In
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
and
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
, dimensional analysis is the analysis of the relationships between different
physical quantities
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
by identifying their
base quantities (such as
length,
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
,
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
, and
electric current) and
units of measure
A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
(such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The
conversion of units
Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors which change the measured quantity value without changing its effects.
Overview
The process ...
from one dimensional unit to another is often easier within the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
or the
SI than in others, due to the regular 10-base in all units.
''Commensurable'' physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measure, e.g. yards and metres, pounds (mass) and kilograms, seconds and years. ''Incommensurable'' physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and kilograms, seconds and kilograms, metres and seconds. For example, asking whether a kilogram is larger than an hour is meaningless.
Any physically meaningful
equation, or
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
, ''must'' have the same dimensions on its left and right sides, a property known as ''dimensional homogeneity''. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on
derived
Derive may refer to:
* Derive (computer algebra system), a commercial system made by Texas Instruments
* ''Dérive'' (magazine), an Austrian science magazine on urbanism
*Dérive, a psychogeographical concept
See also
*
*Derivation (disambiguatio ...
equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
The concept of physical dimension, and of dimensional analysis, was introduced by
Joseph Fourier in 1822.
Formulation
The
Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
describes how every physically meaningful equation involving ''n'' variables can be equivalently rewritten as an equation of dimensionless parameters, where ''m'' is the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the dimensional
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through
nondimensionalization
Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
, which begins with dimensional analysis, and involves scaling quantities by
characteristic units of a system or
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
of nature. This may give insight into the fundamental properties of the system, as illustrated in the examples below.
The dimension of a
physical quantity can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
)
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
. The ''dimension'' of a physical quantity is more fundamental than some ''scale'' or
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
used to express the amount of that physical quantity. For example, ''mass'' is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent.
Natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
, being based on only universal constants, may be thought of as being "less arbitrary".
There are many possible choices of base physical dimensions. The
SI standard selects the following dimensions and corresponding symbols:
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
(T),
length (L),
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
(M),
electric current (I),
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
(Θ),
amount of substance (N) and
luminous intensity
In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human e ...
(J). The symbols are by convention usually written in
roman
Roman or Romans most often refers to:
*Rome, the capital city of Italy
*Ancient Rome, Roman civilization from 8th century BC to 5th century AD
*Roman people, the people of ancient Rome
*'' Epistle to the Romans'', shortened to ''Romans'', a lette ...
sans serif
In typography and lettering, a sans-serif, sans serif, gothic, or simply sans letterform is one that does not have extending features called "serifs" at the end of strokes. Sans-serif typefaces tend to have less stroke width variation than ser ...
typeface. Mathematically, the dimension of the quantity ''Q'' is given by
:
where ''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'' are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
– for instance, one could replace the dimension (I) of
electric current of the SI basis with a dimension (Q) of
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
, since .
As examples, the dimension of the physical quantity
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
''v'' is
:
and the dimension of the physical quantity
force ''F'' is
:
The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have
conversion factors that relate them. For example, ; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity.
There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,
although this does not invalidate the usefulness of dimensional analysis.
Rayleigh's method
In dimensional analysis, Rayleigh's method is a conceptual tool used in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
chemistry, and
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. It expresses a
functional relationship
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
of some
variables in the form of an
exponential equation. It was named after
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
.
The method involves the following steps:
# Gather all the
independent variables that are likely to influence the
dependent variable.
# If ''R'' is a variable that depends upon independent variables ''R''
1, ''R''
2, ''R''
3, ..., ''R''
''n'', then the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
can be written as .
# Write the above equation in the form , where ''C'' is a
dimensionless constant and ''a'', ''b'', ''c'', ..., ''m'' are arbitrary exponents.
# Express each of the quantities in the equation in some
base units in which the solution is required.
# By using
dimensional homogeneity, obtain a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
involving the exponents ''a'', ''b'', ''c'', ..., ''m''.
#
Solve
Solve may refer to:
* Sölve, viking king of Sweden
* SOLVE, an American environmental organization
* Solve (advertising agency)
* "Solve" (song), by Japanese pop band Dream
* HSwMS ''Sölve''
See also
* Equation solving
* Problem solving
* ...
these equations to obtain the value of exponents ''a'', ''b'', ''c'', ..., ''m''.
