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Jerk

Jerk

:This article is about the physics concept of jerk. For other terms of jerk, see Jerk (disambiguation) In physics, jerk (in British English, jolt), also called surge, is the derivative of acceleration with respect to time (or the third derivative of displacement). Yank is mass times jerk, or equivalently, the derivative of force with respect to time. Jerk is a vector, and there is no generally used term to describe its scalar value. The units of jerk are metres per second cubed (m/s³). There is no universal agreement on the symbol for jerk, but j is commonly used. Jerk is used at times in engineering, especially when building roller coasters. Some precision or fragile objects—such as passengers, who need time to sense stress changes and adjust their muscle tension, or suffer e.g. whiplash—can be safely subjected not only to a maximum acceleration, but also to a maximum jerk. Jerk may be considered when the excitation of vibrations is a concern. A device which measures jerk is called a "jerkmeter." Jerk is also important to consider in manufacturing processes. Rapid changes in acceleration of a cutting tool, for example going from zero to 100 percent instantaneously, result in theoretically infinite jerk. This can lead to premature tool wear and result in uneven lines of a cut. This is why modern motion controllers include such features as jerk limitation. Higher derivatives of displacement are rarely necessary, and hence lack agreed names. The fourth derivative of position was considered in development of the Hubble Space Telescope's pointing control system, and called jounce. Many other suggestions have been made, such as jilt, jouse, jolt and delta jerk. As more distinct terms that start with letters other than "j", snap, crackle and pop have been proposed for the 4th, 5th, and 6th derivatives of displacement, respectively, with some nonzero positive value for tongue-in-cheek.
External links

- [http://math.ucr.edu/home/baez/physics/General/jerk.html What is the term used for the third derivative of position?], description of jerk in the [http://math.ucr.edu/home/baez/physics/index.html Usenet Physics FAQ]. Category:Classical mechanics Category: Physical quantity ja:躍度

Jerk (disambiguation)
Jerk may refer to:
- Jerk, in physics, the derivative of acceleration with respect to time (or the third derivative of displacement), ie the "rate of change of acceleration"
- Jerk (band), an industrial metal band from Sydney, Australia
- The Jerk (film), a comedy film starring Steve Martin
- The Jerk (dance), a dance craze
- Jerk, the Jamaican jerk spice, style of cooking native to Jamaica in which meats are dry-rubbed with a fiery spice mixture Note that "Jerk" is also American English slang for a person who is socially inept and deliberately selfish. see The Jerk (film)

Derivative

In mathematics, the derivative is one of the two central concepts of calculus. (The other is the integral; the two are related via the fundamental theorem of calculus.) The simplest type of derivative is the derivative of a real-valued function of a single real variable. It has several interpretations:
- The derivative gives the slope of a tangent to the graph of the function at a point. In this way, derivatives can be used to determine many geometrical properties of the graph, such as concavity or convexity.
- The derivative provides a mathematical formulation of rate of change; it measures the rate at which the function's value changes as the function's argument changes. This derivative is the kind usually encountered in a first course on calculus, and historically was the first to be discovered. However, there are also many generalizations of the derivative. The remainder of this article discusses only the simplest case (real-valued functions of real numbers).

Differentiation and differentiability

In physical terms, differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ to refer to change in a quantity, this rate is defined as a limit of difference quotients :

\frac

as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written :

\frac

suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "dy by dx" or "dy over dx". The form "dy dx" is also used conversationally, although it may be confused with the notation for element of area. Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. The precise definition of this operation (which therefore need not deal with infinitesimal quantities) is given as: :

\lim_\frac.

A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval. If a function is not continuous at x, then there is no tangent line and the function is therefore not differentiable at x; however, even if a function is continuous at x, it may not be differentiable there. In other words, differentiability implies continuity, but not vice versa. One famous example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. The derivative of a differentiable function can itself be differentiable. The derivative of a derivative is called a second derivative. Similarly, the derivative of a second derivative is a third derivative, and so on.

Newton's difference quotient

The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x. Without the concept which we are about to define, it is impossible to directly find the slope of the tangent line to a given function, because we only know one point on the tangent line, namely (x, f(x)). Instead, we will approximate the tangent line with multiple secant lines that have progressively shorter distances between the two intersecting points. When we take the limit of the slopes of the nearby secant lines in this progression, we will get the slope of the tangent line. The derivative is then defined by taking the limit of the slope of secant lines as they approach the tangent line. tangent tangent To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is :

.

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: :

f'(x)=\lim_.

difference quotient If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x. Since immediately substituting 0 for h results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens easily for polynomials; see calculus with polynomials. For almost all functions however, the result is a mess. Fortunately, many guidelines exist.

Notations for differentiation

Lagrange's notation

The simplest notation for differentiation that is in current use is due to Joseph Louis Lagrange and uses the prime mark:

Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Leibniz. For the function whose value at x is the derivative of f at x, we write: : $\frac.$ We can write the derivative of f at the point a in two different ways: : $\frac\left.\right|_ = \left(\frac\right)(a).$ If the output of f(x) is another variable, for example, if y=f(x), we can write the derivative as: : $\frac.$ Higher derivatives are expressed as : $\frac$ or $\frac$ for the n-th derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the 3rd derivative is: : $\frac$ which we can loosely write as: : $\left(\frac\right)^3 \left(f(x)\right) =
\frac \left(f(x)\right).$ Dropping brackets gives the notation above. Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember, because the "du" terms appear symbolically to cancel: : $\frac = \frac \cdot \frac.$ (In the popular formulation of calculus in terms of limits, the "du" terms cannot literally cancel, because on their own they are undefined; they are only defined when used together to express a derivative. In nonstandard analysis, however, they can be viewed as infinitesimal numbers that cancel.)
Newton's notation
Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name: : $\dot = \frac = x'(t)$ : $\ddot = x$ (t) and so on. Newton's notation is mainly used in mechanics, normally for time derivatives such as velocity and acceleration, and in ODE theory. It is usually only used for first and second derivatives.

