JPL]] [[mobile robot]] [[ATHLETE]] is a platform with six serial chain legs ending in wheels.]]
[[File:JSC2001-01725.jpg|thumb|300px|The arms, fingers and head of the [[Lyndon B. Johnson Space Center|JSC]] [[Robonaut]] are modeled as kinematic chains.]]
[[File:SteamEngine Boulton&Watt 1784.png|thumb|right|300px|alt=Boulton & Watt Steam Engine|The movement of the [[Watt steam engine|Boulton & Watt steam engine]] is studied as a system of rigid bodies connected by joints forming a kinematic chain.]]
[[File:Modele cinematique corps humain.svg|thumb|A model of the human skeleton as a kinematic chain allows positioning using forward and inverse kinematics.]]

'''Kinematic chain''' refers to an assembly of [[Rigid body|rigid bodies]] connected by [[Joint (mechanics)|joints]] that is the [[mathematical model]] for a [[mechanical system]].<ref name=Reuleaux1876>[[Franz Reuleaux|Reuleaux, F.]], 1876 [http://books.google.com/books?id=WUZVAAAAMAAJ&printsec=frontcover&dq=kinematics+of+machinery&hl=en&sa=X&ei=qpn4Tse-E9SasgLcsZytDw&ved=0CEQQ6AEwAQ#v=onepage&q=kinematics%20of%20machinery&f=false ''The Kinematics of Machinery,''] (trans. and annotated by A. B. W. Kennedy), reprinted by Dover, New York (1963)</ref> As in the familiar use of the word [[chain]], the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is the [[kinematic]] model for a typical robot [[manipulator]].<ref name="McCarthy2010">J. M. McCarthy and G. S. Soh, 2010, [http://books.google.com/books?id=jv9mQyjRIw4C&pg=PA231&lpg=PA231&dq=geometric+design+of+linkages&source=bl&ots=j6TS1043qE&sig=R5ycw5DximWrQOEVshfiytflD6Q&hl=en&sa=X&ei=0Zj4TuiCFvCGsgKyvO3FAQ&ved=0CGAQ6AEwBQ#v=onepage&q=geometric%20design%20of%20linkages&f=false ''Geometric Design of Linkages,''] Springer, New York.</ref>

Mathematical models of the connections, or joints, between two links are termed [[kinematic pair]]s. Kinematic pairs model the hinged and sliding joints fundamental to [[robotics]], often called ''lower pairs'' and the surface contact joints critical to [[cam]]s and [[gear]]ing, called ''higher pairs.'' These joints are generally modeled as [[holonomic constraints]]. A [[kinematic diagram]] is a schematic of the mechanical system that shows the kinematic chain.

The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in [[compliant mechanism]]s and [[micro-electro-mechanical systems]], and cable compliance in cable robotic and [[tensegrity]] systems.<ref>Larry L. Howell, 2001, [http://books.google.com/books/about/Compliant_mechanisms.html?id=tiiSOuhsIfgC Compliant mechanisms], John Wiley & Sons.</ref>
<ref>Alexander Slocum, 1992, [http://books.google.com/books?id=uG7aqgal65YC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Precision Machine Design], SME</ref>

== Mobility formula ==
The [[degrees of freedom (mechanics)|degrees of freedom]], or ''mobility,'' of a kinematic chain is the number of parameters that define the configuration of the chain.<ref name="McCarthy2010"/><ref name=Uicker2003>J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, '''Theory of Machines and Mechanisms,''' Oxford University Press, New York.</ref>
A system of ''n'' rigid bodies moving in space has ''6n'' degrees of freedom measured relative to a fixed frame. This frame is included in the count of bodies, so that mobility does not depend on link that forms the fixed frame. This means the degree-of-freedom of this system is M=6(N-1), where N=n+1 is the number of moving bodies plus the fixed body.

Joints that connect bodies impose constraints. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints ''c'' that a joint imposes in terms of the joint's freedom ''f'', where ''c=6-f''. In the case of a hinge or slider, which are one degree of freedom joints, have ''f=1'' and therefore ''c=6-1=5''.

The result is that the mobility of a kinematic chain formed from ''n'' moving links and ''j'' joints each with freedom ''fi'', ''i=1, ..., j,'' is given by
:<math> M = 6n - \sum_{i=1}^j\ (6 - f_i) = 6(N-1 - j) + \sum_{i=1}^j\ f_i </math>
Recall that ''N'' includes the fixed link.