#
Substitute the values of exponents in the main equation, and form the
non-dimensional
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s by
grouping
Grouping may refer to:
* Muenchian grouping
* Principles of grouping
* Railways Act 1921, also known as Grouping Act, a reorganisation of the British railway system
* Grouping (firearms), the pattern of multiple shots from a sidearm
See also ...
the variables with like exponents.
As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.
Concrete numbers and base units
Many parameters and measurements in the physical sciences and engineering are expressed as a
concrete number—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
, e.g. 60 km/h. Other relations can involve
multiplication (often shown with a
centered dot or
juxtaposition
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc.
Speech
Juxtaposition in literary terms is the showin ...
), powers (like m
2 for square metres), or combinations thereof.
A set of
base units for a
system of measurement
A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement i ...
is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. For example, units for
length and time are normally chosen as base units. Units for
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
, however, can be factored into the base units of length (m
3), thus they are considered derived or compound units.
Sometimes the names of units obscure the fact that they are derived units. For example, a
newton (N) is a unit of
force, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s
−2). The newton is defined as .
Percentages, derivatives and integrals
Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since .
Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus:
* position (''x'') has the dimension L (length);
* derivative of position with respect to time (''dx''/''dt'',
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
) has dimension T
−1L—length from position, time due to the gradient;
* the second derivative (''d'x''/''dt'' = ''d''(''dx''/''dt'') / ''dt'',
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
) has dimension T
−2L.
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
*
force has the dimension (mass multiplied by acceleration);
* the integral of force with respect to the distance (''s'') the object has travelled (
,
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
) has dimension .
In economics, one distinguishes between
stocks and flows
Stocks are feet restraining devices that were used as a form of corporal punishment and public humiliation. The use of stocks is seen as early as Ancient Greece, where they are described as being in use in Solon's law code. The law describing ...
: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of theis unit divided by one of time (say, dollars/year).
In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example,
debt-to-GDP ratios are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
Dimensional homogeneity
The most basic rule of dimensional analysis is that of dimensional homogeneity.
However, the dimensions form an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under multiplication, so:
For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.
The rule implies that in a physically meaningful ''expression'' only quantities of the same dimension can be added, subtracted, or compared. For example, if ''m''
man, ''m''
rat and ''L''
man denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression is meaningful, but the heterogeneous expression is meaningless. However, ''m''
man/''L''
2man is fine. Thus, dimensional analysis may be used as a
sanity check
A sanity check or sanity test is a basic test to quickly evaluate whether a claim or the result of a calculation can possibly be true. It is a simple check to see if the produced material is rational (that the material's creator was thinking ration ...
of physical equations: the two sides of any equation must be commensurable or have the same dimensions.
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
and energy share the dimension , they are fundamentally different physical quantities.
To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m.
A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. For example,
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between a unit that measures the same dimension must take: multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres.
Conversion factor
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a ''
conversion factor''. For example, kPa and bar are both units of pressure, and . The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to . Since any quantity can be multiplied by 1 without changing it, the expression "" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, because , and bar/bar cancels out, so .
Applications
Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.
Mathematics
A simple application of dimensional analysis to mathematics is in computing the form of the
volume of an ''n''-ball (the solid ball in ''n'' dimensions), or the area of its surface, the
''n''-sphere: being an ''n''-dimensional figure, the volume scales as
while the surface area, being
-dimensional, scales as
Thus the volume of the ''n''-ball in terms of the radius is
for some constant
Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
Finance, economics, and accounting
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the
distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various
financial ratios
A financial ratio or accounting ratio is a relative magnitude of two selected numerical values taken from an enterprise's financial statements. Often used in accounting, there are many standard ratios used to try to evaluate the overall financial ...
, economics ratios, and accounting ratios.
* For example, the
P/E ratio has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid".
* In economics,
debt-to-GDP ratio also has the unit year (debt has a unit of currency, GDP has a unit of currency/year).
*
Velocity of money
image:M3 Velocity in the US.png, 300px, Similar chart showing the logged velocity (green) of a broader measure of money M3 that covers M2 plus large institutional deposits. The US no longer publishes official M3 measures, so the chart only runs thr ...
has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year.
* Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.)
* In financial analysis,
bond duration can be defined as (''dV''/''dr'')/''V'', where ''V'' is the value of a bond (or portfolio), ''r'' is the continuously compounded interest rate and ''dV''/''dr'' is a derivative. From the previous point, the dimension of ''r'' is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because ''dr'' is in the "denominator" of the derivative.
Fluid mechanics
In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, dimensional analysis is performed to obtain dimensionless
pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships. In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:
*
Reynolds number (Re), generally important in all types of fluid problems:
*
Froude number (Fr), modeling flow with a free surface:
*
Euler number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion
:\frac = \frac = \sum_^\infty \frac \cdot t^n,
where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
(Eu), used in problems in which pressure is of interest:
*
Mach number (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound:
where is the local speed of sound.
History
The origins of dimensional analysis have been disputed by historians.
The first written application of dimensional analysis has been credited to an article of
François Daviet at the
Turin
Turin ( , Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital from 1861 to 1865. The ...
Academy of Science. Daviet had the master
Lagrange as teacher.
His fundamental works are contained in acta of the Academy dated 1799.
This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the
Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
.
Simeon Poisson
Simeon () is a given name, from the Hebrew (Biblical Hebrew, Biblical ''Šimʿon'', Tiberian vocalization, Tiberian ''Šimʿôn''), usually transliterated as Shimon. In Greek it is written Συμεών, hence the Latinized spelling Symeon.
Meani ...
also treated the same problem of the
parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). In the second edition of 1833, Poisson explicitly introduces the term ''dimension'' instead of the Daviet ''homogeneity''.
In 1822, the important Napoleonic scientist
Joseph Fourier made the first credited important contributions based on the idea that physical laws like
should be independent of the units employed to measure the physical variables.
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.
Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
in which the
gravitational constant ''G'' is taken as unity, thereby defining .
By assuming a form of
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
in which the
Coulomb constant ''k''
e is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were ,
which, after substituting his equation for mass, results in charge having the same dimensions as mass, viz. .
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book ''The Theory of Sound''.
The original meaning of the word ''dimension'', in Fourier's ''Theorie de la Chaleur'', was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. This was slightly changed by Maxwell, who said the dimensions of acceleration are T
−2L, instead of just the exponents.
Examples
A simple example: period of a harmonic oscillator
What is the period of
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of a mass attached to an ideal linear spring with spring constant suspended in gravity of strength ? That period is the solution for of some dimensionless equation in the variables , , , and .
The four quantities have the following dimensions:
2">/T2 and
2">/T2 From these we can form only one dimensionless product of powers of our chosen variables,
=
, and putting
for some dimensionless constant gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. They are often called
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
s as well.
Note that the variable does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines with , , and , because is the only quantity that involves the dimension L. This implies that in this problem the is irrelevant. Dimensional analysis can sometimes yield strong statements about the ''irrelevance'' of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of : it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way:
, for some dimensionless constant (equal to
from the original dimensionless equation).
When faced with a case where dimensional analysis rejects a variable (, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.
When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as .
A more complex example: energy of a vibrating wire
Consider the case of a vibrating wire of
length ''ℓ'' (L) vibrating with an
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
''A'' (L). The wire has a
linear density
Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in textile engineering, the amount of mass per unit length) and linear charge density (the amount of electric charge per unit ...
''ρ'' (M/L) and is under
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
''s'' (LM/T
2), and we want to know the energy ''E'' (L
2M/T
2) in the wire. Let ''π''
1 and ''π''
2 be two dimensionless products of
powers of the variables chosen, given by
:
The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
:
where ''F'' is some unknown function, or, equivalently as
:
where ''f'' is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function ''f''. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ''ℓ'', and so infer that . The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.
The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
such as the
Reynolds number, which may be interpreted by dimensional analysis.
