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'''''Commensurable''''' physical quantities are of the same [[Kind of quantity|kind]] and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. ''Incommensurable'' physical [[Quantity|quantities]] are of different [[Kind of quantity|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 grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless.
 
Any physically meaningful [[equation]], or [[inequality (mathematics)|inequality]], ''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 [[Formal proof|derived]] equations and [[Computation|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.<ref name="Bolster">{{cite journal|lastlast1=Bolster|firstfirst1=Diogo|last2=Hershberger|first2=Robert E.|last3=Donnelly|first3=Russell E.|title=Dynamic similarity, the dimensionless science|url=https://pubs.aip.org/physicstoday/article-abstract/64/9/42/413713/Dynamic-similarity-the-dimensionless|workjournal=Physics Today|doi=10.1063/PT.3.1258|date=September 2011|volume=64|issue=9|pages=42-4742–47|url-access=subscription}}</ref>{{rp|42}}
 
== Formulation ==
{{redirect|Dimension (physics)|physical dimensions|Size}}
 
The [[Buckingham π theorem]] describes how every physically meaningful equation involving {{math|''n''}} variables can be equivalently rewritten as an equation of {{nowrapmath|''n'' − ''m''}} dimensionless parameters, where ''m'' is the [[rank of a matrix|rank]] of the dimensional [[matrix (mathematics)|matrix]]. 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]], which begins with dimensional analysis, and involves scaling quantities by [[characteristic units]] of a system or [[naturalphysical unitsconstant]]s of nature.<ref name="Bolster"/>{{rp|43}} 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 number|rational]]) [[power (mathematics)|power]]. The ''dimension'' of a physical quantity is more fundamental than some ''scale'' or [[units of measurement|unit]] 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]], being based on only universal constants, may be thought of as being "less arbitrary".
 
There are many possible choices of base physical dimensions. The [[International System of Units|SI standard]] selects the following dimensions and corresponding '''dimension symbols''':{{anchor|Dimension symbol}}
: [[time]] (T), [[length]] (L), [[mass]] (M), [[electric current]] (I), [[absolute temperature]] (Θ), [[amount of substance]] (N) and [[luminous intensity]] (J).
The symbols are by convention usually written in [[roman type|roman]] [[sans serif]] typeface.<ref name=SIBrochure9th>{{cite book |author1=BIPM |author1-link=BIPM |title=SI Brochure: The International System of Units (SI) |date=2019 |isbn=978-92-822-2272-0 |pages=136–137 |edition=v. 1.08, 9th |url=https://www.bipm.org/en/publications/si-brochure |access-date=1 September 2021 |language=en, fr |format=PDF |chapter=2.3.3 Dimensions of quantities}}</ref> Mathematically, the dimension of the quantity {{math|''Q''}} is given by
: <math>\operatorname{dim}Q = \mathsf{T}^a\mathsf{L}^b\mathsf{M}^c\mathsf{I}^d\mathsf{\Theta}^e\mathsf{N}^f\mathsf{J}^g</math>
where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}}, {{math|''e''}}, {{math|''f''}}, {{math|''g''}} are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a [[linearly independent]] [[basis (linear algebra)|basis]] – for instance, one could replace the dimension (I) of [[electric current]] of the SI basis with a dimension (Q) of [[electric charge]], since {{nowrap|1=Q = TI}}.
 
A quantity that has only <{{math>|''b'' \ne≠ 0</math>}} (with all other indicesexponents zero) is known as a '''[[geometry|geometric]] quantity'''. A quantity that has only both <{{math>|''a'' \ne≠ 0</math>}} and <{{math>|''b'' \ne≠ 0</math>}} is known as a '''[[kinematics|kinematic]] quantity'''. A quantity that has only all of <{{math>|''a'' \ne≠ 0</math>}}, <{{math>|''b'' \ne≠ 0</math>}}, and <{{math>|''c'' \ne≠ 0</math> with the rest zero}} is known as a '''[[dynamics (mechanics)|dynamic]] quantity'''.<ref>{{cite book | chapter-url=https://link.springer.com/chapter/10.1007/978-1-349-00245-0_1 | doi=10.1007/978-1-349-00245-0_1 | chapter=Principles of the Theory of Dimensions | title=Theory of Hydraulic Models | year=1971 | last1=Yalin | first1=M. Selim | pages=1–34 | isbn=978-1-349-00247-4 }}</ref>
A quantity that has all exponents null is said to have '''dimension one'''.<ref name=SIBrochure9th/>
 
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There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,<ref name="duff">{{Citation |last1=Duff |first1=M.J. |last2=Okun |first2=L.B. |last3=Veneziano |first3=G. |title=Trialogue on the number of fundamental constants |journal=Journal of High Energy Physics |volume=2002 |page=023 |date=September 2002 |doi=10.1088/1126-6708/2002/03/023 |arxiv=physics/0110060|bibcode = 2002JHEP...03..023D |issue=3 |s2cid=15806354 }}</ref> although this does not invalidate the usefulness of dimensional analysis.
 
=== Simple cases ===
As examples, the dimension of the physical quantity [[speed]] {{math|''v''}} is
: <math>\operatorname{dim}v
= \frac{\text{length}}{\text{time}}
= \frac{\mathsf{L}}{\mathsf{T}}
= \mathsf{T}^{-1}\mathsf{L} .</math>
and the dimension of the physical quantity [[force (physics)|force]] ''F'' is
:<math>\operatorname{dim}F = \text{mass} \times \text{acceleration} = \text{mass} \times \frac{\text{length}}{\text{time}^2} = \frac{\mathsf{L}\mathsf{M}}{\mathsf{T}^2} = \mathsf{T}^{-2}\mathsf{L}\mathsf{M}</math>.
The dimension of the physical quantity [[pressure]] ''P'' is
 
The dimension of the physical quantity [[acceleration]] {{math|''a''}} is
:<math>\begin{align} \operatorname{dim}P & = \frac{\text{force}}{\text{area}} \\ & = \frac{\text{mass acceleration}}{\text{area}} \\ & = \frac{\mathsf{M^1 L^1 T^{-2}}}{\mathsf{L^2}} \\ & = \mathsf{M^1 L^{1-2} T^{-2}} \\ & = \mathsf{M^1 L^{-1} T^{-2}} \\& = \mathsf{T^{-2} L^{-1} M^1}\\ \end{align} </math>.
: <math>\operatorname{dim}a
= \frac{\text{speed}}{\text{time}}
= \frac{\mathsf{T}^{-1}\mathsf{L}}{\mathsf{T}}
= \mathsf{T}^{-2}\mathsf{L} .</math>
 
The dimension of the physical quantity [[energyforce (physics)|force]] {{math|''EF''}} is
: <math>\operatorname{dim}F
= \text{mass} \times \text{acceleration}
= \mathsf{M} \times \mathsf{T}^{-2}\mathsf{L}
= \mathsf{T}^{-2}\mathsf{L}\mathsf{M} .</math>
 
The dimension of the physical quantity [[pressure]] {{math|''P''}} is
:<math>
: <math>\operatorname{dim}P
\begin{align} \operatorname{dim}E & = \text{force displacement} \\ & = \text{mass acceleration displacement} \\ & = \frac{\text{mass velocity displacement}}{\text{time}} \\ & = \frac{\text{mass displacement displacement}}{\text{time time}} \\ & = \frac {\mathsf{{mass}^1}\mathsf{displacement}^2}{\mathsf{time^2}} \\ & =\frac {\mathsf{{M}^1}\mathsf{L}^2}{\mathsf{T^2}} \\ & = \mathsf{M^1}\mathsf{L^2}\mathsf{T^{-2}} \\ & = \mathsf{T^{-2}}\mathsf{L^2}\mathsf{M^1} \\ \end{align} </math>.
= \frac{\text{force}}{\text{area}}
= \frac{\mathsf{T}^{-2}\mathsf{L}\mathsf{M}}{\mathsf{L}^2}
= \mathsf{T}^{-2}\mathsf{L}^{-1}\mathsf{M} .</math>
 