Euler's notation

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function with the variable as a subscript of the operator: This notation can also be abbreviated when taking derivatives of expressions that contain a single variable. The subscript to the operator is dropped and is assumed to be the only variable present in the expression. In the following examples, u represents any expression of a single variable: Euler's notation is useful for stating and solving linear differential equations.

Critical points

Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero). If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum. Taking derivatives and solving for critical points is often a simple way to find local minima or maxima, which can be useful in optimization. In fact, local minima and maxima can only occur at critical points. This is related to the extreme value theorem.

Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative"—the rate of change over time—which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:
- Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
- Acceleration is the derivative (with respect to time) of an object's velocity.
- Jerk is the derivative (with respect to time) of an object's acceleration. For example, if an object's position

p(t) = -16t^2 + 16t + 32

; then, the object's velocity is

\dot p(t) = p'(t) = -32t + 16

; the object's acceleration is

\ddot p(t) = p

(t) = -32; and the object's jerk is $p(t) = 0.$ If the velocity of a car is given, as a function of time, then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.
Algebraic manipulation
Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.
- Constant rule: The derivative of any constant is zero.
- Constant multiple rule: If c is some real number; then, the derivative of $cf(x)$ equals c multiplied by the derivative of f(x) (a consequence of linearity below).
- Linearity: $(af + bg)' = af' + bg'$ for all functions f and g and all real numbers a and b.
- Power rule: If $f(x) = x^r$ , for some real number r; $f'(x) = rx^$ .
- Product rule: $(fg)' = f'g + fg'$ for all functions f and g.
- Quotient rule: $(f/g)' = (f'g - fg')/(g^2)$ unless g is zero.
- Chain rule: If $f(x) = h(g(x))$ , then $f'(x) = h'(g(x)) g'(x)$ .
- Inverse function: If $g(x) = f^(x)$ , and $f(x)$ is injective, then $g'(x) = 1/f'(f^(x))$ .
- Derivative of one variable with respect to another when both are functions of a third variable: Let $x = f(t)$ and $y = g(t)$ . Now $d y/d x = (d y/d t)/(d x/d t)$ . This is the chain rule in the Leibniz notation.
- Implicit differentiation: If $f(x,y) = 0$ is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y). In addition, the derivatives of some common functions are useful to know. See the table of derivatives. As an example, the derivative of : $f(x) = 2x^4 + \sin (x^2) - \ln (x)\;e^x + 7$ is : $f'(x) = 8x^3 + 2x\cos (x^2) - \frac\;e^x - \ln (x)\;e^x.$
Using derivatives to graph functions
Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither. In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3). Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.
Generalizations
Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Some people pronounce the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'. The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles. In order to differentiate all continuous functions and much more, one defines the concept of distribution. For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument. For example, the function f(x + iy) = x + 2iy satisfies the latter, but not the first. See also the article on holomorphic functions.
See also

- Derivative (examples)
- Derivative (generalizations)
- Partial derivative
- Total derivative
- Table of derivatives
- Smooth function
- Differintegral
- Automatic differentiation
External links

- [http://wims.unice.fr/wims/wims.cgi?module=tool/analysis/function.en WIMS Function Calculator] makes online calculation of derivatives; this software enables also interactive exercises.
References

- Spivak, Michael; Calculus (3rd edition, 1994) Publish or Perish Press. ISBN 0914098896. Explains why all this works.
- Thompson, Silvanus Phillips, Calculus made easy : being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus New York : St. Martin's Press, 1998 ISBN 0312185480. Introduced by Martin Gardner. "What one fool can do, another can."
- Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.
- Anton, Howard (1980). Calculus with analytical geometry.. New York:John Wiley and Sons. ISBN 0-471-03248-4.
- ko:미분 ja:微分 simple:Derivative th:อนุพันธ์

Displacement

The term displacement can have one of several meanings, depending on context:
- Displacement (distance), a physical quantity in kinematics
- another term for a congruent transformation in geometry
- Electric displacement field, a physical quantity in electrodynamics
- Engine displacement, a property of an internal combustion engine
- Displacement (fencing)
- Displacement (fluid), a different physical quantity, used in fluid mechanics and navigation; used as a measure of a ship's size
- Displacement hull, where the moving hull's weight is supported by buoyancy alone and it must displace water from its path rather than planing on the water's surface
- A ship's displacement, also known as tonnage
- Displacement aka offset used in relative addressing of computer memory
- Particle displacement, acoustics of sound in air
- Displacement of people by persecution or violence
- Displacement (psychology)
- Single or double displacement reaction, a chemical reaction concerning the exchange of ions
- Displacement in Orthopedic surgery refers to change in alignment of the fracture fragments.

Force (physics)

:For other senses of this word, see force (disambiguation). In physics, a force is an external cause responsible for any change of a physical system. For instance, a person holding a dog by a rope is experiencing the force applied by the rope on their hand, and the cause for its pulling forward is the force exercised by the rope. The kinetic expression of this change is, according to Newton's second law, acceleration, non kinetic expressions such as deformation can also occur. The SI unit for force is the newton.