==Analysis of kinematic chains==
The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form [[algebraic equations]] that are solved to determine the configuration of the chain associated with specific values of input parameters, called [[degrees of freedom (mechanics)|degrees of freedom]].

The constraint equations for a kinematic chain are obtained using [[rigid transformation]]s [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the dimensions of each link. In the case of a serial open chain, the result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link. A chain of ''n'' links connected in series has the kinematic equations,
:<math>[T] = [Z_1][X_1][Z_2][X_2]\ldots[X_{n-1}][Z_n],\!</math>
where [T] is the transformation locating the end-link---notice that the chain includes a "zeroth" link consisting of the ground frame to which it is attached. These equations are called the [[forward kinematics]] equations of the serial chain.<ref>J. M. McCarthy, 1990, ''Introduction to Theoretical Kinematics,'' MIT Press, Cambridge, MA.</ref>

Kinematic chains of a wide range of complexity are analyzed by equating the kinematics equations of serial chains that form loops within the kinematic chain. These equations are often called ''loop equations''.

The complexity (in terms of calculating the [[forward kinematics|forward]] and [[inverse kinematics]]) of the chain is determined by the following factors:
* Its [[topology]]: a serial chain, a [[parallel manipulator]], a [[tree (graph theory)|tree]] structure, or a [[graph theory|graph]].
* Its [[Euclidean geometry|geometrical]] form: how are neighbouring [[kinematic pair|joints]] spatially connected to each other?
'''Explanation:-''' 
Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system.<ref name="Uicker2003"/>

==Synthesis of kinematic chains==
The constraint equations of a kinematic chain can be used in reverse to determine the dimensions of the links from a specification of the desired movement of the system. This is termed ''kinematic synthesis.''<ref name="Hartenberg1964">R. S. Hartenberg and J. Denavit, 1964, ''Kinematic Synthesis of Linkages,'' McGraw-Hill, New York.</ref>

Perhaps the most developed formulation of kinematic synthesis is for [[four-bar linkage]]s, which is known as [[Burmester theory]].<ref>Suh, C. H., and Radcliffe, C. W., '''Kinematics and Mechanism Design,''' John Wiley and Sons, New York, 1978.</ref><ref>Sandor,G.N.,andErdman,A.G.,1984,AdvancedMechanismDesign:AnalysisandSynthesis, Vol. 2. Prentice-Hall, Englewood Cliffs, NJ.</ref><ref>Hunt, K. H., '''Kinematic Geometry of Mechanisms,''' Oxford Engineering Science Series, 1979</ref>

[[Ferdinand Freudenstein]] is often called the father of modern kinematics for his contributions to the kinematic synthesis of [[Linkage (mechanical)|linkages]] beginning in the 1950s. His use of the newly developed computer to solve ''Freudenstein's equation'' became the prototype of [[computer-aided design]] systems.<ref name="Hartenberg1964"/>

This work has been generalized to the synthesis of spherical and spatial mechanisms.<ref name="McCarthy2010"/>

==See also==
* [[Denavit-Hartenberg parameters]]
* [[Chebychev–Grübler–Kutzbach criterion]]
* [[Configuration space]]
* [[Machine (mechanical)]]
* [[Mechanism (engineering)]]
* [[Six-bar linkage]]
* [[Simple machines]]
* [[Six degrees of freedom]]

==References==
{{reflist}}

{{DEFAULTSORT:Kinematic Chain}}
[[Category:Computer graphics]]
[[Category:3D computer graphics]]
[[Category:Computational physics]]
[[Category:Robot kinematics]]
[[Category:Virtual reality]]
[[Category:Mechanical engineering]]
[[Category:Mechanisms]]
[[Category:Diagrams]]
[[Category:Classical mechanics]]
[[Category:Kinematics]]

Differential operator

2013-12-12T02:00:08Z

27.252.64.22: /* Formal adjoint in one variable */

{{Unreferenced|date=December 2009}}
In [[abstract algebra]], a branch of [[mathematics]], an '''Archimedean group''' is an [[algebraic structure]] consisting of a [[Set (mathematics)|set]] together with a [[binary operation]] and [[binary relation]] satisfying certain axioms detailed below. We can also say that an Archimedean group is a [[linearly ordered group]] for which the [[Archimedean property]] holds. For example, the set '''R''' of [[real number]]s together with the operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after [[Archimedes]].