A third example: demand versus capacity for a rotating disc
![Dimensional analysis 01](https://upload.wikimedia.org/wikipedia/commons/8/88/Dimensional_analysis_01.jpg)
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness ''t'' (L) and radius ''R'' (L). The disc has a density ''ρ'' (M/L
3), rotates at an angular velocity ''ω'' (T
−1) and this leads to a stress ''S'' (T
−2L
−1M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups:
: demand/capacity = ''ρR'ω''/''S''
: thickness/radius or aspect ratio = ''t''/''R''
Through the use of numerical experiments using, for example, the
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs
Properties
Mathematical properties
The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
: The
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
is written as 1; , and the inverse of L is 1/L or L
−1. L raised to any integer power ''p'' is a member of the group, having an inverse of L
−''p'' or 1/L
''p''. The operation of the group is multiplication, having the usual rules for handling exponents (). Physically, 1/L can be interpreted as
reciprocal length, and 1/T as reciprocal time (see
reciprocal second
The inverse second or reciprocal second (s−1) is a unit of frequency, defined as the multiplicative inverse of the second (a unit of time). It is dimensionally equivalent to:
* the unit hertz – the SI unit for frequency
* the unit radian per s ...
).
An abelian group is equivalent to a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over the integers, with the dimensional symbol corresponding to the tuple . When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
in the module.
A basis for such a module of dimensional symbols is called a set of
base quantities, and all other vectors are called derived units. As in any module, one may choose different
bases, which yields different systems of units (e.g.,
choosing whether the unit for charge is derived from the unit for current, or vice versa).
The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module,
.
In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like
. However, it is not possible to take arbitrary fractional powers of units, due to
representation-theoretic obstructions.
One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions M and L, one has the vector spaces
and
, and can define
as the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. Similarly, the dual space can be interpreted as having "negative" dimensions. This corresponds to the fact that under the
natural pairing between a vector space and its dual, the dimensions cancel, leaving a
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
scalar.
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The
nullity describes some number (e.g., ''m'') of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, . (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and
exponentiating) together the measured quantities to produce something with the same unit as some derived quantity ''X'' can be expressed in the general form
:
Consequently, every possible
commensurate equation for the physics of the system can be rewritten in the form
:
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
Mechanics
The dimension of physical quantities of interest in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
: they must
span
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan ester ...
the space, and be
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
.
For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as
2"> = LM/T2 L, M, while the latter can be expressed as
1/2"> = (LM/F)1/2 L, M.
On the other hand, length, velocity and time do not form a set of base dimensions for mechanics, for two reasons:
* There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not ''span the space'').
* Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not ''linearly independent'').
Other fields of physics and chemistry
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
. In
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the
amount of substance (the number of molecules divided by the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
, ≈ ) is also defined as a base dimension, N.
In the interaction of
relativistic plasma Relativistic plasmas in physics are plasmas for which relativistic corrections to a particle's mass and velocity are important. Such corrections typically become important when a significant number of electrons reach speeds greater than 0.86 c (Lo ...
with strong laser pulses, a dimensionless
relativistic similarity parameter
In relativistic laser-plasma physics the relativistic similarity parameter ''S'' is a dimensionless parameter defined as
: S=\frac,
where is the electron plasma density, is the critical plasma density and is the normalized vector potential. H ...
, connected with the symmetry properties of the collisionless
Vlasov equation The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma ...
, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.
Polynomials and transcendental functions
Scalar arguments to
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
s such as
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
,
trigonometric
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
and
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
ic functions, or to
inhomogeneous polynomials, must be
dimensionless quantities
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(''a''/''b'') = log ''a'' − log ''b'', where the logarithm is taken in any base, holds for dimensionless numbers ''a'' and ''b'', but it does ''not'' hold if ''a'' and ''b'' are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.
Similarly, while one can evaluate
monomials
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
(''x''
''n'') of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for ''x''
2, the expression (3 m)
2 = 9 m
2 makes sense (as an area), while for ''x''
2 + ''x'', the expression (3 m)
2 + 3 m = 9 m
2 + 3 m does not make sense.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,
:
This is the height to which an object rises in time ''t'' if the acceleration of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
is 9.8 and the initial upward speed is 500 . It is not necessary for ''t'' to be in ''seconds''. For example, suppose ''t'' = 0.01 minutes. Then the first term would be
:
Incorporating units
The value of a dimensional physical quantity ''Z'' is written as the product of a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
'Z''within the dimension and a dimensionless numerical factor, ''n''.
[For a review of the different conventions in use see: ]
: