Análisis dimensional

As examples, the dimension of the physical quantity velocityv is $${\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.}$$

The dimension of the physical quantity accelerationa is $${\displaystyle \operatorname {dim} a={\frac {\text{velocity}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.}$$

The dimension of the physical quantity forceF is $${\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.}$$

The dimension of the physical quantity pressureP is $${\displaystyle \operatorname {dim} P={\frac {\text{force}}{\text{area}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}}{{\mathsf {L}}^{2}}}={\mathsf {T}}^{-2}{\mathsf {L}}^{-1}{\mathsf {M}}.}$$

The dimension of the physical quantity energyE is $${\displaystyle \operatorname {dim} E={\text{force}}\times {\text{displacement}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}\times {\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}.}$$

The dimension of the physical quantity powerP is $${\displaystyle \operatorname {dim} P={\frac {\text{energy}}{\text{time}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {T}}}={\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}.}$$

The dimension of the physical quantity electric chargeQ is $${\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.}$$

The dimension of the physical quantity voltageV is $${\displaystyle \operatorname {dim} V={\frac {\text{power}}{\text{current}}}={\frac {{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {I}}}={\mathsf {T^{-3}}}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}.}$$

The dimension of the physical quantity capacitanceC is $${\displaystyle \operatorname {dim} C={\frac {\text{electric charge}}{\text{electric potential difference}}}={\frac {{\mathsf {T}}{\mathsf {I}}}{{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}}}={\mathsf {T^{4}}}{\mathsf {L^{-2}}}{\mathsf {M^{-1}}}{\mathsf {I^{2}}}.}$$

En el análisis dimensional, el método de Rayleigh es una herramienta conceptual utilizada en física , química e ingeniería . Expresa la relación funcional de algunas variables mediante una ecuación exponencial . Recibe su nombre de Lord Rayleigh .

El método consta de los siguientes pasos:

1. Reúna todas las variables independientes que puedan influir en la variable dependiente .
2. Si R es una variable que depende de las variables independientes R ₁ , R ₂ , R ₃ , ..., R _n , entonces la ecuación funcional se puede escribir como R = F ( R ₁ , R ₂ , R ₃ , ..., R _n ) .
3. Escribe la ecuación anterior en la forma R = C R ₁ ^a R ₂ ^b R ₃ ^c ... R _n ^m , donde C es una constante adimensional y a , b , c , ..., m son exponentes arbitrarios.
4. Exprese cada una de las cantidades de la ecuación en algunas unidades base en las que se requiere la solución.
5. Utilizando la homogeneidad dimensional , obtenga un conjunto de ecuaciones simultáneas que involucren los exponentes a , b , c , ..., m .
6. Resuelve estas ecuaciones para obtener los valores de los exponentes a , b , c , ..., m .
7. Sustituya los valores de los exponentes en la ecuación principal y forme los parámetros adimensionales agrupando las variables con exponentes iguales.

Como desventaja, el método de Rayleigh no proporciona ninguna información sobre el número de grupos adimensionales que se obtendrán como resultado del análisis dimensional.

Los porcentajes son cantidades adimensionales, ya que son razones de dos cantidades con las mismas dimensiones. En otras palabras, el  signo % se puede leer como "centésimas", puesto que 1% = 1/100 .

Al derivar una cantidad, se divide la dimensión por la dimensión de la variable con respecto a la cual se deriva. Por lo tanto:

• La posición ( x ) tiene la dimensión L (longitud);
• La derivada de la posición con respecto al tiempo ( dx / dt , velocidad ) tiene dimensión T ⁻¹ L—longitud desde la posición, tiempo debido a la derivada;
• La segunda derivada ( d ² x / dt ² = d ( dx / dt ) / dt , aceleración ) tiene dimensión T ⁻² L .

De igual modo, al tomar una integral se añade la dimensión de la variable con respecto a la cual se está integrando, pero en el numerador.

• La fuerza tiene la dimensión T ⁻² L M (masa multiplicada por aceleración);
• la integral de fuerza con respecto a la distancia ( s ) que ha recorrido el objeto ( ⁠${\displaystyle \textstyle \int F\ ds}$⁠ , trabajo ) tiene dimensión T ⁻² L ² M .

En economía, se distingue entre existencias y flujos : una existencia tiene una unidad (por ejemplo, productos o dólares), mientras que un flujo es un derivado de una existencia y tiene una unidad de la forma de esta unidad dividida por una unidad de tiempo (por ejemplo, dólares/año).

En algunos contextos, las cantidades dimensionales se expresan como cantidades adimensionales o porcentajes omitiendo algunas dimensiones. Por ejemplo, los ratios deuda/PIB se suelen expresar como porcentajes: deuda total pendiente (dimensión monetaria) dividida por el PIB anual (dimensión monetaria). Sin embargo, se podría argumentar que, al comparar un stock con un flujo, el PIB anual debería tener dimensiones monetarias/temporales (dólares/año, por ejemplo) y, por lo tanto, la relación deuda/PIB debería tener la unidad anual, lo que indica que esta relación representa el número de años necesarios para que un PIB constante pague la deuda, si todo el PIB se destina al pago de la deuda y esta permanece invariable.

Una aplicación sencilla del análisis dimensional a las matemáticas consiste en calcular la forma del volumen de una n -bola (la esfera sólida en n dimensiones) o el área de su superficie, la n -esfera : al ser una figura n -dimensional, el volumen escala como x ⁿ , mientras que el área de la superficie, al ser ( n -1) -dimensional, escala como x ^{n -1} . Así, el volumen de la n -bola en función del radio es C _n r ⁿ , para alguna constante C _n . Determinar la constante requiere matemáticas más complejas, pero la forma puede deducirse y comprobarse únicamente mediante el análisis dimensional.

En finanzas, economía y contabilidad, el análisis dimensional se suele entender como la distinción entre existencias y flujos . En términos más generales, el análisis dimensional se utiliza para interpretar diversos ratios financieros , económicos y contables.

• Por ejemplo, la relación precio/beneficio (P/E) tiene dimensiones de tiempo (unidad: año) y puede interpretarse como "años de ganancias para obtener el precio pagado".
• En economía, la relación deuda/PIB también se expresa en años (la deuda se mide en unidades monetarias, el PIB en unidades monetarias/año).
• La velocidad del dinero tiene una unidad de 1/año (PIB/oferta monetaria tiene una unidad de moneda/año sobre moneda): con qué frecuencia circula una unidad monetaria por año.
• Los tipos de interés anuales con capitalización continua y los tipos de interés simples suelen expresarse como un porcentaje (cantidad adimensional), mientras que el tiempo se expresa como una cantidad adimensional que consiste en el número de años. Sin embargo, si el tiempo incluye el año como unidad de medida, la dimensión del tipo es 1/año. Por supuesto, no hay nada especial (aparte de la convención habitual) en usar el año como unidad de tiempo: se puede usar cualquier otra unidad de tiempo. Además, si el tipo y el tiempo incluyen sus unidades de medida, el uso de unidades diferentes para cada uno no supone ningún problema. En cambio, si el tipo y el tiempo son adimensionales, deben referirse a un período común. (Cabe señalar que los tipos de interés efectivos solo pueden definirse como cantidades adimensionales).
• 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.

In fluid mechanics, 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.^[9] 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: $${\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}$$
• Froude number (Fr), modeling flow with a free surface: $${\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}$$
• Euler number (Eu), used in problems in which pressure is of interest: $${\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}$$
• Mach number (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound: $${\displaystyle \mathrm {Ma} ={\frac {u}{c}},}$$ where c is the local speed of sound.

What is the period of oscillationT of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g? That period is the solution for T of some dimensionless equation in the variables T, m, k, and g. The four quantities have the following dimensions: T (T); m (M); k (M/T²); and g (L/T²). From these we can form only one dimensionless product of powers of our chosen variables, G₁ = T²k/m(T² · M/T² / M = 1), and putting G₁ = C for some dimensionless constant 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. They are often called dimensionless numbers as well.

The variable 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 g with k, m, and T, because g is the only quantity that involves the dimension L. This implies that in this problem the 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 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: ⁠${\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}}$⁠, for some dimensionless constant κ (equal to ${\displaystyle {\sqrt {C}}}$ from the original dimensionless equation).

When faced with a case where dimensional analysis rejects a variable (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 κ.

Consider the case of a vibrating wire of lengthℓ (L) vibrating with an amplitudeA (L). The wire has a linear densityρ (M/L) and is under tensions (LM/T²), and we want to know the energy E (L²M/T²) in the wire. Let π₁ and π₂ be two dimensionless products of powers of the variables chosen, given by

${\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}}$

The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation

${\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}$

where F is some unknown function, or, equivalently as

${\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}$

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 E = ℓs. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

El poder del análisis dimensional se hace realmente evidente cuando se aplica a situaciones, distintas a las mencionadas anteriormente, que son más complejas, donde el conjunto de variables involucradas no es evidente y las ecuaciones subyacentes son sumamente complejas. Consideremos, por ejemplo, una pequeña piedra sobre el lecho de un río. Si el río fluye con la suficiente rapidez, la piedra se elevará y será arrastrada por la corriente. ¿A qué velocidad crítica ocurrirá esto? Determinar las variables supuestas no es tan sencillo como antes. Sin embargo, el análisis dimensional puede ser una herramienta poderosa para comprender problemas como este, y suele ser la primera que se aplica a problemas complejos donde las ecuaciones y restricciones subyacentes no se comprenden bien. En tales casos, la respuesta puede depender de un número adimensional , como el número de Reynolds , que puede interpretarse mediante el análisis dimensional.

Consideremos el caso de un disco delgado, sólido, de caras paralelas y giratorio, con un espesor axial t (L) y un radio ^R ( L). El disco tiene una densidad ρ (M/L³ ⁾ , gira a una velocidad angular ω (T⁻¹ ⁾ y esto produce una tensión S (T⁻²L⁻¹M ⁾ en el material. Existe una solución elástica lineal teórica, propuesta por Lamé, para este problema cuando el disco es delgado en relación con su radio, las caras del disco pueden moverse axialmente libremente y se pueden asumir las relaciones constitutivas de tensión plana. A medida que el disco se vuelve más grueso en relación con el radio, la solución de tensión plana deja de ser válida. Si el disco está restringido axialmente en sus caras libres, se producirá un estado de deformación plana. Sin embargo, si este no es el caso, el estado de tensión solo puede determinarse mediante la consideración de la elasticidad tridimensional y no existe una solución teórica conocida para este caso. Por lo tanto, un ingeniero podría estar interesado en establecer una relación entre las cinco variables. El análisis dimensional para este caso conduce a los siguientes grupos adimensionales ( 5 − 3 = 2 ):

demanda/capacidad = ρR ² ω ² / S

espesor/radio o relación de aspecto = t / R

Mediante experimentos numéricos, utilizando, por ejemplo, el método de elementos finitos , se puede obtener la naturaleza de la relación entre los dos grupos adimensionales, como se muestra en la figura. Dado que este problema solo involucra dos grupos adimensionales, se proporciona una imagen completa en un solo gráfico, que puede utilizarse como diagrama de diseño/evaluación para discos giratorios. ^{[ 22 ]}

Las dimensiones que se pueden formar a partir de un conjunto dado de dimensiones físicas básicas, como T, L y M, forman un grupo abeliano : la identidad se escribe como 1; L ⁰ = 1 , y el inverso de L es 1/L o L ⁻¹ . L elevado a cualquier potencia entera p es un miembro del grupo, teniendo un inverso de L ^{− p} o 1/L ^p . La operación del grupo es la multiplicación, con las reglas habituales para manejar exponentes ( L ⁿ × L ^m = L ^{n + m} ). Físicamente, 1/L puede interpretarse como longitud recíproca , y 1/T como tiempo recíproco (véase segundo recíproco ).

Un grupo abeliano es equivalente a un módulo sobre los enteros, donde el símbolo dimensional T ⁱ L ^j M ^k corresponde a la tupla ( i , j , k ) . Cuando cantidades físicas medidas (ya sean de la misma dimensión o de dimensiones diferentes) se multiplican o dividen entre sí, sus unidades dimensionales también se multiplican o dividen; esto corresponde a la suma o resta en el módulo. Cuando cantidades medibles se elevan a una potencia entera, se realiza la misma operación con los símbolos dimensionales asociados a dichas cantidades; esto corresponde a la multiplicación escalar en el módulo.

La base de un módulo de símbolos dimensionales se denomina conjunto de cantidades base , y todos los demás vectores se denominan unidades derivadas. Como en cualquier módulo, se pueden elegir diferentes bases , lo que da lugar a diferentes sistemas de unidades (por ejemplo, elegir si la unidad de carga se deriva de la unidad de corriente, o viceversa).

La identidad del grupo, la dimensión de cantidades adimensionales, corresponde al origen en este módulo, (0, 0, 0) .

En ciertos casos, se pueden definir dimensiones fraccionarias, específicamente definiendo formalmente potencias fraccionarias de espacios vectoriales unidimensionales, como V ^{L 1/2} . ^{[ 23 ]} Sin embargo, no es posible tomar potencias fraccionarias arbitrarias de unidades, debido a obstrucciones de la teoría de la representación . ^{[ 24 ]}

Se puede trabajar con espacios vectoriales de dimensiones dadas sin necesidad de usar unidades (que corresponden a los sistemas de coordenadas de los espacios vectoriales). Por ejemplo, dadas las dimensiones M y L , se tienen los espacios vectoriales V ^M y V ^L , y se puede definir V ^ML  := V ^M ⊗ V ^L como el producto tensorial . De manera similar, el espacio dual puede interpretarse como si tuviera dimensiones "negativas". ^{[ 25 ]} Esto corresponde al hecho de que, bajo el emparejamiento natural entre un espacio vectorial y su dual, las dimensiones se cancelan, dejando un escalar adimensional .

El conjunto de unidades de las magnitudes físicas involucradas en un problema corresponde a un conjunto de vectores (o una matriz). La nulidad describe un número (por ejemplo, m ) de maneras en que estos vectores pueden combinarse para producir un vector cero. Esto corresponde a producir (a partir de las mediciones) una serie de magnitudes adimensionales, {π ₁ , ..., π _m } . (De hecho, estas maneras abarcan completamente el subespacio nulo de otro espacio diferente, el de las potencias de las mediciones). Cada forma posible de multiplicar (y exponenciar ) las magnitudes medidas para producir algo con la misma unidad que alguna magnitud derivada X puede expresarse en la forma general

${\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.}$

En consecuencia, toda ecuación conmensurable posible para la física del sistema puede reescribirse en la forma

${\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.}$

Conocer esta restricción puede ser una herramienta poderosa para obtener nuevos conocimientos sobre el sistema.

The dimension of physical quantities of interest in mechanics 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. 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: they must span the space, and be linearly independent.

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/T²], L, M, while the latter can be expressed as [T = (LM/F)^1/2], L, M.

On the other hand, length, velocity and time (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 (V = L/T), is redundant (the set is not linearly independent).

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. In thermodynamics, 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, ≈ 6.02×10²³ mol⁻¹) is also defined as a base dimension, N. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, 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.

Bridgman's theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor.^[26][27] This excludes polynomials of more than one term or transcendental functions not of that form.

Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities. (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) = loga − logb, 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.^[28]

Similarly, while one can evaluate monomials (xⁿ) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x², the expression (3 m)² = 9 m² makes sense (as an area), while for x² + x, the expression (3 m)² + 3 m = 9 m² + 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,

${\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.}$

This is the height to which an object rises in time t if the acceleration of gravity is 9.8 metres per second per second and the initial upward speed is 500 metres per second. It is not necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be

${\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}}$

The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical value or numerical factor, n.^[29]

${\displaystyle Z=n\times [Z]=n[Z]}$

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 factor, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:

${\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} }$ is identical to ${\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.}$

The factor 0.3048 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.

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

A quantity equation, also sometimes called a complete equation, is an equation that remains valid independently of the unit of measurement used when expressing the physical quantities.^[30]

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 displacementd as speeds multiplied by time difference t would be:

d = st

for s = 5 m/s, where t and d may be expressed in any units, converted if necessary. In contrast, a corresponding numerical-value equation would be:

D = 5 T

where T is the numeric value of t when expressed in seconds and D is the numeric value of d when expressed in metres.

Generally, the use of numerical-value equations is discouraged.^[30]

The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the 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.

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, 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 ~ 1/χ^d, where d is the dimension of the lattice.

It has been argued by some physicists, e.g., Michael J. Duff,^[5][31] 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: c, ħ, and 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 ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c → ∞, ħ → 0 and G → 0. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.

Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors, but this is not the case. Numbers are often used to represent things that are not elements of vector spaces, and dimensional analysis should not be applied to such things. In physics, scalars are positive quantities like weight or distance with magnitude but not direction (though many authors loosely allow scalars to be negative), 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. 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 position from displacement, date from duration, bearing from angle, and absolute temperature from temperature change.

Consider points on a timeline, each with a date with respect to a given origin, and durations of between them. The dates themselves are labels of points, while the durations of time between points have a dimension of time:

• adding two durations should yield a new duration (waiting ten days then twenty days gets you thirty days forward),
• adding a duration to a date should yield a new date (Waiting one month from March 1st, 1900 take you to April 1st.),
• subtracting two dates should yield a duration,
• but one may not add two dates (Adding March 1st to April 1st does not make sense.)

This illustrates the subtle type-distinction between affine values (elements of an affine space, such as date) and vector values (elements of a vector space, such as duration).

• Vectors may be added to each other, yielding a new vector, and a vector may be added to a suitable affine quantity (a vector space acts on an affine space), yielding a new affine quantity.
• Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity.

Properly then dates and positions are not dimensional quantities but affine quantities. Durations and displacements are dimensional quantities. To represent dimensional quantities with numbers, one must only select a reference unit. To represent affine quantities like date and position in terms of dimensional quantity, one must also select a point of reference and coordinate system in addition to the reference unit.

While some quantities are well-represented by vector quantities to which dimensional analysis can be applied, others are more naturally represented in an affine space, and still others like bearings and probabilities are elements of other spaces.

This distinction is particularly important in the case of temperature, for which the numeric value of absolute zero is not the origin 0 in some scales. For absolute zero,

−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F,

where the symbol ≘ means corresponds to, since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated.

For temperature differences,

1 K = 1 °C ≠ 1 °F = 1 °R.

(Here °R refers to the Rankine scale, not the Réaumur scale). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K. But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature corresponding to the Celsius temperature −272.15 °C, or a temperature difference equal to 1 °C.

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. (In 1 dimension, this issue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in multi-dimensional Euclidean space, one also needs a bearing: they need to be compared to a frame of reference.

This leads to the extensions discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank ${\displaystyle m}$ of the dimensional matrix.^[54]

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_x 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.

Angles are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested).^[55] As an example, consider again the projectile problem in which a point mass is launched from the origin (x, y) = (0, 0) at a speed v and angle θ above the x-axis, with the force of gravity directed along the negative y-axis. It is desired to find the range R, at which point the mass returns to the x-axis. Conventional analysis will yield the dimensionless variable π = Rg/v², but offers no insight into the relationship between R and θ.

Siano has suggested that the directed dimensions of Huntley be replaced by using orientational symbols1_x 1_y 1_z to denote vector directions, and an orientationless symbol 1₀.^[56] Thus, Huntley's L_x becomes L1_x with L specifying the dimension of length, and 1_x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1_i⁻¹ = 1_i, the following multiplication table for the orientation symbols results:

The 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 1_z. For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1_x and the side opposite has an orientation 1_y. Since (using ~ to indicate orientational equivalence) tan(θ) = θ + ... ~ 1_y/1_x we conclude that an angle in the xy-plane must have an orientation 1_y/1_x = 1_z, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1_z while cos(θ) has orientation 1₀. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ), where a and b are real scalars. An expression such as ${\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}$ is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:

${\displaystyle \sin \left(a\,1_{\text{z}}+b\,1_{\text{z}}\right)=\sin \left(a\,1_{\text{z}})\cos(b\,1_{\text{z}}\right)+\sin \left(b\,1_{\text{z}})\cos(a\,1_{\text{z}}\right),}$

which for ${\displaystyle a=\theta }$ and ${\displaystyle b=\pi /2}$ yields ⁠${\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})}$⁠. 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 ⁠${\displaystyle 1_{0}}$⁠.

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 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, θ, being in the xy-plane will thus have dimension 1_z and the range of the projectile R will be of the form:

${\displaystyle R=g^{a}\,v^{b}\,\theta ^{c}{\text{ which means }}{\mathsf {L}}\,1_{\mathrm {x} }\sim \left({\frac {{\mathsf {L}}\,1_{\text{y}}}{{\mathsf {T}}^{2}}}\right)^{a}\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{b}\,1_{\mathsf {z}}^{c}.\,}$

Dimensional homogeneity will now correctly yield a = −1 and b = 2, and orientational homogeneity requires that ⁠${\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1}$⁠. In other words, that c must be an odd integer. In fact, the required function of theta will be sin(θ)cos(θ) which is a series consisting of odd powers of θ.

It is seen that the Taylor series of sin(θ) and cos(θ) are orientationally homogeneous using the above multiplication table, while expressions like cos(θ) + sin(θ) and exp(θ) are not, and are (correctly) deemed unphysical.

Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the radian may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis.

• Covariance and contravariance of vectors
• Exterior algebra
• Geometric algebra
• Quantity calculus