Elementary concepts

Force in its most primitive definition can be thought of as that which when acting alone causes an object to accelerate. In a practical sense forces can be divided into two groups: contact forces and field forces. Contact forces require the physical contact of one object with other such as a hammer striking a nail or the force exerted by a gas under pressure - gas produced by exploding gunpowder forces a heavy ball out of a cannon. Field forces on the other hand need no physical medium of contact. Gravity and magnetism are examples of such forces. It should be noted however, that fundamentally all forces are in fact field forces. The force of hammer striking the nail in the previous example turns out to be a clash of the electric forces in both hammer and nail. Nevertheless it is appropriate in some cases to maintain these two classifications for ease of understanding.

Quantitative definition

In physics models, the point-like system is used, where objects are represented as one-dimensional points at their centre of mass. The only change the system can experience is a change of its momentum (its speed). Since the rise of the atomic theory, any physical system has been considered in classical physics as composed of point-like systems called atoms or molecules. Therefore, all forces can be defined by their effect; that is, by the change of movement they induce on point-like systems. This change of movement can be quantified by the acceleration (the derivative of velocity). The discovery by Isaac Newton that a given force will induce an acceleration in inverse proportion to a quantity called the mass of inertia or inertial mass which is independent of the speed of the system is Newton's second law. This law allows us to predict the effect of a force on any point-like system whose mass is known. It is usually written as: :F = dp/dt = d(m·v)/dt = m·a (in the case where m does not depend on t) where :F is the force (a vector quantity), :p is the momentum, :t is the time, :v is the velocity, :m is the mass, and :a=d²x/dt² is the acceleration, the second derivative with respect to t of the position vector x. If the mass m is measured in kilograms and the acceleration a is measured in metres per second squared, then the unit of force is kilogram × metre/second squared. This unit is called the newton: 1 N = 1 kg x 1 m/s². This equation is a system of three second-order differential equations with respect to the three-dimensional position vector which is an unknown function of time. This equation can be solved if F is a known function of x and some of its derivatives and if the mass m is known. Morevover the boundary conditions are required; for example, the values of the position vector and x and the velocity v at the starting time, say t=0. Of course, this formula is only useful if one knows the numerical values of F and m. The definition above is an implicit definition, arrived at as follows. One defines a reference system (one litre of water) and a reference force (the gravitational force applied by the Earth on it at the altitude of Paris). One takes Newton's second law for granted (one postulates that it is true) and measures the acceleration induced by the reference force on the reference system. This gives us a mass unit (1 kg) and a force unit (the older unit of 1 kilogram-force = 9.81 N). Once this is done, one can measure any force by the acceleration it induces on the reference system and measure the inertial mass of any system by measuring the acceleration induced on this system by the reference force. Force is often considered a fundamental quantity in physics, but there are more fundamental quantities, such as momentum (p = mass m x velocity v). Energy, measured in joules, is still less fundamental than force and momentum, because it is defined as work, and work is defined in terms of force. The two most fundamental theories of nature - quantum electrodynamics and general relativity - do not contain the concept of force at all. Although not the most fundamental quantity in physics, force is an important basic mathematical concept from which other concepts, such work and pressure (measured in pascals), are derived. Force is sometimes confused with stress.

Types of force

There are four known fundamental forces in nature.
- Nuclear forces acting between subatomic particles
- Electromagnetic forces between electric charges
- Weak forces arising from radioactive decay
- Gravitational forces between masses Quantum field theory accurately models the first three fundamental forces, but does not model quantum gravity. Quantum gravity on a large scale can, however, be described by general relativity. The four fundamental forces describe every observable phenomenon including the many other forces observed such as: Coulomb's force (the force between electrical charges), gravitational force (force between masses), magnetic force, frictional forces, centripetal, centrifugal, impact force, and spring force, to name a few. Forces can also be classified into conservative forces and nonconservative forces. Conservative forces are equivalent to the gradient of a potential, and include gravity, electromagnetic force, and spring force. Nonconservative forces include friction and drag.

Properties of force

Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction. Forces can be added together using the parallelogram of force. When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from zero to the sum of the magnitudes of the two forces, depending on the angle between their lines of action. If the two forces are equal, but opposite, the resultant is zero. This condition is called static equilibrium, with the result that the object remains at rest or moves with a constant velocity. As well as being added, forces can be can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.

Forces in theory

The total (Newtonian) force, in newtons, on an object at any given time is defined as the rate of change of the object's velocity multiplied by the object's mass: :

\mathbf = \lim_ \frac

where :m is the inertial mass of the particle (measured in kilograms) :v_o is its initial velocity (measured in metres per second) :v is its final velocity (measured in metres per second) :T is the time from the initial state to the final state (measured in seconds); :Lim T→0 is the limit as T tends towards zero. Force was so defined to explain the effects of superimposing situations: if in one situation, a force is experienced by a particle, and if in another situation another force is experienced by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, and the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion. There are other ways to look at the above definition of force. First, the mass of a body multiplied by its velocity is called its momentum, p, so the above definition is equivalent to: :

\textbf=

If F is not constant over Δt, then this is the definition of average force over the time interval. To apply it at an instant we apply an idea from calculus. If we graph p as a function of time, the average force will be the slope of the line connecting the momentum at two times. Taking the limit as the two times get closer together gives the slope at an instant, which is called the derivative: :

\textbf=

Many forces are associated with a potential energy field. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. The potential field is defined as that field whose gradient is equal and opposite to the force produced at every point: :

\textbf=-\nabla U

The derivative of force with respect to time is called yank. Higher order derivatives are sometimes used, but they lack names because of their rarity. In most explanations of mechanics, force is usually defined only implicitly, in terms of the equations that work with it. Some physicists, philosophers and mathematicians, such as Ernst Mach, Clifford Truesdell and Walter Noll, have found this problematic and sought a more explicit definition of force. According to the Special theory of relativity the mass of an object increases as it's velocity and therefore it's energy increases. The law of force must then be modified to the following: :

\textbf=

where :v is the mass's velocity :c is the speed of light. Note that the equation is undefined if the mass's speed is equal to c because one then has to divide by zero. This is one reason most physicists believe an object with nonzero rest mass can not be accelerated to the speed of light, as this would require an infinite force.

Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s⁻².

Non-SI units of force and mass

The F=m·a relationship can be used with any consistent units (SI or CGS). If these units are not consistent, a more general form, F=k·m·a, can be used, where the constant k is a conversion factor dependent upon the units being used. For example, in imperial engineering units, F is measured in "pounds force" or "lbf", m in "pounds mass" or "lb", and a in feet per second squared. In this particular system, one needs to use the more general form above, usually written F=m·a/g_c with the constant normally used for this purpose g_c = 32.174 lb·ft/(lbf·s²) equal to the reciprocal of the k above. As with the kilogram, the pound is colloquially used as both a unit of mass and a unit of force. 1 lbf is the force required to accelerate 1 lb at 32.174 ft per second squared, since 32.174 ft per second squared is the standard acceleration due to terrestrial gravity. Another imperial unit of mass is the slug, defined as 32.174 lb. It is the mass that accelerates by one foot per second squared when a force of one lbf is exerted on it. When the standard gee (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf. The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard gee which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity. By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the hyl, TME (from a German acronym), and mug (from metric slug). Another unit of force called the poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl. The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern SI system, but is still used in applications such as:
- Thrust of jet and rocket engines
- Spoke tension of bicycles
- Draw weight of bows
- Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
- Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
- Pressure gauges in "kg/cm²" or "kgf/cm²" In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force. The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to disintinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.

Conversions

Below are several coversion factors between various mesurements of force:
- 1 kgf (kilopond kp) = 9.80665 newtons
- 1 metric slug = 9.80665 kg
- 1 lbf = 32.174 poundals
- 1 slug = 32.174 lb
- 1 kgf = 2.2046 lbf

Forces in everyday life

Forces are part of everyday life, with examples such as:
- gravity: objects fall, even after being thrown upwards, or slide and roll down
- friction: floors and objects that are not extremely slippery
- spring force, objects resist tensile stress, compressive stress and/or shear stress, objects bounce back.
- electromagnetic force: attraction of magnets
- movement created by force: the movement of objects when force is applied.

Forces in the laboratory

Founding experiments

- Galileo Galilei used rolling balls to disprove the Aristotelian theory of motion (1602 - 1607)
- Henry Cavendish's torsion bar experiment measured the force of gravity between two masses (1798)

Instruments to measure forces

- spring balance
- pivot balance
- forcemeter

History

Force was first described by Archimedes.

References

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External links

- [http://jumk.de/calc/force.shtml Calculation: force F - English and American units to metric units]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.patbelford.com/gallery/web3d/education/forceworkpower/index.html Interactive demonstration of Force-Work-Power Relationship] Category:Introductory physics ko:힘 ms:Daya (fizik) ja:力 simple:Force (physics)

SI base unit

The SI system of units defines seven SI base units: fundamental physical units defined by an operational definition. All other physical units can be derived from these base units: these are known as SI derived units. Derivation is by dimensional analysis. Use SI prefixes to abbreviate long numbers. The following are the fundamental units from which all others are derived, they are dimensionally independent. The definitions stated below are widely accepted.

External links

- [http://www1.bipm.org/en/si/base_units/ BIPM]
- [http://www.npl.co.uk/mass/faqs/kilogram.html NPL - Kilogram]
- [http://physics.nist.gov/cuu/Units/index.html NIST -SI] Category:SI units Category:Dimensional analysis ja:SI基本単位 ko:SI 기본 단위 simple:SI base unit zh-cn:国际标准基准单位 zh-tw:國際標準基準單位

Metre

:This article is about the unit of length. For other uses of metre or meter, see meter (disambiguation). The metre (Commonwealth English) or meter (American English) (symbol: m) is the SI base unit of length. It is defined as the length of the path travelled by light in absolute vacuum during a time interval of 1/299,792,458 of a second. Adding SI prefixes to metre creates multiples and submultiples; for example kilometre (1000 metres; kilo- = 1000) and millimetre (one thousandth of a metre; milli- = 1 / 1 000).

Conversions

1 metre is equivalent to:
- exactly 1/0.9144 yards (approximately 1.0936 yards)
- exactly 1/0.3048 feet (approximately 3.2808 feet)
- exactly 10000/254 inches (approximately 39.370 inches)

