Dimension of the quantity

The dimension (りょうのじげん British: dimension of a quantity) of the quantity is expression to show relations with the quantity included in a certain quantity system and the basic quantity of the quantity system as the product of 冪乗 of basic quantity and the factor to cope [¹]; [²]. I bind it with the square bracket that it is often [¹] that it is prescribed that I write a dimension of quantity $Q$ in $dim Q$ and am transcribed in the standards such as ISO or JIS in $[Q]$ [^{note 1}].

The dimension is a method to express relations between the quantity and keeps the multiplication of the quantity equation. When a certain quantity $Q$ is expressed in quantity equation $Q = q 1 q 2$ by two quantity $q 1, q 2$ , the relations between dimensions of each quantity reflect the form of the quantity equation

[Q] = [q 1] [q 2]

となる. Basic quantity $A, B, C,$ ... It is $[A],[B],[C],$ in a と-adaptive factor to do... When expressed で; the dimension of quantity $Q$

[Q] = [A] a [B] b [C] c X...

I am expressed with の form uniquely. This time 冪指数 $a, b, c,$ ... I am called は dimension index. The dimension of the quantity that all dimension indexes become zero is 1 by law of exponent. The quantity of dimension 1 is called the quantity of no dimension (British: dimensionless quantity) [¹]; [²] [³].

Summary

The rectangular area is expressed by the product of the length of two sides. The triangular area is referred to with half of the product of the height length of the base. The area of Japanese yen is referred to with half of the product of the radius the length of the circumference. There is difference in "the length of the side" or "the length of the circumference", but the area of various plane figures is expressed all [length] as an X [length] in this way if I ignore the coefficient. In addition, the volume of the prism and the column is expressed by the area of the base and the product of the height, and a pyramid and the conic volume are expressed with area of the base and a one-third of the product of the height. These are referred to [area] an X [length]. Furthermore, the angle by the circular measure is expressed in ratios for the radius of the length of the circumference. Between quantity geometric in this way

[area] = [length] X [length] = [length] ²
[volume] = [area] X [length] = [length] ³
[angle] = [length ]/[ length] = [length] ⁰

Relations as 冪 of the のような length are found. Such relations are the expression by the dimension of the quantity.

The balance of the power to depend on a lever is expressed by a moment of the power that is the product of the power to suffer from length and the points of action from the fulcrum to a point of action. When I move an object using the simple machines such as a lever or the pulley, the work expressed by the product (inner product) with the component force to hang over the movement distance of the object and the movement direction does not change. Pressure to appear on principles of the Pascal is power per area. In a Hooke's law, the spring constant that is the ratio of the length that power and the spring hanging to the spring transform decides the characteristic of the spring. The frictional force in the contact surface of the object is expressed by a coefficient of friction that is the ratio for the vertical drag. The relations between the quantity in such a statics

[moment of the power] = [length] X [power]
[work] = [length] X [power]
[pressure] = [power ]/[ area] = [length] ^{- 2}* [power]
[spring constant] = [power ]/[ length] = [length] ^{- 1}* [power]
[coefficient of friction] = [power ]/[ power] = [length] ⁰* [power] ⁰

となる. It is necessary to include power without expressing it only in 冪 of the length. In this way, the basic quantity to be necessary to express a level varies according to a quantity system to treat.

I can use the dimensional concept of the quantity for a social science-like quantity system not only physical quantity. For example,

[the number of the visitors] = [the number of people]
[sales] = [amount of money]
[visitor unit price] = [the number of sales ]/[ visitors] = [amount of money] X [the number of people] ^-1

などが is managed.

Dimension in the international quantity system

Basic quantity of ISQ and the dimension
Basic quantity	Dimensional sign	SI base unit [^{note 2}]
Length	L	Meter (m)
Mass	M	Kilogram (kg)
Time	T	Second (s)
Electric current	I	Ampere (A)
Thermodynamic temperature	Θ	Kelvin (K)
Quantity of material	N	Molar (mol)
The intensity of light	J	Cd (cd)

In the international quantity system (ISQ), seven physical quantity is prescribed for basic quantity, and each basic quantity is given an independent dimension, and a more dimensional sign is prescribed, too [³]. The dimensional sign is written in capital letter one character of the block letter roman [³]. In addition, in the international unit system (SI), I assume a unit of basic quantity of ISQ SI base unit.

Nothing is defining, and, in ISQ, the countable quantity such as the number of the particles or the number of the states is treated for counted quantity and are done with no dimension quantity. The dimension of the material quantity converted by Avogadro's constant from the nothing number of the particles which are defining, and are introduced is common to all chemical species. Therefore, I can let you add it without a conversion factor by the calculation of the quantity of all pro-mixture materials. In addition, it is no dimension quantity because a molar fraction to express pro-mixture composition is the ratio of the quantity of the same dimension.

In ISQ, an angle or the level are treated for quantity of no dimension. Therefore, the variable of a trigonometric function and the exponential function is quantity of no dimension. But only the entropy that is a level of the number of the states has a dimension.

Basic quantity

If it is one of the quantity admitted that it is independent functionally each other [²] and chooses a combination of basic quantity in [⁴], a quantity system thinking about by a decision in a certain quantity system with the basic quantity appropriately, the expression by the dimension of one quantity is decided in just what alone. There is some arbitrariness which quantity you choose for basic quantity.

For example, the quantity of dynamics assumes length, mass and time basic quantity in ISQ, but based on an exercise of Newton equation the dimension of the power

[power] = [length] X [mass] X [time] ^-2

I am connected with として, length, mass and a dimension of the time. There is not the inevitability to choose these three quantity as basic quantity and I change it to the mass and can choose power (weight) as basic quantity.

In addition, speed of light $c$ is often fixed to 1 (the fixed number) when I treat the special theory of relativity. Then

[time] = [ $c$ ^-1] X [length] = [length]

であり, the dimension of the time equal in a dimension of the length and cannot choose time as length and independent basic quantity anymore. The fixed number to characterize quantum mechanics is Planck's constant $ħ$ . The Planck's constant has a dimension of the action and is the product of a dimension of the energy and the dimension of the time. Therefore, when I fix a Planck's constant to 1

[time] = [ $ħ$ ]/[ energy] = [energy] ^-1

であり, the dimension of the time equal in dimensional reverse of the energy. Length, energy, all the dimensions of the time are connected when I fix speed of light and a Planck's constant to 1 at the same time, and quantity of dynamics will be expressed in 冪 of one basics quantity. Fixation to 1 of the basic physical fixed number is often carried out as well as speed of light and a Planck's constant, and such a system is called a natural unit system.

with the space dimension of relationships

The dimension of the quantity associates with the dimension of the space closely. The dimension of the volume is expressed by the cube of the dimension of the length in the three-dimensional space, but generally the dimension of the volume is expressed in dimensional $d$ 乗 of the length in the space of dimension $d$ . As for the area, the area of the hypersurface with various kinds of dimensions is thought about from dimensional square of the length to $d -1$ 乗. As well as quantity of geometry such as a volume or the area, the dimensions such as the density per volume or the density per area will depend on dimension $d$ of the space.

There is the fixed number that the dimension depends on a space-time dimension for with the basic physical fixed number. Gravitational constant $κ$ of Einstein who is the fixed number to characterize the general theory of relativity that is space-time geometry is a coefficient to bind the space-time curvature that is geometry quantity and an energy tensor together. Because an energy tensor is density, the dimension depends on the space-time dimension. In addition, microstructure fixed number $α$ which is famous for the fixed number of no dimension is not no dimension in the space (space-time of dimension $d +1$ ) of dimension $d$ .

I choose the temperature for basic quantity at length, energy, time, and the dimension of the representative physics fixed number is expressed as follows when I express each dimension in L, E, T.

Velocity of light： [ $c$ ] = LT $-1$
Conversion Planck's constant： [ $ħ$ ] = ET
Gravitational constant of Einstein： [ $κ$ ] = L $d -2$ E $-1$
The microstructure fixed number： [ $α$ ] = L $d -3$

Footnote

In ^ ISO or JIS, $[Q]$ using the square bracket is used as a sign to express a unit. In addition, the dimension is a unit and a concept with much confusion, but is a concept not to depend on how to choose units.
In ^ ISQ using SI is not necessary, but list it as reference.

The source

^ ^a ^b ^c Quantity of JIS Z8000-1:2014 and unit - Part 1: The public
^ ^a ^b ^c JIS Z8103:2000 measurement term
The ^ ^a ^b ^c "international unit system eighth edition," it is pp.15–16, a dimension of the quantity of 1.3.
The ^ "international unit system eighth edition," it is p.13, quantity of 1.1 and a unit.

References

"Quantity of JIS Z8000-1 and unit - Part 1 is published by public" Japan Standards Association, 2014.
"JIS Z8103 measurement term" Japan Standards Association publication, 2000.
It is the international instrument eighth edition "an international unit system" (SI) (PDF) for reason supervision measurement standard synthesis center, 2,006 years.

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2016년 8월 13일 토요일