The dimension of the physical quantity [[Power (physics)|powerenergy]] P{{math|''E''}} is
: <math>\operatorname{dim}E
= \text{force} \times \text{displacement}
= \mathsf{T}^{-2}\mathsf{L}\mathsf{M} \times \mathsf{L}
= \mathsf{T}^{-2}\mathsf{L}^2\mathsf{M} .</math>
 
The dimension of the physical quantity [[Power (physics)|power]] {{math|''P''}} is
:<math>
: <math>\operatorname{dim}P
\begin{align} \operatorname{dim}P & = \frac{\text{energy}}{\text{time}} \\ & = \frac{\mathsf{T^{-2}}\mathsf{L^2}\mathsf{M^1}}{\mathsf{T}} \\ & =\mathsf{T^{-2-1}}\mathsf{L^2}\mathsf{M^1} \\ & = \mathsf{T^{-3}}\mathsf{L^2}\mathsf{M^1} \\ \end{align} </math>.
= \frac{\text{energy}}{\text{time}}
= \frac{\mathsf{T}^{-2}\mathsf{L}^2\mathsf{M}}{\mathsf{T}}
= \mathsf{T}^{-3}\mathsf{L}^2\mathsf{M} .</math>
 
The dimension of the physical quantity [[electric charge]] q{{math|''Q''}} is
: <math>\operatorname{dim}Q
= \text{current} \times \text{time}
= \mathsf{T}\mathsf{I} .</math>
 
The dimension of the physical quantity [[voltage]] {{math|''V''}} is
:<math>
: <math>\operatorname{dim}V
\begin{align} \operatorname{dim}q & = \mathsf{T^{1}}\mathsf{I^1} \\ \end{align} </math>.
= \frac{\text{power}}{\text{current}}
= \frac{\mathsf{T}^{-3}\mathsf{L}^2\mathsf{M}}{\mathsf{I}}
= \mathsf{T^{-3}}\mathsf{L}^2\mathsf{M} \mathsf{I}^{-1} .</math>
 
The dimension of the physical quantity [[electric potential differencecapacitance]] V{{math|''C''}} is
: <math>\operatorname{dim}C
 
= \frac{\text{electric charge}}{\text{electric potential difference}}
:<math>
\begin{align} \operatorname{dim}V & = \frac {\mathsf{T^{-3}}\mathsf{L^2I}\mathsf{M^1} }{\mathsf{I^1T}} \\ & =\mathsf{T^{-3}}\mathsf{L}^2}\mathsf{M^1} \mathsf{I}^{-1}} \\ \end{align} </math>.
= \mathsf{T^4}\mathsf{L^{-2}}\mathsf{M^{-1}}\mathsf{I^2} .</math>
 
The dimension of the physical quantity [[capacitance]] C is
 
:<math>\begin{alignat}{2} \operatorname{dim}C & = \frac{\text{electric charge}}{\text{electric potential difference}} \\ & =
\frac {\mathsf{T^{1}}\mathsf{I^1}}{\mathsf{T^{-3}}\mathsf{L^2}\mathsf{M^1} \mathsf{I^{-1}}}
\\ & = \frac{\mathsf{T^{1- -3}}\mathsf{I^{1- -1}}}{\mathsf{L^2}\mathsf{M^1}}
\\ & = \frac{\mathsf{T^4}\mathsf{I^2}}{\mathsf{L^2}\mathsf{M^1}}
\\ & = \mathsf{T^4}\mathsf{I^2}\mathsf{L^{-2}}\mathsf{M^{-1}}
\\ & = \mathsf{T^4}\mathsf{L^{-2}}\mathsf{M^{-1}}\mathsf{I^2}\\ \end{alignat}</math>.
 
=== Rayleigh's method ===
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The method involves the following steps:
# Gather all the [[independent variable]]s that are likely to influence the [[dependent variable]].
# If {{math|''R''}} is a variable that depends upon independent variables {{math|''R''<sub>1</sub>}},&nbsp; {{math|''R''<sub>2</sub>}},&nbsp; {{math|''R''<sub>3</sub>}},&nbsp; ...,&nbsp; {{math|''R''<sub>''n''</sub>}}, then the [[functional equation]] can be written as {{nowrapmath|1=''R'' = ''F''(''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, ..., ''R''<sub>''n''</sub>)}}.
# Write the above equation in the form {{nowrapmath|1=''R'' = ''C'' ''R''<sub>1</sub><sup>''a''</sup> ''R''<sub>2</sub><sup>''b''</sup> ''R''<sub>3</sub><sup>''c''</sup> ... ''R''<sub>''n''</sub><sup>''m''</sup>}}, where {{math|''C''}} is a [[dimensionless constant]] and {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}} are arbitrary exponents.
# Express each of the quantities in the equation in some [[Base unit (measurement)|base unit]]s in which the solution is required.
# By using [[#Dimensional homogeneity|dimensional homogeneity]], obtain a [[set (mathematics)|set]] of [[simultaneous equations]] involving the exponents {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}}.
# [[Equation solving|Solve]] these equations to obtain the valuevalues of the exponents {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, ..., {{math|''m''}}.
# [[Simultaneous equations#Substitution method|Substitute]] the values of exponents in the main equation, and form the [[non-dimensional]] [[parameter]]s by [[Combining like terms|grouping]] the variables with like exponents.
 
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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&nbsp;kilometres per hour or 1.4&nbsp;kilometres per second. Compound relations with "per" are expressed with [[Division (mathematics)|division]], e.g. 60&nbsp;km/h. Other relations can involve [[multiplication]] (often shown with a [[centered dot]] or [[Juxtaposition#Mathematics|juxtaposition]]), powers (like m<sup>2</sup> for square metres), or combinations thereof.
 
A set of [[Base unit (measurement)|base unit]]s for a [[system of measurement]] 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.<ref>{{Cite bookcitation |lastauthor=JCGM 200|author-link=Joint Committee for Guides in Metrology |url=https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |title=JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM) |year=2012 |edition=3rd |access-date=2 June 2015 |archive-url=https://web.archive.org/web/20150923224356/http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf |archive-date=23 September 2015 |url-status=dead}}</ref> For example, units for [[length]] and time are normally chosen as base units. Units for [[volume]], however, can be factored into the base units of length (m<sup>3</sup>), 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 (unit)|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<sup>−2</sup>). The newton is defined as {{nowrap|1=1 N = 1 kg⋅m⋅s<sup>−2</sup>}}.
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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 ({{math|''x''}}) has the dimension L (length);
* derivative of position with respect to time ({{math|''dx''/''dt''}}, [[velocity]]) has dimension T<sup>−1</sup>L—length from position, time due to the gradient;
* the second derivative {{math|1=(''d''{{i sup|2}}''x''/''dt''{{i sup|2}} = ''d''(''dx''/''dt'') / ''dt''}}, [[acceleration]]) has dimension T<sup>{{dimanalysis|length=1|time=−2</sup>L}}.
Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.
* [[force]] has the dimension {{dimanalysis|mass=1|length=1|time=−2}} (mass multiplied by acceleration);
* the integral of force with respect to the distance ({{math|''s''}}) the object has travelled (<math>{{tmath|\textstyle\int F\ ds</math>}}, [[Work (physics)#Mathematical calculation|work]]) has dimension {{dimanalysis|mass=1|length=2|time=−2}}.
 
In economics, one distinguishes between [[stocks and flows]]: 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 this unit divided by one of time (say, dollars/year).
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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&nbsp;km in 2 hours, one may divide these and conclude that the object's average speed was 50&nbsp;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 {{math|''m''<sub>man</sub>}}, {{math|''m''<sub>rat</sub>}} and {{math|''L''<sub>man</sub>}} denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression {{nowrapmath|''m''<sub>man</sub> + ''m''<sub>rat</sub>}} is meaningful, but the heterogeneous expression {{nowrapmath|''m''<sub>man</sub> + ''L''<sub>man</sub>}} is meaningless. However, {{math|''m''<sub>man</sub>/''L''<sup>2</sup><sub>man</sub>}} is fine. Thus, dimensional analysis may be used as a [[sanity check]] 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]] and energy share the dimension {{dimanalysis|length=2|mass=1|time=−2}}, 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 {{nowrap|1=1&nbsp;yard = 0.9144&nbsp;m}} to convert 35 yards to 32.004&nbsp;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.<ref>{{Cite book |last1=de Jong |first1=Frits J. |url=https://archive.org/details/dimensionalanaly0000jong |title=Dimensional analysis for economists |last2=Quade |first2=Wilhelm |publisher=North Holland |year=1967 |page=[https://archive.org/details/dimensionalanaly0000jong/page/28 28] |url-access=registration}}</ref> For example, [[Newton's laws of motion]] must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between atwo unitunits that measuresmeasure 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 ==
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=== Mathematics ===
A simple application of dimensional analysis to mathematics is in computing the form of the [[N-sphere#Volume of the n-ball|volume of an {{math|''n''}}-ball]] (the solid ball in ''n'' dimensions), or the area of its surface, the [[n-sphere|{{math|''n''}}-sphere]]: being an {{math|''n''}}-dimensional figure, the volume scales as <{{math>|''x^''{{sup|''n''}}}},</math> while the surface area, being <{{math>|(''n-'' − 1)</math>}}-dimensional, scales as <{{math>|''x^''{{sup|''n-1''−1}}}}.</math> Thus the volume of the {{math|''n''}}-ball in terms of the radius is <{{math>C_nr^|''C''{{sub|''n''}}''r''{{isup|''n''}}}},</math> for some constant <{{math>C_n|''C''{{sub|''n''}}}}.</math> Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone.
 
=== Finance, economics, and accounting ===
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* [[Velocity of money]] 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 {{math|(''dV''/''dr'')/''V''}}, where {{math|''V''}} is the value of a bond (or portfolio), {{math|''r''}} is the continuously compounded interest rate and {{math|''dV''/''dr''}} is a derivative. From the previous point, the dimension of {{math|''r''}} is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because {{math|''dr''}} is in the "denominator" of the derivative.
 
=== Fluid mechanics ===
In [[fluid mechanics]], dimensional analysis is performed to obtain dimensionless [[Buckingham π theorem|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.<ref>{{Cite book |last1=Waite |first1=Lee |title=Applied Biofluid Mechanics |url=https://archive.org/details/appliedbiofluidm00wait |url-access=limited |last2=Fine |first2=Jerry |date=2007 |publisher=McGraw-Hill |isbn=978-0-07-147217-3 |location=New York |page=[https://archive.org/details/appliedbiofluidm00wait/page/n278 260]}}</ref> 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]] ({{math|Re}}), generally important in all types of fluid problems: <math display="block">\mathrm{Re} = \frac{\rho\,ud}{\mu}.</math>
 
* [[ReynoldsFroude number]] (Re{{math|Fr}}), generallymodeling importantflow inwith alla typesfree of fluid problemssurface: <math display="block">\mathrm{ReFr} = \frac{u}{\rhosqrt{g\,udL}{\mu}.</math>
* [[FroudeEuler number (physics)|Euler number]] (Fr{{math|Eu}}), modelingused flowin withproblems ain freewhich surfacepressure is of interest: <math display="block">\mathrm{FrEu} = \frac{u\Delta p}{\sqrt{g\,L}rho u^2}.</math>
* [[Euler number (physics)|EulerMach number]] (Eu{{math|Ma}}), usedimportant in problemshigh inspeed whichflows pressurewhere isthe velocity approaches or exceeds the local speed of interestsound: <math display="block">\mathrm{EuMa} = \frac{\Delta pu}{\rho u^2c}.,</math> where {{math|''c''}} is the local speed of sound.
* [[Mach number]] (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound: <math display="block">\mathrm{Ma} = \frac{u}{c},</math> where {{mvar|c}} is the local speed of sound.
 
== History ==
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In 1822, the important Napoleonic scientist [[Joseph Fourier]] made the first credited important contributions<ref>{{Citation |last=Mason |first=Stephen Finney |title=A history of the sciences |page=169 |year=1962 |place=New York |publisher=Collier Books |isbn=978-0-02-093400-4}}</ref> based on the idea that physical laws like [[Newton's second law|{{nowrap|1=''F'' = ''ma''}}]] should be independent of the units employed to measure the physical variables.
 
[[James Clerk Maxwell]] 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.<ref name="maxwell">{{Citation |last=Roche |first=John J |title=The Mathematics of Measurement: A Critical History |page=203 |year=1998 |publisher=Springer |isbn=978-0-387-91581-4 |url= https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA203|quote = Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.}}</ref> 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]] in which the [[gravitational constant]] {{math|''G''}} is taken as [[1|unity]], thereby defining {{nowrap|1=M = T<sup>−2</sup>L<sup>3</sup>}}.<ref name="maxwell2">{{Citation |last=Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |page=4 |year=1873}}</ref> By assuming a form of [[Coulomb's law]] in which the [[Coulomb constant]] ''k''<sub>e</sub> is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were {{nowrap|1=Q = T<sup>−1</sup>L<sup>3/2</sup>M<sup>1/2</sup>}},<ref name="maxwell3">{{Citation |last= Maxwell |first=James Clerk |title=A Treatise on Electricity and Magnetism |series=Clarendon Press series |page=45 |year=1873 |publisher=Oxford |hdl=2027/uc1.l0065867749 |hdl-access=free}}</ref> which, after substituting his {{nowrap|1=M = T<sup>−2</sup>L<sup>3</sup>}} equation for mass, results in charge having the same dimensions as mass, viz. {{nowrap|1=Q = T<sup>−2</sup>L<sup>3</sup>}}.
 
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]], who was trying to understand why the sky is blue.<ref>{{harv|Pesic|2005}}</ref> Rayleigh first published the technique in his 1877 book ''The Theory of Sound''.<ref>{{Citation |last=Rayleigh |first=Baron John William Strutt |title=The Theory of Sound |url=https://books.google.com/books?id=kvxYAAAAYAAJ |year=1877 |publisher=Macmillan}}</ref>
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=== A simple example: period of a harmonic oscillator ===
What is the period of [[Harmonic oscillator|oscillation]] {{mvarmath|''T''}} of a mass {{mvar|m}} attached to an ideal linear spring with spring constant {{mvarmath|''k''}} suspended in gravity of strength {{mvarmath|''g''}}? That period is the solution for {{mvarmath|''T''}} of some dimensionless equation in the variables {{mvarmath|''T''}}, {{mvarmath|''m''}}, {{mvarmath|''k''}}, and {{mvarmath|''g''}}.
The four quantities have the following dimensions: {{mvar|T}} [T]; {{mvar|m}} [M]; {{mvar|k}} [M/T<sup>2</sup>]; and {{mvarmath|''g''}} [L/T<sup>2</sup>]. From these we can form only one dimensionless product of powers of our chosen variables, <{{math>G_1</math>|1=''G''{{sub|1}} = <math>''T^''{{isup|2 }}''k''/''m</math>''}} {{nowrap|1=[T<sup>2</sup> · M/T<sup>2</sup> / M = 1]}}, and putting <{{math>G_1|1=''G''{{sub|1}} = ''C</math>''}} for some dimensionless constant {{mvarmath|''C''}} 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 (mathematics)|group]]. They are often called [[dimensionless number]]s as well.
 
Note that theThe variable {{mvar|g}} does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines {{mvar|g}} with {{mvar|k}}, {{mvar|m}}, and {{mvar|T}}, because {{mvar|g}} is the only quantity that involves the dimension L. This implies that in this problem the {{mvarmath|''g''}} 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 {{mvarmath|''g''}}: 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: <math>{{tmath|1=T = \kappa \sqrt\tfrac{m}{k}</math> }}, for some dimensionless constant {{math|''κ''}} (equal to <math>\sqrt{C}</math> from the original dimensionless equation).
 
When faced with a case where dimensional analysis rejects a variable ({{mvarmath|''g''}}, 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 {{mvarmath|''κ''}}.
 
=== A more complex example: energy of a vibrating wire ===
Consider the case of a vibrating wire of [[length]] {{math|''ℓ''}} (L) vibrating with an [[amplitude]] {{math|''A''}} (L). The wire has a [[linear density]] {{math|''ρ''}} (M/L) and is under [[Tension (physics)|tension]] {{math|''s''}} (LM/T<sup>2</sup>), and we want to know the energy {{math|''E''}} (L<sup>2</sup>M/T<sup>2</sup>) in the wire. Let {{math|''π''<sub>1</sub>}} and {{math|''π''<sub>2</sub>}} be two dimensionless products of [[Power (mathematics)|powers]] of the variables chosen, given by
: <math>\begin{align}
\pi_1 &= \frac{E}{As} \\
\pi_2 &= \frac{\ell}{A}.
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The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation
: <math>F\left(\frac{E}{As}, \frac{\ell}{A}\right) = 0 ,</math>
where {{math|''F''}} is some unknown function, or, equivalently as
 
: <math>E = As f\left(\frac{\ell}{A}\right) ,</math>
where ''F'' is some unknown function, or, equivalently as
where {{math|''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&nbsp;{{math|''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&nbsp;{{math|''ℓ''}}, and so infer that {{math|1=''E'' = ''ℓs''}}. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.
:<math>E = As f\left(\frac{\ell}{A}\right) ,</math>
 
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 {{nowrap|1=''E'' = ''ℓs''}}. 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]] such as the [[Reynolds number]], which may be interpreted by dimensional analysis.
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=== A third example: demand versus capacity for a rotating disc ===
[[File:Dimensional analysis 01.jpg|thumb|upright=1.5|Dimensional analysis and numerical experiments for a rotating disc]]
Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness {{math|''t''}} (L) and radius {{math|''R''}} (L). The disc has a density {{math|''ρ''}} (M/L<sup>3</sup>), rotates at an angular velocity {{math|''ω''}} (T<sup>−1</sup>) and this leads to a stress {{math|''S''}} (T<sup>−2</sup>L<sup>−1</sup>M) 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 ({{nowrap|1=5&nbsp; −&nbsp; 3&nbsp; =&nbsp; 2}}) non-dimensional groups:
 
: demand/capacity = {{math|''ρR''{{i sup|2}}''ω''{{i sup|2}}/''S''}}
: thickness/radius or aspect ratio = {{math|''t''/''R''}}
 
Through the use of numerical experiments using, for example, the [[finite element method]], 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.<ref>{{cite web|last1=Ramsay|first1=Angus|title=Dimensional Analysis and Numerical Experiments for a Rotating Disc|url=http://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/dimensional-analysis--numerical-experiments-for-a-rotating-disc/|website=Ramsay Maunder Associates|access-date=15 April 2017}}</ref>
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{{details|Buckingham π theorem}}
 
The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an [[abelian group]]: The [[identity element|identity]] is written as 1;{{citation needed|reason=Both the new SI and the as-yet unpublished VIM4 make no such statement.|date=May 2021}} {{nowrap|1=L<sup>0</sup> = 1}}, and the inverse of L is 1/L or L<sup>−1</sup>. L raised to any integer power {{math|''p''}} is a member of the group, having an inverse of L<sup>{{math|−''p''}}</sup> or 1/L<sup>{{math|''p''}}</sup>. The operation of the group is multiplication, having the usual rules for handling exponents ({{nowrap|1=L<sup>{{math|''n''}}</sup> × L<sup>{{math|''m''}}</sup> = L<sup>{{math|''n''+''m''}}</sup>}}). Physically, 1/L can be interpreted as [[reciprocal length]], and 1/T as reciprocal time (see [[reciprocal second]]).
 
An abelian group is equivalent to a [[module (mathematics)|module]] over the integers, with the dimensional symbol {{gaps|T<sup>{{math|''i''}}</sup>|L<sup>{{math|''j''}}</sup>|M<sup>{{math|''k''}}</sup>}} corresponding to the tuple {{nowrapmath|(''i'', ''j'', ''k'')}}. 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 the module.
 
A basis for such a module of dimensional symbols is called a set of [[Base quantity|base quantities]], and all other vectors are called derived units. As in any module, one may choose different [[Basis (linear algebra)|bases]], which yields different systems of units (e.g., [[ampere#Proposed future definition|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, <{{math>|(0, 0, 0)</math>}}.
 
In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like <{{math>|''V^''{{isup|''L^''{{sup|1/2}}</math>}}}}.{{sfn|Tao|2012|loc="With a bit of additional effort (and taking full advantage of the one-dimensionality of the vector spaces), one can also define spaces with fractional exponents &nbsp;..."}} However, it is not possible to take arbitrary fractional powers of units, due to [[representation theory|representation-theoretic]] obstructions.{{sfn|Tao|2012|loc="However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units &nbsp;..."}}
 
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 {{math|''M''}} and {{math|''L''}}, one has the vector spaces <{{math>|''V^''{{isup|''M</math>''}}}} and <{{math>|''V^''{{isup|''L</math>''}}}}, and can define <{{math>|1=''V^''{{isup|''ML''}} := ''V^''{{isup|''M''}} \otimes⊗ ''V^''{{isup|''L</math>''}}}} as the [[tensor product]]. Similarly, the dual space can be interpreted as having "negative" dimensions.<ref>{{harvnb|Tao|2012}} "Similarly, one can define <{{math>|''V^''{{isup|''T^''{-1{isup|−1}}}}}}</math> as the dual space to <{{math>|''V^''{{isup|''T</math>''}}}} ..."</ref> This corresponds to the fact that under the [[natural pairing]] between a vector space and its dual, the dimensions cancel, leaving a [[dimensionless]] scalar.
 
The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The [[Kernel (linear algebra)#nullity|nullity]] describes some number (e.g., {{math|''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, {{math|{{mset|π<sub>1</sub>, ..., π<sub>''m''</sub>}}}}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and [[Exponent (mathematics)|exponentiating]]) together the measured quantities to produce something with the same unit as some derived quantity {{math|''X''}} can be expressed in the general form
: <math>X = \prod_{i=1}^m (\pi_i)^{k_i}\,.</math>
 
:<math>X = \prod_{i=1}^m (\pi_i)^{k_i}\,.</math>
 
Consequently, every possible [[#Commensurability|commensurate]] equation for the physics of the system can be rewritten in the form
: <math>f(\pi_1,\pi_2, ..., \pi_m)=0\,.</math>
 
:<math>f(\pi_1,\pi_2, ..., \pi_m)=0\,.</math>
 
Knowing this restriction can be a powerful tool for obtaining new insight into the system.
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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 [F = LM/T<sup>2</sup>], L, M, while the latter can be expressed as [T = (LM/F)<sup>1/2</sup>], L, M.
 
On the other hand, length, velocity and time {{nowrap|(T, L, V)}} 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 ({{nowrap|1=V = L/T}}), is redundant (the set is not ''linearly independent'').
 
=== Other fields of physics and chemistry ===
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<!--see discussion page/transcendental functions This requirement is clear when one observes the [[Taylor expansion]]s for these functions (a sum of various powers of the function argument). For example, the logarithm of 3&nbsp;kg is undefined even though the logarithm of&nbsp;3 is nearly&nbsp;0.477. An attempt to compute ln&nbsp;3&nbsp;kg would produce, if one naively took ln&nbsp;3&nbsp;kg to mean the dimensionally meaningless "ln(1&nbsp;+&nbsp;2&nbsp;kg)",
: <math>\mathrm{2\,kg} - \frac{\mathrm{4\,kg}^2}{2} + \cdots ,</math>
 
: <math>2\,\mathrm{kg} - \frac{4\,\mathrm{kg}^2}{2} + \cdots</math>
 
which is dimensionally incompatible – the sum has no meaningful dimension – requiring the argument of transcendental functions to be dimensionless.
 
Another way to understand this problem is that the different coefficients ''scale'' differently under change of unit – were one to reconsider this in grams as "ln&thinsp;3000&nbsp;g" instead of "ln&thinsp;3&nbsp;kg", one could compute ln&thinsp;3000, but in terms of the [[Taylor series]], the degree 1 term would scale by 1000, the degree-2 term would scale by 1000<sup>2</sup>, and so forth – the overall output would not scale as a particular dimension.
-->
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 {{math|1=log(''a''/''b'') = log&thinsp;''a'' − log&thinsp;''b''}}, where the logarithm is taken in any base, holds for dimensionless numbers {{math|''a''}} and {{math|''b''}}, but it does ''not'' hold if {{math|''a''}} and {{math|''b''}} are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.<ref>{{harnvb|Berberan-Santos|Pogliani|1999|page=256}}</ref>
 
Similarly, while one can evaluate [[monomials]] ({{math|''x''<sup>''n''</sup>}}) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for {{math|''x''<sup>2</sup>}}, the expression {{nowrap|1=(3&nbsp; m)<sup>2</sup>&nbsp; =&nbsp; 9&nbsp; m<sup>2</sup>}} makes sense (as an area), while for {{math|''x''<sup>2</sup>&nbsp; +&nbsp; ''x''}}, the expression {{nowrap|1=(3&nbsp; m)<sup>2</sup>&nbsp; +&nbsp; 3&nbsp; m&nbsp; =&nbsp; 9&nbsp; m<sup>2</sup>&nbsp; +&nbsp; 3&nbsp; 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,
: <math> \tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot t^2 + (\mathrm{500~m/s}) \cdot t. </math>
 
This is the height to which an object rises in time&nbsp;{{math|''t''}} if the acceleration of [[gravity]] is 9.8 {{nowrap|metres per second per second}} and the initial upward speed is 500 {{nowrap|metres per second}}. It is not necessary for {{math|''t''}} to be in ''seconds''. For example, suppose {{math|''t''}}&nbsp;=&nbsp;0.01&nbsp;minutes. Then the first term would be
:<math> \tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot t^2 + (\mathrm{500~m/s}) \cdot t. </math>
: <math>\begin{align}
 
This is the height to which an object rises in time&nbsp;''t'' if the acceleration of [[gravity]] is 9.8 {{nowrap|metres per second per second}} and the initial upward speed is 500 {{nowrap|metres per second}}. It is not necessary for ''t'' to be in ''seconds''. For example, suppose ''t''&nbsp;=&nbsp;0.01&nbsp;minutes. Then the first term would be
 
:<math>\begin{align}
&\tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot (\mathrm{0.01~min})^2 \\[10pt]
={} &\tfrac{1}{2} \cdot -9.8 \cdot \left(0.01^2\right) (\mathrm{min/s})^2 \cdot \mathrm{m} \\[10pt]
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{{main|Physical quantity#Components}}
 
The value of a dimensional physical quantity {{math|''Z''}} is written as the product of a [[Unit of measurement|unit]] [{{math|''Z''}}] within the dimension and a dimensionless numerical value or numerical factor, {{math|''n''}}.<ref name=Pisanty13>For a review of the different conventions in use see: {{cite web |url=http://physics.stackexchange.com/q/77690 |title=Square bracket notation for dimensions and units: usage and conventions |last1=Pisanty |first1= E|date=17 September 2013 |website=Physics Stack Exchange |access-date=15 July 2014}}</ref>
: <math>Z = n \times [Z] = n [Z]</math>
 
:<math>Z = n \times [Z] = n [Z]</math>
 
When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A [[Conversion of units|conversion factor]], which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
: <math> \mathrm{1\,ft} = \mathrm{0.3048\,m}</math> is identical to <math> 1 = \frac{\mathrm{0.3048\,m}}{\mathrm{1\,ft}}.</math>
 
The factor 0.3048&nbsp;m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.
:<math> 1 \ \mbox{ft} = 0.3048 \ \text{m} \ </math> is identical to <math> 1 = \frac{0.3048 \ \text{m}}{1 \ \text{ft}}.\ </math>
 
The factor <math> 0.3048 \ \frac{\text{m}}{\mbox{ft}} </math> is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.
 
Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.
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In contrast, in a ''numerical-value equation'', just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit.
 
For example, a quantity equation for [[displacement (geometry)|displacement]] {{math|''d''}} as [[speed]] {{math|''s''}} multiplied by time difference {{math|''t''}} would be:
: {{math|1=''d'' = ''s'' ''t''}}
for {{math|''s''}} = 5&nbsp;m/s, where {{math|''t''}} and {{math|''d''}} may be expressed in any units, [[conversion of units|converted]] if necessary.
In contrast, a corresponding numerical-value equation would be:
: {{math|1=''D'' = 5 ''T''}}
where {{math|''T''}} is the numeric value of {{math|''t''}} when expressed in seconds and {{math|''D''}} is the numeric value of {{math|''d''}} when expressed in metres.
 
Generally, the use of numerical-value equations is discouraged.<ref name ="nist">{{Cite book|url=https://physics.nist.gov/cuu/pdf/sp811.pdf|title=Guide for the Use of the International System of Units (SI): The Metric System|last=Thompson|first=Ambler|date=November 2009 |publisher=DIANE Publishing|isbn=9781437915594|language=en}}</ref>
Line 309 ⟶ 314:
=== Constants ===
{{Main|Dimensionless quantity}}
The dimensionless constants that arise in the results obtained, such as the {{math|''C''}} in the Poiseuille's Law problem and the <{{math>\kappa</math>|''κ''}} in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "[[back of the envelope]]" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
 
=== Formalisms ===
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the [[Ising model]] can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, <{{math>\xi</math> |''χ''}}) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be <{{math>\sim|~ 1/\xi^''χ''{{sup|''d''}</math>}}}, where <{{math>|''d</math>''}} is the dimension of the lattice.
 
It has been argued by some physicists, e.g., [[Michael Duff (physicist)|Michael J. Duff]],<ref name="duff" /><ref>{{cite arXiv |last=Duff |first=Michael James |eprint=hep-th/0208093v3 |title=Comment on time-variation of fundamental constants |date=July 2004}}</ref> that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: ''[[Speed of light|{{math|''c]]''}}]], ''[[Planck constant|{{math|''ħ]]''}}]], and ''[[Gravitational constant|{{math|''G]]''}}]], in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
 
Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants {{math|''ħ''}}, {{math|''c''}}, and {{math|''G''}} (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit <{{math>|''c\rightarrow'' \infty</math>→ ∞}}, <{{math>\hbar\rightarrow|''ħ'' → 0</math>}} and <{{math>|''G\rightarrow'' → 0</math>}}. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.
 
== Dimensional equivalences ==
Line 326 ⟶ 331:
{| class="wikitable"
|-
! scope="col" style="width:100px;" | Energy, {{math|''E'' }}
T<sup>−2</sup>L<sup>2</sup>M
! scope="col" style="width:100px;" | Expression
Line 333 ⟶ 338:
|rowspan="4"| Mechanical
| <math> Fd </math>
| {{math|''F''}} = [[force]], {{math|''d''}} = [[distance]]
|-
| <math> S/t \equiv Pt </math>
| {{math|''S''}} = [[action (physics)|action]], {{math|''t''}} = time, {{math|''P''}} = [[power (physics)|power]]
|-
| <math> mv^2 \equiv pv \equiv p^2/m </math>
| {{math|''m''}} = [[mass]], {{math|''v''}} = [[velocity]], {{math|''p''}} = [[momentum]]
|-
| <math> I\omega^2 \equiv L\omega \equiv L^2/I </math>
| {{math|''L''}} = [[angular momentum]], {{math|''I''}} = [[moment of inertia]], {{math|''ω''}} = [[angular velocity]]
|-
| Ideal gases
| <math> p V \equiv NT </math>
| {{math|''p''}} = pressure, {{math|''V''}} = volume, {{math|''T''}} = temperature, {{math|''N''}} = [[amount of substance]]
|-
| Waves
| <math> AIt \equiv ASt </math>
| {{math|''A''}} = [[area]] of [[Huygens–Fresnel principle|wave front]], {{math|''I''}} = wave [[intensity (physics)|intensity]], {{math|''t''}} = [[time]], {{math|''S''}} = [[Poynting vector]]
|-
| rowspan="3" | Electromagnetic
| <math> q\phi </math>
| {{math|''q''}} = [[electric charge]], {{math|''ϕ''}} = [[electric potential]] (for changes this is [[voltage]])
|-
| <math> \varepsilon E^2V \equiv B^2V/\mu </math>
| {{math|''E''}} = [[electric field]], {{math|''B''}} = [[magnetic field]], <br /> {{math|''ε''}} = [[permittivity]], {{math|''μ''}} = [[permeability (electromagnetism)|permeability]], <br />{{math|''V''}} = 3d [[volume]]
|-
| <math> pE \equiv m B \equiv I A B </math>
| {{math|''p''}} = [[electric dipole moment]], {{math|''m''}} = magnetic moment, <br /> {{math|''A''}} = area (bounded by a current loop), ''I'' = [[electric current]] in loop
|}
 
{| class="wikitable"
|-
! scope="col" style="width:100px;" | Momentum, {{math|''p''}}
T<sup>−1</sup>LM
! scope="col" style="width:100px;" | Expression
Line 372 ⟶ 377:
| rowspan="2" | Mechanical
| <math> mv \equiv Ft </math>
| {{math|''m''}} = mass, {{math|''v''}} = velocity, {{math|''F''}} = force, {{math|''t''}} = time
|-
| <math> S/r \equiv L/r </math>
| {{math|''S''}} = action, {{math|''L''}} = angular momentum, {{math|''r''}} = [[displacement (vector)|displacement]]
|-
| Thermal
Line 383 ⟶ 388:
| Waves
| <math> \rho V v </math>
| {{math|''ρ''}} = [[density]], {{math|''V''}} = [[volume]], {{math|''v''}} = [[phase velocity]]
|-
| Electromagnetic
| <math> q A </math>
| {{math|''A''}} = [[magnetic vector potential]]
|}
 
{| class="wikitable"
|-
! scope="col" style="width:100px;" | Force, {{math|''F''}}
T<sup>−2</sup>LM
! scope="col" style="width:100px;" | Expression
Line 399 ⟶ 404:
| Mechanical
| <math> ma \equiv p/t </math>
| {{math|''m''}} = mass, {{math|''a''}} = acceleration
|-
| Thermal
| <math> T \delta S/\delta r </math>
| {{math|''S''}} = entropy, {{math|''T''}} = temperature, {{math|''r''}} = displacement (see [[entropic force]])
|-
| Electromagnetic
| <math> Eq \equiv Bqv </math>
| {{math|''E''}} = electric field, {{math|''B''}} = magnetic field, {{math|''v''}} = velocity, {{math|''q''}} = charge
|}
 
=== Natural units ===
{{main|Natural units}}
 
If {{nowrap|1=''c'' = ''ħ'' = 1}}, where ''c'' is the [[speed of light]] and ''ħ'' is the [[reduced Planck constant]], and a suitable fixed unit of energy is chosen, then all quantities of time ''T'', length ''L'' and mass ''M'' can be expressed (dimensionally) as a power of energy ''E'', because length, mass and time can be expressed using speed ''v'', action ''S'', and energy ''E'':<ref name=Martin08/>
 
:<math>t = \frac{S}{E},\quad L = \frac{Sv}{E}, \quad M = \frac{E}{v^2}</math>
 
though speed and action are dimensionless ({{nowrap|1=''v'' = ''c'' = 1}} and {{nowrap|1=''S'' = ''ħ'' = 1}}) – so the only remaining quantity with dimension is energy. In terms of powers of dimensions:
 
:<math>\mathsf{E}^n = \mathsf{T}^p\mathsf{L}^q\mathsf{M}^r = \mathsf{E}^{-p-q+r} </math>
 
This is particularly useful in particle physics and high energy physics, in which case the energy unit is the electron volt (eV). Dimensional checks and estimates become very simple in this system.
 
However, if electric charges and currents are involved, another unit to be fixed is for electric charge, normally the [[electron charge]] ''e'' though other choices are possible.
 
{| class="wikitable"
|-
! scope="col" style="width:100px;" rowspan="2"| Quantity
! scope="col" style="width:90px;" colspan="3"| ''p'', ''q'', ''r'' powers of energy
! scope="col" style="width:90px;"| ''n''<br />power of energy
|-
! scope="col" style="width:25px;"| ''p''
! scope="col" style="width:25px;"| ''q''
! scope="col" style="width:25px;"| ''r''
! ''n''
|-
| Action, ''S''
| −1
| 2
| 1
| 0
|-
| Speed, ''v''
| −1
| 1
| 0
| 0
|-
| Mass, ''M''
| 0
| 0
| 1
| 1
|-
| Length, ''L''
| 0
| 1
| 0
| −1
|-
| Time, ''t''
| 1
| 0
| 0
| −1
|-
| Momentum, ''p''
| −1
| 1
| 1
| 1
|-
| Energy, ''E''
| −2
| 2
| 1
| 1
|}
 
Line 490 ⟶ 426:
== Geometry: position vs. displacement ==
 
=== Affine quantities ===
{{Mainfurther|Affine space}}
Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. (In mathematics scalars are considered a special case of vectors;{{citation needed|date=September 2013}} vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an [[origin (mathematics)|origin]]. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change).
 
Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable:
Line 517 ⟶ 453:
 
=== Orientation and frame of reference ===
Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a ''direction''. (This issue does not arise inIn 1 dimension, orthis ratherissue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in a multi-dimensional Euclidean space, one also needs ana orientationbearing: they need to be compared to a [[frame of reference]].
 
This leads to the [[#Extensions|extensions]] discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.
Line 525 ⟶ 461:
 
He introduced two approaches:
 
* The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have L<sub>x</sub> represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
* Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.
 
==== Directed dimensions ====
As an example of the usefulness of the first approach, suppose we wish to calculate the [[trajectory#Range and height|distance a cannonball travels]] when fired with a vertical velocity component <math>v_\text{y}</math> and a horizontal velocity component {{tmath|v_\text{x} }}, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then {{mvar|R}}, the distance travelled, with dimension L, {{tmath|v_\text{x} }}, {{tmath|v_\text{y} }}, both dimensioned as T<sup>−1</sup>L, and {{mvar|g}} the downward acceleration of gravity, with dimension T<sup>−2</sup>L.
 
As an example of the usefulness of the first approach, suppose we wish to calculate the [[trajectory#Range and height|distance a cannonball travels]] when fired with a vertical velocity component <math>V_\mathrm{y}</math> and a horizontal velocity component <math>V_\mathrm{x}</math>, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then {{mvar|R}}, the distance travelled, with dimension L, <math>V_\mathrm{x}</math>, <math>V_\mathrm{y}</math>, both dimensioned as T<sup>−1</sup>L, and {{mvar|g}} the downward acceleration of gravity, with dimension T<sup>−2</sup>L.
 
With these four quantities, we may conclude that the equation for the range {{mvar|R}} may be written:
: <math>R \propto v_\text{x}^a\,v_\text{y}^b\,g^c .</math>
 
:<math>R \propto V_\text{x}^a\,V_\text{y}^b\,g^c.\,</math>
 
Or dimensionally
: <math>\mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c</math>
from which we may deduce that <math>a + b + c = 1</math> and {{tmath|1=a + b + 2c = 0}}, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.
 
However, if we use directed length dimensions, then <math>v_\mathrm{x}</math> will be dimensioned as T<sup>−1</sup>L<sub>{{math|x}}</sub>, <math>v_\mathrm{y}</math> as T<sup>−1</sup>L<sub>{{math|y}}</sub>, {{mvar|R}} as L<sub>{{math|x}}</sub> and {{mvar|g}} as T<sup>−2</sup>L<sub>{{math|y}}</sub>. The dimensional equation becomes:
:<math>\mathsf{L} = \left(\frac{\mathsf{L}}{\mathsf{T}}\right)^{a+b} \left(\frac{\mathsf{L}}{\mathsf{T}^2}\right)^c\,</math>
: <math>
 
from which we may deduce that <math>a + b + c = 1</math> and <math>a + b + 2c = 0</math>, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.
 
However, if we use directed length dimensions, then <math>V_\mathrm{x}</math> will be dimensioned as T<sup>−1</sup>L<sub>{{math|x}}</sub>, <math>V_\mathrm{y}</math> as T<sup>−1</sup>L<sub>{{math|y}}</sub>, {{mvar|R}} as L<sub>{{math|x}}</sub> and {{mvar|g}} as T<sup>−2</sup>L<sub>{{math|y}}</sub>. The dimensional equation becomes:
 
:<math>
\mathsf{L}_\mathrm{x} =
\left(\frac{\mathsf{LT}_\mathrm^{x-1}}{\mathsf{TL}_\mathrm{x}}\right)^a
\left(\frac{\mathsf{LT}_\mathrm^{y-1}}{\mathsf{TL}_\mathrm{y}}\right)^b
\left(\frac{\mathsf{LT}_\mathrm^{y-2}}{\mathsf{TL}_\mathrm{y}^2}\right)^c
</math>
and we may solve completely as {{math|1=''a'' = 1}}, {{math|1=''b'' = 1}} and {{math|1=''c'' = −1}}. The increase in deductive power gained by the use of directed length dimensions is apparent.
 
and we may solve completely as <math>a = 1</math>, <math>b = 1</math> and <math>c = -1</math>. The increase in deductive power gained by the use of directed length dimensions is apparent.
 
Huntley's concept of directed length dimensions however has some serious limitations:
 
* It does not deal well with vector equations involving the ''[[cross product]]'',
* nor does it handle well the use of ''angles'' as physical variables.
Line 564 ⟶ 492:
 
==== Quantity of matter ====
 
In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (''inertial mass''), and mass as a measure of the quantity of matter. '''Quantity of matter''' is defined by Huntley as a quantity only {{em|proportional}} to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.
 
Line 583 ⟶ 510:
 
There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
: <math>\pi_1 = \frac{\dot{m}}{\eta r}</math>
: <math>\pi_2 = \frac{p_\mathrm{x}\rho r^5}{\dot{m}^2}</math>
 
If we distinguish between inertial mass with dimension <math>M_\text{i}</math> and quantity of matter with dimension <math>M_\text{m}</math>, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
: <math>C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}}</math>
 
:<math>C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}}</math>
 
where now only {{mvar|C}} is an undetermined constant (found to be equal to <math>\pi/8</math> by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield [[Poiseuille's law]].
Line 595 ⟶ 521:
 
=== Siano's extension: orientational analysis ===
{{see also|Angle#Dimensional analysis}}
 
[[Angle]]s are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested <ref>{{ cite journal | title=Angles in the SI: a detailed proposal for solving the problem | year=2021 | pages=053002 | journal=Metrologia | doi=10.1088/1681-7575/ac023f | volume=58 | issue=5 | url=http://dx.doi.org/10.1088/1681-7575/ac023f | last1=Quincey | first1= Paul | arxiv=2108.05704 }}</ref>) . As an example, consider again the projectile problem in which a point mass is launched from the origin {{math|1=(''x'', ''y'') = (0, 0)}} at a speed {{math|''v''}} and angle {{math|''θ''}} above the ''x''-axis, with the force of gravity directed along the negative ''y''-axis. It is desired to find the range {{math|''R''}}, at which point the mass returns to the ''x''-axis. Conventional analysis will yield the dimensionless variable {{math|1=''π'' = ''R'' ''g''/''v''<sup>2</sup>}}, but offers no insight into the relationship between {{math|''R''}} and {{math|''θ''}}.
 
Siano has suggested that the directed dimensions of Huntley be replaced by using ''orientational symbols'' {{math|1<sub>x</sub>&nbsp;1<sub>y</sub>&nbsp;1<sub>z</sub>}} to denote vector directions, and an orientationless symbol 1<sub>0</sub>.<ref>{{harvs|txt=yes|last=Siano|year1=1985-I|year2=1985-II}}</ref> Thus, Huntley's L<sub>{{math|x}}</sub> becomes L1<sub>{{math|x}}</sub> with L specifying the dimension of length, and {{math|1<sub>x</sub>}} specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that {{math|1=1<sub>''i''</sub><sup>−1</sup> = 1<sub>''i''</sub>}}, the following multiplication table for the orientation symbols results:
Line 616 ⟶ 543:
|}
 
Note that theThe orientational symbols form a group (the [[Klein four-group]] or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of {{math|1<sub>z</sub>}}. For angles, consider an angle {{mvar|θ}} that lies in the z-plane. Form a right triangle in the z-plane with {{mvar|θ}} being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation {{math|1<sub>x</sub>}} and the side opposite has an orientation {{math|1<sub>y</sub>}}. Since (using {{math|~}} to indicate orientational equivalence) {{math|1=tan(''θ'') = ''θ''&nbsp;+&nbsp;... ~ 1<sub>y</sub>/1<sub>x</sub>}} we conclude that an angle in the xy-plane must have an orientation {{math|1=1<sub>y</sub>/1<sub>x</sub> = 1<sub>z</sub>}}, which is not unreasonable. Analogous reasoning forces the conclusion that {{math|1=sin(''θ'')}} has orientation {{math|1<sub>z</sub>}} while {{math|cos(''θ'')}} has orientation 1<sub>0</sub>. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form {{math|''a'' cos(''θ'') + ''b'' sin(''θ'')}}, where {{mvar|a}} and {{mvar|b}} are real scalars. Note that anAn expression such as <math>\sin(\theta+\pi/2)=\cos(\theta)</math> is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:
: <math>
 
:<math>
\sin\left(a\,1_\text{z} + b\,1_\text{z}\right) =
\sin\left(a\,1_\text{z}) \cos(b\,1_\text{z}\right) +
\sin\left(b\,1_\text{z}) \cos(a\,1_\text{z}\right),
</math>
which for <math>a = \theta</math> and <math>b = \pi/2</math> yields {{tmath|1=\sin(\theta\,1_\text{z} + [\pi/2]\,1_\text{z}) = 1_\text{z}\cos(\theta\,1_\text{z})}}. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is {{tmath|1_0}}.
 
which for <math>a = \theta</math> and <math>b = \pi/2</math> yields <math>\sin(\theta\,1_\text{z} + [\pi/2]\,1_\text{z}) = 1_\text{z}\cos(\theta\,1_\text{z})</math>. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is <math>1_0</math>.
 
The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into [[Canonical form|normal form]]. The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd.
 
As an example, for the projectile problem, using orientational symbols, {{math|''θ''}}, being in the xy-plane will thus have dimension {{math|1<sub>z</sub>}} and the range of the projectile {{mvar|R}} will be of the form:
: <math>R = g^a\,v^b\,\theta^c\text{ which means }\mathsf{L}\,1_\mathrm{x} \sim
 
:<math>R = g^a\,v^b\,\theta^c\text{ which means }\mathsf{L}\,1_\mathrm{x} \sim
\left(\frac{\mathsf{L}\,1_\text{y}}{\mathsf{T}^2}\right)^a \left(\frac{\mathsf{L}}{\mathsf{T}}\right)^b\,1_\mathsf{z}^c.\,</math>
 
Dimensional homogeneity will now correctly yield {{math|1=''a'' = −1}} and {{math|1=''b'' = 2}}, and orientational homogeneity requires that <math>{{tmath|1=1_x /(1_y^a 1_z^c)=1_z^{c+1} = 1</math>}}. In other words, that {{mvar|c}} must be an odd integer. In fact, the required function of theta will be {{math|sin(''θ'')cos(''θ'')}} which is a series consisting of odd powers of {{mvar|θ}}.
 
It is seen that the Taylor series of {{math|sin(''θ'')}} and {{math|cos(''θ'')}} are orientationally homogeneous using the above multiplication table, while expressions like {{math|cos(''θ'') + sin(''θ'')}} and {{math|exp(''θ'')}} are not, and are (correctly) deemed unphysical.
Line 658 ⟶ 582:
 
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| title = On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems
| journal = Journal of Applied Mechanics
| volume = 68 | issue = A–34
}}
* {{Citationcitation
| first = H. | last = Whitney
| year = 1968
| title = The Mathematics of Physical Quantities, Parts I and II
| journal = American Mathematical Monthly
| volume = 75 | issue = 2 | pages = 115–138, 227–256
| doi = 10.2307/2315883
| issue = 2
| jstor = 2315883
}}
* [[Edwin Bidwell Wilson|Wilson, Edwin B.]] (1920) [https://archive.org/details/aeronautics00wilsgoog/page/n197/mode/2up "Theory of Dimensions"], chapter XI of ''Aeronautics'', via [[Internet Archive]]
* {{Citation
* {{citation
|first = GA
| first = GA | last = Vignaux
| chapter = Dimensional Analysis in Data Modelling
|editor editor1 = Erickson, Gary J.
| editor2 = Neudorfer, Paul O.
| title = Maximum entropy and Bayesian methods: proceedings of the Eleventh International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Seattle, 1991
| publisher = Kluwer Academic
| year = 1992
| isbn = 978-0-7923-2031-9
}}
* {{Citationcitation
| first1 = Wacław | last1 = Kasprzak
| first2 = Bertold | last2 = Lysik
|last1 = Kasprzak
| first3 = Marek | last3 = Rybaczuk
|first2 = Bertold
| title = Dimensional Analysis in the Identification of Mathematical Models
|last2 = Lysik
| publisher = World Scientific
|first3 = Marek
|last3 year = Rybaczuk1990
| isbn = 978-981-02-0304-7
|title = Dimensional Analysis in the Identification of Mathematical Models
}}
|publisher = World Scientific
|year = 1990
|isbn = 978-981-02-0304-7 }}
 
== Further reading ==
* {{cite book
| first=Douglas C. | last=Giancoli
| title=Physics: Principles with Applications
| edition=7th
| chapter=1. Introduction, Measurement, Estimating §1.8 Dimensions and Dimensional Analysis
| year=2014
| publisher=Pearson
| isbn=978-0-321-62592-2 | oclc=853154197
}}
 
== External links ==
*{{cite book |first=Douglas C. |last=Giancoli |title=Physics: Principles with Applications |edition=7th |chapter=1. Introduction, Measurement, Estimating §1.8 Dimensions and Dimensional Analysis |year=2014 |isbn=978-0-321-62592-2 |oclc=853154197}}
 
==External links==
{{Wikibooks|Fluid Mechanics|Ch4|Dimensional analysis}}
{{Commons category}}
* [https://web.archive.org/web/20100410142839/http://www.roymech.co.uk/Related/Fluids/Dimension_Analysis.html List of dimensions for variety of physical quantities]
* [http://www.calchemy.com/uclive.htm Unicalc Live web calculator doing units conversion by dimensional analysis]
* [http://www.boost.org/doc/libs/1_66_0/doc/html/boost_units.html A C++ implementation of compile-time dimensional analysis in the Boost open-source libraries]
* [http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf Buckingham's pi-theorem]
* [http://QuantitySystem.CodePlex.com Quantity System calculator for units conversion based on dimensional approach] {{Webarchive|url=https://web.archive.org/web/20171224025732/http://quantitysystem.codeplex.com/ |date=24 December 2017 }}
* [http://www.outlawmapofphysics.com Units, quantities, and fundamental constants project dimensional analysis maps]
* {{cite web|last=Bowley|first=Roger|title=Dimensional Analysis|url=http://www.sixtysymbols.com/videos/dimensional.htm|website=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2009}}
* {{cite thesis|last=Dureisseix|first=David|title=An introduction to dimensional analysis|url=https://cel.archives-ouvertes.fr/cel-01380149|year=2019|publisher=INSA Lyon|type=lecture}}
 
{{systems of measurement}}
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