History

The word metre is from the Greek metron (μετρον), "a measure" via the French mètre. Its first recorded usage in English is from 1797. In the 18th century, there were two favoured approaches to the definition of the standard unit of length. One suggested defining the metre as the length of a pendulum with a half-period of one second. The other suggested defining the metre as one ten-millionth of the length of the earth's meridian along a quadrant (one-fourth the polar circumference of the earth). In 1791, the French Academy of Sciences selected the meridional definition over the pendular definition because of the slight variation of the force of gravity over the surface of the earth, which affects the period of a pendulum. In 1793, France adopted the metre, with this definition, as its official unit of length. Although it was later determined that the first prototype metre bar was short by a fifth of a millimetre due to miscalculation of the flattening of the earth, this length became the standard. So, the circumference of the Earth through the poles is approximately forty million metres. Earth in a vacuum.]] In the 1870s and in light of modern precision, a series of international conferences were held to devise new metric standards. The Metre Convention (Convention du Mètre) of 1875 mandated the establishment of a permanent International Bureau of Weights and Measures (BIPM: Bureau International des Poids et Mesures) to be located in Sèvres, France. This new organisation would preserve the new prototype metre and kilogram when constructed, distribute national metric prototypes, and would maintain comparisons between them and non-metric measurement standards. This organisation created a new prototype bar in 1889 at the first General Conference on Weights and Measures (CGPM: Conférence Générale des Poids et Mesures), establishing the International Prototype Metre as the distance between two lines on a standard bar of an alloy of ninety percent platinum and ten percent iridium, measured at the melting point of ice. In 1893, the standard metre was first measured with an interferometer by Albert A. Michelson, the inventor of the device and an advocate of using some particular wavelength of light as a standard of distance. By 1925, interferometry was in regular use at the BIPM. However, the International Prototype Metre remained the standard until 1960, when the eleventh CGPM defined the metre in the new SI system as equal to 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. The original international prototype of the metre is still kept at the BIPM under the conditions specified in 1889. To further reduce uncertainty, the seventeenth CGPM of 1983 replaced the definition of the metre with its current definition, thus fixing the length of the metre in terms of time and the speed of light: :The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. Note that this definition exactly fixes the speed of light in a vacuum at 299,792,458 metres per second. Definitions based on the physical properties of light are more precise and reproducible because the properties of light are considered to be universally constant.

Timeline of definition

- 1790 May 8 — The French National Assembly decides that the length of the new metre would be equal to the length of a pendulum with a half-period of one second.
- 1791 March 30 — The French National Assembly accepts the proposal by the French Academy of Sciences that the new definition for the metre be equal to one ten-millionth of the length of the earth's meridian along a quadrant (one-fourth the polar circumference of the earth).
- 1795 — Provisional metre bar constructed of brass.
- 1799 December 10 — The French National Assembly specifies that the platinum metre bar, constructed on 23 June 1799 and deposited in the National Archives, as the final standard.
- 1889 September 28 — The first CGPM defines the length as the distance between two lines on a standard bar of an alloy of platinum with ten percent iridium, measured at the melting point of ice.
- 1927 October 6 — The seventh CGPM adjusts the definition of the length to be the distance, at 0 °C, between the axes of the two central lines marked on the prototype bar of platinum-iridium, this bar being subject to one standard atmosphere of pressure and supported on two cylinders of at least one centimetre diameter, symmetrically placed in the same horizontal plane at a distance of 571 millimetres from each other.
- 1960 October 20 — The eleventh CGPM defines the length to be equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the 2p¹⁰ and 5d⁵ quantum levels of the krypton-86 atom.
- 1983 October 21 — The seventeenth CGPM defines the length to be distance travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

External links

- [http://www.unitconversion.org/unit_converter/length.html?unit=meter&value=1 Length Converter: convert metre to other units, such as yard, mile, and so on]
- [http://physics.nist.gov/cuu/Units/meter.html History of the metre at the U.S. National Institute of Standards and Technology (NIST)]
- [http://www.mel.nist.gov/div821/museum/timeline.htm Timeline of history of the metre at the NIST]
- [http://www1.bipm.org/en/scientific/length/ Bureau International des Poids et Measures - Lengths] Category:SI base units Category:Units of length ko:미터 ms:Meter ja:メートル simple:Metre th:เมตร

Second

:This article is about the unit of time. For other uses, see second (disambiguation). The second (symbol: s) is the SI base unit of time.

Definition

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

Origin

Originally, the second was known as a "second minute", meaning the second minute (i.e. small) division of an hour. The first division was known as a "prime minute" and is equivalent to the minute we know today.

Conversions

- 60 seconds = 1 minute
- 3 600 seconds = 1 hour
- 86.4 kiloseconds (86 400 seconds) = 1 day (in the SI sense)

Explanation

The factor of 60 may have been influenced by the Babylonians who used factors of 60 in their counting system. The hour had previously been defined by the Egyptians in terms of the rotation of the Earth as 1/24 of a mean solar day. This made the second 1/86,400 of a mean solar day. In 1956 the second was defined in terms of the period of revolution of the Earth around the Sun for a particular epoch, because by then it had become recognized that the Earth's rotation on its own axis was not sufficiently uniform as a standard of time. The Earth's motion was described in Newcomb's Tables of the Sun, which provides a formula for the motion of the Sun at the epoch 1900 based on astronomical observations made during the eighteenth and nineteenth centuries. The second thus defined is :the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time. This definition was ratified by the Eleventh General Conference on Weights and Measures in 1960. Reference to the year 1900 does not mean that this is the epoch of a mean solar day of 86,400 seconds. Rather, it is the epoch of the tropical year of 31,556,925.9747 seconds of ephemeris time. Ephemeris Time (ET) was defined as the measure of time that brings the observed positions of the celestial bodies into accord with the Newtonian dynamical theory of motion. With the development of the atomic clock, it was decided to use atomic clocks as the basis of the definition of the second, rather than the rotation of the earth. Following several years of work, two astronomers at the United States Naval Observatory (USNO) and two astronomers at the National Physical Laboratory (Teddington, England) determined the relationship between the hyperfine transition frequency of the caesium atom and the ephemeris second. Using a common-view measurement method based on the received signals from radio station WWV, they determined the orbital motion of the Moon about the Earth, from which the apparent motion of the Sun could be inferred, in terms of time as measured by an atomic clock. As a result, in 1967 the Thirteenth General Conference on Weights and Measures defined the second of atomic time in the International System of Units (SI) as :the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. The ground state is defined at zero magnetic field. The second thus defined is equivalent to the ephemeris second. The definition of the second was later refined at the 1997 meeting of the BIPM to include the statement :This definition refers to a caesium atom at rest at a temperature of 0 K. In practice, this means that high-precision realizations of the second should compensate for the effects of ambient radiation to try to extrapolate to the value of the second as defined above.