==Definition==
In the subsequent, we use the notation <math>na</math> (where <math>n</math> is in the set '''N''' of [[natural number]]s) for the sum of ''a'' with itself ''n'' times.

An '''Archimedean group''' (''G'', +, ≤) is a [[linearly ordered group]] subject to the following condition:

for any ''a'' and ''b'' in ''G'' which are greater than ''0'', the inequality ''na'' ≤ ''b'' holding for every ''n'' in '''N''' implies ''a'' = 0.

==Examples of Archimedean groups==
The sets of the [[integer]]s, the [[rational number]]s, the [[real number]]s, together with the operation of addition and the usual ordering (≤), are Archimedean groups.

==Examples of non-Archimedean groups==
An ordered group (''G'', +, ≤) defined as follows is not Archimedean:
* ''G'' = '''R''' × '''R'''.
* Let ''a'' = (''u'', ''v'') and ''b'' = (''x'', ''y'') then ''a'' + ''b'' = (''u'' + ''x'', ''v'' + ''y'')
* ''a'' ≤ ''b'' [[iff]] ''v'' < ''y'' or (''v'' = ''y'' and ''u'' ≤ ''x'') ([[lexicographical order]] with the least-significant number on the left).

'''Proof:''' Consider the elements (1, 0) and (0, 1). For all ''n'' in '''N''' one evidently has ''n'' (1, 0) < (0, 1).

For another example, see [[p-adic number]].

==Theorems==
For each ''a'', ''b'' in ''G'' there exist ''m'', ''n'' in '''N''' such that ''ma'' ≤ ''b'' and ''a'' ≤ ''nb''.

{{DEFAULTSORT:Archimedean Group}}
[[Category:Ordered groups]]

Lomo LC-A

2013-08-20T06:28:05Z

27.252.58.170: /* Operation */

{{Calculus |Differential}}
[[File:4 fonctions du second degré.svg|right|thumb|200px|The second derivative of a [[quadratic function]] is [[constant function|constant]].]]

In [[calculus]], the '''second derivative''', or the '''second order derivative''', of a [[function (mathematics)|function]] ''f'' is the [[derivative]] of the derivative of ''f''. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous [[acceleration]] of the vehicle, or the rate at which the [[velocity]] of the vehicle is changing.

On the [[graph of a function]], the second derivative corresponds to the [[curvature]] or concavity of the graph. The graph of a function with positive second derivative curves upwards, while the graph of a function with negative second derivative curves downwards.

== Second derivative power rule ==
The power rule for the first derivative, if solved down a bit, will produce the second derivative power rule. The rule is given below:

<math>\frac{d^2}{dx^2}[x^n]=n(n-1)x^{(n-2)}=(n^2-n)x^{(n-2)}.</math>

== Notation ==
{{Details|Notation for differentiation}}
The second derivative of a function <math>f(x)\!</math> is usually denoted <math>f''(x)\!</math>. That is:
:<math>f'' = (f')'\!</math>
When using [[Leibniz's notation]] for derivatives, the second derivative of a dependent variable ''y'' with respect to an independent variable ''x'' is written
:<math>\frac{d^2y}{dx^2}.</math>
This notation is derived from the following formula:
:<math>\frac{d^2y}{dx^2} \,=\, \frac{d}{dx}\left(\frac{dy}{dx}\right).</math>

== Example ==
Given the function
:<math>f(x) = x^3,\!</math>
the derivative of ''f'' is the function
:<math>f'(x) = 3x^2.\!</math>
The second derivative of ''f'' is the derivative of ''f''′, namely
:<math>f''(x) = 6x.\!</math>

== Relation to the graph ==
[[File:Animated illustration of inflection point.gif|500px|thumb|A plot of <math>f(x) = \sin(2x)</math> from <math>-\pi/4</math> to <math>5\pi/4</math>. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, <math>\pi</math>/2, and <math>\pi</math>).]]

=== Concavity ===
The second derivative of a function ''f'' measures the '''concavity''' of the graph of ''f''. A function whose second derivative is positive will be [[concave up]] (sometimes referred to as convex), meaning that the [[tangent]] line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be [[concave down]] (sometimes called simply “concave”), and its tangent lines will lie above the graph of the function.

=== Inflection points ===
{{main|Inflection point}}
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an '''inflection point'''. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

=== Second derivative test ===
{{main|Second derivative test}}
The relation between the second derivative and the graph can be used to test whether a [[stationary point]] for a function (i.e. a point where <math>f'(x)=0\!</math>) is a [[local maximum]] or a [[local minimum]]. Specifically,
* If <math>\ f^{\prime\prime}(x) < 0</math> then <math>\ f</math> has a local maximum at <math>\ x</math>.
* If <math>\ f^{\prime\prime}(x) > 0</math> then <math>\ f</math> has a local minimum at <math>\ x</math>.
* If <math>\ f^{\prime\prime}(x) = 0</math>, the second derivative test says nothing about the point <math>\ x</math>, a possible inflection point.
The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.

== Limit ==
It is possible to write a single [[Limit (mathematics)|limit]] for the second derivative:
:<math>f''(x) = \lim_{h \to 0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.</math>
The expression on the right can be written as a [[difference quotient]] of difference quotients:
:<math>\frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = \frac{\frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h}}{h}.</math>
This limit can be viewed as a continuous version of the [[second difference]] for [[sequence (mathematics)|sequences]].

Please note that the existence of the above limit does not mean that the function <math>f</math> has a second derivative. The limit above just gives a possibility for calculating the second derivative but does not provide a definition. As a counterexample look on the [[sign function]] <math>\sgn(x)</math> which is defined through

:<math>\sgn(x) = \begin{cases}
-1 & \text{if } x < 0, \\
0 & \text{if } x = 0, \\
1 & \text{if } x > 0. \end{cases}</math>

The sign function is not continuous at zero and therefore the second derivative for <math>x=0</math> does not exist. But the above limit exists for <math>x=0</math>:

:<math>\begin{align}
\lim_{h \to 0} \frac{\sgn(0+h) - 2\sgn(0) + \sgn(0-h)}{h^2} &= \lim_{h \to 0} \frac{1 - 2\cdot 0 + (-1)}{h^2} \\
&= \lim_{h \to 0} \frac{0}{h^2} \\
&= 0 \end{align}</math>

== Quadratic approximation ==
Just as the first derivative is related to [[linear approximation]]s, the second derivative is related to the best [[quadratic approximation]] for a function ''f''. This is the [[quadratic function]] whose first and second derivatives are the same as those of ''f'' at a given point. The formula for the best quadratic approximation to a function ''f'' around the point ''x'' = ''a'' is
:<math>f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2.</math>
This quadratic approximation is the second-order [[Taylor polynomial]] for the function centered at ''x'' = ''a''.

== Eigenvalues and eigenvectors of the second derivative ==

For many combinations of [[boundary conditions]] explicit formulas for [[eigenvalues and eigenvectors of the second derivative]] can be obtained. For example, assuming <math>x \in [0,L]</math> and homogeneous [[Dirichlet boundary conditions]], i.e., <math> v(0)=v(L)=0</math>, the [[eigenvalues]] are <math> \lambda_j = -\frac{j^2 \pi^2}{L^2}</math> and the corresponding [[eigenvectors]] (also called [[eigenfunctions]]) are <math> v_j(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{j \pi x}{L}\right) </math>. Here, <math> v''_j(x) = \lambda_j v_j(x), \, j=1,\ldots,\infty.</math>

For other well-known cases, see the main article [[eigenvalues and eigenvectors of the second derivative]].

== Generalization to higher dimensions ==

=== The Hessian ===
{{main|Hessian matrix}}
The second derivative generalizes to higher dimensions through the notion of second [[partial derivative]]s. For a function ''f'':'''R'''3 → '''R''', these include the three second-order partials

:<math>\frac{\part^2 f}{\part x^2}, \; \frac{\part^2 f}{\part y^2}, \text{ and }\frac{\part^2 f}{\part z^2}</math>

and the mixed partials

:<math>\frac{\part^2 f}{\part x \, \part y}, \; \frac{\part^2 f}{\part x \, \part z}, \text{ and }\frac{\part^2 f}{\part y \, \part z}.</math>

If the function's image and domain both have a potential, then these fit together into a [[symmetric matrix]] known as the '''Hessian'''. The [[eigenvalue]]s of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the [[second partial derivative test]].)

=== The Laplacian ===
{{main|Laplace operator}}
Another common generalization of the second derivative is the '''Laplacian'''. This is the differential operator <math>\nabla^2</math> defined by
:<math>\nabla^2 f = \frac{\part^2 f}{\part x^2}+\frac{\part^2 f}{\part y^2}+\frac{\part^2 f}{\part z^2}.</math>
The Laplacian of a function is equal to the [[divergence]] of the [[gradient]].

== References ==
=== Print ===
*{{Citation
| last = Anton
| first = Howard
| last2 = Bivens
| first2 = Irl
| last3 = Davis
| first3 = Stephen
| date = February 2, 2005
| title = Calculus: Early Transcendentals Single and Multivariable
| place = New York
| publisher = Wiley
| edition = 8th
| isbn = 978-0-471-47244-5
}}
*{{Citation
| last = Apostol
| first = Tom M.
| date = June 1967
| title = Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra
| publisher = Wiley
| edition = 2nd
| volume = 1
| isbn = 978-0-471-00005-1
}}
*{{Citation
| last = Apostol
| first = Tom M.
| date = June 1969
| title = Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications
| publisher = Wiley
| edition = 2nd
| volume = 1
| isbn = 978-0-471-00007-5
}}
*{{Citation
| last = Eves
| first = Howard
| date = January 2, 1990
| title = An Introduction to the History of Mathematics
| edition = 6th
| publisher = Brooks Cole
| isbn = 978-0-03-029558-4
}}
*{{Citation
| last = Larson
| first = Ron
| last2 = Hostetler
| first2 = Robert P.
| last3 = Edwards
| first3 = Bruce H.
| date = February 28, 2006
| title = Calculus: Early Transcendental Functions
| edition = 4th
| publisher = Houghton Mifflin Company
| isbn = 978-0-618-60624-5
}}
*{{Citation
| last = Spivak
| first = Michael
| author-link = Michael Spivak
| date = September 1994
| title = Calculus
| publisher = Publish or Perish
| edition = 3rd
| isbn = 978-0-914098-89-8
}}
*{{Citation
| last = Stewart
| first = James
| date = December 24, 2002
| title = Calculus
| publisher = Brooks Cole
| edition = 5th
| isbn = 978-0-534-39339-7
}}
*{{Citation
| last = Thompson
| first = Silvanus P.
| date = September 8, 1998
| title = Calculus Made Easy
| edition = Revised, Updated, Expanded
| place = New York
| publisher = St. Martin's Press
| isbn = 978-0-312-18548-0
}}

=== Online books ===

*{{Citation
| last = Crowell
| first = Benjamin
| title = Calculus
| year = 2003
| url = http://www.lightandmatter.com/calc/
}}
*{{Citation
| last = Garrett
| first = Paul
| year = 2004
| title = Notes on First-Year Calculus
| url = http://www.math.umn.edu/~garrett/calculus/
}}
*{{Citation
| last = Hussain
| first = Faraz
| year = 2006
| title = Understanding Calculus
| url = http://www.understandingcalculus.com/
}}
*{{Citation
| last = Keisler
| first = H. Jerome
| year = 2000
| title = Elementary Calculus: An Approach Using Infinitesimals
| url = http://www.math.wisc.edu/~keisler/calc.html
}}
*{{Citation
| last = Mauch
| first = Sean
| year = 2004
| title = Unabridged Version of Sean's Applied Math Book
| url = http://www.its.caltech.edu/~sean/book/unabridged.html
}}
*{{Citation
| last = Sloughter
| first = Dan
| year = 2000
| title = Difference Equations to Differential Equations
| url = http://synechism.org/drupal/de2de/
}}
*{{Citation
| last = Strang
| first = Gilbert
| year = 1991
| title = Calculus
| url = http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
}}
*{{Citation
| last = Stroyan
| first = Keith D.
| year = 1997
| title = A Brief Introduction to Infinitesimal Calculus
| url = http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm
}}
*{{Citation
| last = Wikibooks
| title = Calculus
| url = http://en.wikibooks.org/wiki/Calculus
}}

== External links ==
*[http://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/ Discrete Second Derivative from Unevenly Spaced Points]

[[Category:Mathematical analysis]]
[[Category:Differential calculus]]
[[Category:Functions and mappings]]
[[Category:Linear operators in calculus]]