External links

- [http://www.bipm.fr/en/si/si_brochure/chapter2/2-1/second.html Official BIPM definition of the second]
- [http://tycho.usno.navy.mil/leapsec.html Public domain article about definition of seconds and leap seconds] Category:SI base units Category:Units of time Category:CGS units ja:秒 simple:Second

Roller coaster

:For the Scott Cain album, see Roller Coaster Roller Coaster The roller coaster is a popular amusement ride developed for amusement parks and modern theme parks. LaMarcus Adna Thompson patented the first roller coaster on January 20, 1865. In essence a specialised railroad system, a coaster consists of a track that rises and falls in specially designed patterns, sometimes with one or more inversions (the most common being loops) that turns the rider briefly upside down. The track does not necessarily have to be a complete circuit (the antonym of complete circuit is "shuttle"), though some purists insist that it must to be a true coaster. (Not all thrill rides that run on a track are roller coasters). Most coasters have cars for two, four, or six passengers each, in which the passengers sit to travel around the circuit. An entire set of cars hooked together is called a train.
Mechanics
train]] The cars on a typical roller coaster are not self-powered. A standard full-circuit lift-powered coaster works like this: after leaving the boarding area (station), the train is pulled up with a chain or cable along the lift hill to the first peak of the coaster track. Then potential energy becomes kinetic energy as the cars race down the first downward slope. Kinetic energy is converted back into potential energy as the train moves up again to the second peak. This is necessarily lower as some mechanical energy is lost due to friction. Then the train goes down again, and up, and so on. However, not all coasters run this way. The train may be set into motion by a launch mechanism (flywheel launch, linear induction motors, linear synchronous motors, hydraulic launch, compressed air launch, drive tire, etc.). Some coasters move back and forth along the same section of track; these roller coasters are called shuttles because of this motion and usually run the circuit once with riders moving forwards and then backwards through the same course. And there are even roller coasters which are powered by a kind of locomotive. A properly designed roller coaster under good conditions will have enough kinetic, or moving, energy to complete the entire course, at the end of which brakes bring the train to a complete stop and it is pushed into the station. A brake run at the end of the circuit is the most common method of bringing the roller coaster ride to a stop.
Blocking
Most large roller coasters have the ability to run two or more trains at once. These rides use a block system, which prevents the trains from colliding. In a block system, the track is divided into multiple sections or blocks. Only one train at a time is permitted in each block. At the end of each block, there is a section of track where a train can be stopped if necessary (either by preventing dispatch from the station, closing brakes, or stopping a lift). Sensors at the end of each block detect when a train passes so that the computer running the ride is aware of which blocks are occupied. When the computer detects a train about to travel into an already occupied block, it uses whatever method is available to keep it from entering. The above can cause a cascade effect when multiple trains become stopped at the end of each block. In order to prevent this problem, ride operators follow set procedures regarding when to release a newly-loaded train from the station. One common pattern, used on rides with 2 trains, is to do the following: hold train #1 (which has just finished the ride) right outside the station, release train #2 (which has loaded while #1 was running), and then allow #1 into the station to unload.
History
The earliest roller coasters descended from Russian winter sled rides held on specially constructed hills of ice, especially around St Petersburg. By the late 1700s their popularity was such that entrepreneurs elsewhere began copying the idea, using wheeled cars built on tracks. One such company was 'Les Montagnes Russes à Belleville' which constructed and operated a gravity track in Paris from 1812. The first loop track was probably also built in Paris from an English design in 1846, with a single-person wheeled sled running through a 13-foot diameter loop. None of these tracks were complete circuits. The first roller coasters in the USA were based on gravity switchback trains developed in the 1880s. These primitive coasters were run to provide amusement by railroad companies on weekends when ridership was lower. The earliest complete circuit track appeared in 1884, and in 1885 Phillip Hinkle introduced the concept of the "lift hill." By 1912, the first underfriction coaster was developed by John Miller, often called the Thomas Edison of roller coasters. Soon, roller coasters spread to amusement parks all around the United States and the rest of the world. Perhaps the most well known historical roller coaster, The Cyclone, was opened at Coney Island in Brooklyn, New York in 1927. Like The Cyclone, all early roller coasters were made of wood. Many old wooden roller coasters are still operational, at parks such as Kennywood near Pittsburgh, Pennsylvania and Blackpool Pleasure Beach, England, UK. The Great Depression marked the end of the first golden age of roller coasters. Theme parks in general went into a decline that lasted until 1972, when the Racer was built at Kings Island in Mason, Ohio (near Cincinnati). Designed by John Allen, the instant success of the Racer began a second golden age, which continues through this writing (2003). In 1959, the Disneyland theme park introduced a new design breakthrough in roller coasters with the Matterhorn Bobsleds. This was the first roller coaster to use a tubular steel track. Unlike conventional wooden rails, tubular steel can be bent in any direction, which allows designers to incorporate loops, corkscrews, and many other manoeuvres into their designs. Most modern roller coasters are made of steel but wooden roller coasters are still being built. Some of the major variations in contemporary roller coaster design involve the modification of the car. Some seat the passenger in a bodyless frame, with the passenger's legs dangling in the air and providing a less obstructed view of the ground, thus providing an extra scare to the passengers. Another variation involves cars that have the riders in a standing position (though still heavily strapped in). Finally, some roller coasters spend some or all of their travel time with the passengers sitting in the opposite direction to their travel, so they cannot see what direction the coaster will travel next. New roller coaster designs and state of the art technology push the physical limits on what type of experiences can be had on the newest coasters. For example, coasters like the Incredible Hulk Coaster feature launch lift hills to create an unique experience. Hulk Coaster
Safety
Because roller coasters are intended to feel risky, accidents, such as the September 5, 2003 fatality at the seemingly tame Disneyland Big Thunder Mountain Railroad, attract public attention. Statistically, roller coasters are very safe. The U.S. Consumer Product Safety Commission estimates that 134 park guests required hospitalization in 2001 and that fatalities related to amusement rides average two per year. According to a study commissioned by Six Flags, 319 million people visited parks in 2001. The study concluded that a visitor has a one in one-and-a-half billion chance of being fatally injured, and that the injury rates for children's wagons, golf, and folding lawn chairs are higher than for amusement rides. In fact, driving to the amusement park has a higher risk of injury than riding the rides at the amusement park. Nevertheless, accidents do occur. Regulations vary from one authority to another. Thus in the USA, California requires amusement parks to report any ride-related accident that requires an emergency room visit, while Florida exempts parks whose parent companies employ more than 1000 people from having to report any accidents at all. Rep. Ed Markey of Massachusetts has introduced legislation that would give oversight of rides to the Consumer Product Safety Commission (CPSC). In 1999, a rider who weighed more than 400 pounds (180 kg) was unable to close his lap bar properly and was thrown from the Superman coaster at Six Flags Darien Lake, sustaining serious injuries. Despite this, a similar accident occurred in 2004 when a 230 pound (100 kg) man with cerebral palsy was permitted to board the Superman coaster at Six Flags New England and, on the last turn of the ride, was thrown from his seat and killed. Critics maintain that, despite the generally good safety record, accidents are occurring that are preventable. In recent years, controversy has arisen about the safety of the increasingly extreme rides. There have been suggestions that these may be subjecting passengers to translational and rotational accelerations that may be capable of causing brain injuries. In 2003 the [http://www.biausa.org Brain Injury Association of America] concluded in a [http://www.biausa.org/Pages/blue_final_report.html report] that "There is evidence that roller coaster rides pose a health risk to some people some of the time. Equally evident is that the overwhelming majority of riders will suffer no ill effects."
Types of roller coasters
Today, there are two main types of roller coasters: Steel roller coasters and Wooden roller coasters (also called 'Woodies'). Steel coasters are known for their smooth ride and often convoluted shapes that frequently turn riders upside-down via inversions known as loops, corkscrews, pretzels, and other descriptive names. Wooden coasters are fondly looked at by coaster enthusiasts for their more rough ride and "air-time" produced by negative G-forces when the coaster car reaches the top of some hills along the ride. Much debate can be had regarding which coaster type is better, as they both have their pros and cons. Regardless of the type of roller coaster being built, coasters come in a multitude of designs. Some designs take their cue from how the rider is positioned to experience the ride. Traditionally, coaster riders sit facing forward in the coaster car, while newer coaster designs have ignored this tradition in the quest for building more exciting, unique ride experiences for the riders. In addition to changing the rider's viewpoint, coaster designs also focus on track styles to make the ride fresh and different from other coasters. One method of designing a coaster is to select one item from each of the different coaster options: height, rider experience, and track design. These three elements combine to make a unique coaster for the park.
Designs

- 4th Dimension roller coaster
- Bobsled roller coaster
- Flying roller coaster
- Inverted roller coaster
- Pipeline roller coaster
- Sit Down roller coaster
- Stand Up roller coaster
- Suspended roller coaster
Height-specific
Some names originated by Cedar Point, a pioneer in breaking height records, have become general parlance for coasters in certain height categories:
- Strata Coaster - between 400-499 feet tall
- Giga Coaster - between 300-399 feet tall
- Hypercoaster - between 200-299 feet tall
Catagories

- Enclosed roller coaster
- Family roller coaster
- Floorless roller coaster
- Hybrid roller coaster
- Indoor roller coaster
- Kiddie roller coaster
- LIM roller coaster
- LSM roller coaster
- Mine Train roller coaster
- Moebius roller coaster
- Powered roller coaster
- Shuttle roller coaster
- Side Friction roller coaster
- Spinning Cars roller coaster
- Twin roller coaster
- Water roller coaster
Designers and Manufacturers

- Anton Schwarzkopf
- Arrow Dynamics
- Bolliger & Mabillard (B&M)
- Chance Rides
- Custom Coasters International
- Dinn Corporation
- Gerstlauer
- Giovanola
- Great Coasters International
- Intamin
- Morgan Manufacturing
- Philadelphia Toboggan Company
- S&S Power
- Setpoint
- Werner Stengel
- Togo
- Ron Toomer
- Vekoma
- Zamperla
- Zierer
See also

- Notable roller coasters
- List of roller coasters
- Roller coaster elements
- Types of roller coasters
- Rollercoaster, a movie featuring a number of roller coasters
External links

- [http://www.personal.psu.edu/faculty/v/a/vac3/table.html A list of roller coaster patents, with links to the U.S. Patent office]
- [http://rcdb.com/ Roller Coaster Database]
- [http://www.vast.org/vip/book/home.htm Roller Coaster Physics]
- [http://www.aceonline.org/ American Coaster Enthusiasts]
- [http://www.rccgb.co.uk/ Roller Coaster Club of Great Britain]
- [http://www.coasterclub.org/ European Coaster Club]
- [http://www.ceemr.org/ Spanish Coaster Club]
- [http://www.realcoasters.com/ Realcoasters.com - Roller Coaster Photography]
- [http://www.coasterbuzz.com/ Coasterbuzz enthusiast news and forum]
- [http://www.coasterradio.com/ Coaster Radio podcast website]
- [http://www.akidnews.com/christianscoasters.html A Ride Design Website]
- [http://groups.yahoo.com/group/RollerCoasterTalk/ Rollercoaster Talk, a forum] Category:Amusement rides ko:롤러코스터 ja:ローラーコースター

Manufacturing

: This is a specific reference to the process of Manufacturing. For the wider business elements of Manufacturing see the definitions in the manufacturing overview. Manufacturing is the transformation, by means of a tool and/or processing medium, of raw materials into finished goods for sale, or intermediate processes involving the production or finishing of semi-manufactures. It is a large branch of industry and of secondary production. Some industries, like semiconductor and steel manufacturers use the term fabrication. Although handicraft production has been with us for many millennia, modern-style manufacturing is generally regarded as beginning around 1780 with the British Industrial Revolution, spreading thereafter to Continental Europe and North America, and subsequently around the world. Originally, the term applied to commodities or artifacts which were "made by hand". While it remains a huge part of the modern world economy—perhaps a quarter of aggregate world production of goods and services—many of the world's wealthier nations devote an ever smaller proportion of their workforce to manufacturing activity owing to relocation of enterprises to lower-wage countries and the rising proportion of economic activity devoted to service activity.

Manufacturing topics

Lists of related topics

- list of engineering topics
- list of management topics
- list of production topics
- list of marketing topics
- list of economics topics
- list of international trade topics
- list of finance topics
- list of accounting topics
- list of information technology management topics
- list of business law topics
- list of human resource management topics
- list of business theorists
- list of economists
- list of corporate leaders

External links

- http://www.coolstuffbeingmade.com - Watch Cool Manufacturered Products Being Made
- http://www.nam.org - The National Association of Manufacturers Web site
- http://manufacturing.stanford.edu - A site with videos showing the manufacturing process for many everyday things
- http://www.themanufacturer.com/uk - A site consisting of news and articles about the UK manufacturing industry
- http://www.themanufacturer.com/us - A site consisting of news and articles about the US manufacturing industry
- http://www.selectingsoftware.com - Manufacturing Software Directory Category:Manufacturing Category:Industry Category:Technology ja:製造業

Infinite

Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes. In mathematics, infinity is relevant to or the subject matter of articles such as mathematical limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the Absolute Infinite. By some, infinity is considered to be not a number but a concept of increase beyond bounds. In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals. For a discussion about infinity and the physical universe, see Universe.

History

Ancient view of infinity

The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. Jaina mathematicians were the first to conceive of different orders of infinity, including one they called unenumerable (innumerable).[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm] [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. In Europe, the traditional view derives from Aristotle: :"... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8] This is often called potential infinity; however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example, ∀n∈Z(∃m∈Z[m>n∧P(m)]), which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham: :"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.) The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.

Views from the Renaissance to modern times

Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers with the natural numbers as follows: :1, 2, 3, 4, ... :2, 4, 6, 8, ... It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite. :"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", "greater", and "less", are not applicable to infinite, but only to finite, quantities." [On two New Sciences, 1638] The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets. Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative. :"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 7., author's emphasis) Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also Logic of antinomies :"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks § 141, cf Philosophical Grammar p. 465) Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. :"... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room." :"... what is infinite about endlessness is only the endlessness itself."

Infinity symbol

The precise origins of the infinity symbol

\infty

are unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon. A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. This possible explanation is probably incorrect, however, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858. John Wallis is usually credited with introducing

\infty

as a symbol for infinity in 1655 in his De sectionibus conicus. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many". Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet. The infinity symbol is represented in Unicode by the character ∞ (∞).

Mathematical infinity

Infinity in real analysis

In real analysis, the symbol

\infty

, called "infinity", denotes an unbounded limit.

x \rightarrow \infty

means that x grows beyond any assigned value, and

x \rightarrow -\infty

means x is eventually less than any assigned value. Points labeled

\infty

and

-\infty

can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat

\infty

and

-\infty

as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then
-

\int_^ \, f(t) dt \  = \infty

means that f(t) does not bound a finite area from 0 to 1
-

\int_^ \, f(t) dt \  = \infty

means that the area under f(t) is not finite
-

\int_^ \, f(t) dt \  = 1

means that the area under f(t) approaches 1

Infinity in complex analysis

As in real analysis, in complex analysis the symbol

\infty

, called "infinity", denotes an unbounded limit.

x \rightarrow \infty

means that the magnitude

|x|

of x grows beyond any assigned value. A point labeled

\infty

can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of

\infty

at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Arithmetic properties of infinity

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed.

Infinity with itself

\infty + \infty = \infty \cdot \infty = (-\infty) \cdot (-\infty) = \infty

(-\infty) + (-\infty) = \infty \cdot (-\infty) = (-\infty) \cdot \infty = (-\infty)

Operations involving infinity and real numbers

-\infty < x < \infty

x + \infty = \infty

and

x + (-\infty) = (-\infty)

x - \infty = -\infty

x - (-\infty) = \infty

= 0

and

= 0

# If

0 then x \cdot \infty = \infty and x \cdot (-\infty) = (-\infty) .
# If -\infty then x \cdot \infty = -\infty and x \cdot (-\infty) = \infty .

Undefined operations

0 \cdot \infty

and

0 \cdot (-\infty)

\infty + (-\infty)

and

(-\infty) + \infty

^0

1^

Notice that

[ = 0] \not\equiv [0 \cdot \infty = x]

. This is because zero times infinity is undefined.

Infinity in set theory

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (

\aleph_0

), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite

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