Dimensional analysis

3893:(TLM) 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 ( 6998:). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). 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 equal to −272.15 °C, or the temperature difference equal to 1 °C. 13677: 7850:, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? 3839:

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

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that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations. Mathematica can also factor out certain dimension with UnitDimensions by specifying an argument to the function UnityDimensions. For example, you can use UnityDimensions to factor out angles. In addition to UnitDimensions, Mathematica can find the dimensions of a QuantityVariable with the function QuantityVariableDimensions.

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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.)

1839: 4015:. 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 4714: 9990: 7782: 5410:, 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: 1637: 1657: 1098: 4422:, 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. 1393: 1243: 3489:, 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. 3228:, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time. This was slightly changed by Maxwell, who said the dimensions of acceleration are TL, instead of just the exponents. 4983:{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}} 943: 9808: 9561: 6871:. 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). 7589: 9782:. 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. 434: 812: 2404:

are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have

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Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the

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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

2171:—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with 4430:

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

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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

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must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same

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Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given

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Mathematica 13.2 has a function for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation. Mathematica also has a function to find the dimensions of a unit such as 1 J named UnitDimensions. Mathematica also has a function

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played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass

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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

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Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors; vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to

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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

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The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a

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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

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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

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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

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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

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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

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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

3933:, 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. 8205:, 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: 5383:

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,

1490: 1834:{\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}}}.} 690: 9189:

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

7402: 963: 1263: 1118: 6979:, 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. 3663: 10666:

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

9985:{\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}.\,} 10812: 832: 9381: 7777:{\displaystyle {\mathsf {L}}_{\mathrm {x} }=\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {x} }}\right)^{a}\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{b}\left({{\mathsf {T}}^{-2}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{c}} 4669: 1470: 7010:. (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 5390:) 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 4385:

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

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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

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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

312: 10800: 9743:. 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 5437:

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

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Klinkenberg, A. (1955), "Dimensional systems and systems of units in physics with special reference to chemical engineering: Part I. The principles according to which dimensional systems and systems of units are constructed",

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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.

3819:. 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 9778:

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

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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

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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

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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

1632:{\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}.} 7060:

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

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are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested ) . As an example, consider again the projectile problem in which a point mass is launched from the origin

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is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed. For example, units for

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There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, although this does not invalidate the usefulness of dimensional analysis.

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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:

6427: 4339:. 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 6467: 5789: 10107: 4354:

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 , L, M, while the latter can be expressed as , L, M.

2963: 2907: 11485: 8330:, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable. 7292: 4223: 1093:{\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}}.} 4317: 5126: 3801: 1388:{\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}}.} 243:

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.

6091: 1238:{\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}}.} 88:

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.

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quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h.

9373: 7881:. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables: 582:; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity. 3446: 10821:, "However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units ...". 6960:

Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis.

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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

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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

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dimensional relationships. In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include:

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the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.

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pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

6191: 3577: 2371: 12719: 6702: 938:{\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.} 9556:{\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),} 8634: 8512: 7555: 7520: 4431:

functions are grouped together in the dimensionless numeric multiplying factor. This excludes polynomials of more than one term or transcendental functions not of that form.

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Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since

10809:, "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 ...". 4057:

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

2397:: 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). 6356: 5931: 4557: 4461:. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.) 2588:

A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables. For example,

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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

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This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the

6285: 5858: 3478: 7924: 3406:: 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: 9631: 9590: 12354:

Maximum entropy and Bayesian methods: proceedings of the Eleventh International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Seattle, 1991

9624: 7486: 5371:" 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. 4120:) 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, 7862:

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 (

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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

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Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the

429:{\displaystyle \operatorname {dim} Q={\mathsf {T}}^{a}{\mathsf {L}}^{b}{\mathsf {M}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}} 10782: 11267: 6568: 5544: 5379:

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

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Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well.

807:{\displaystyle \operatorname {dim} a={\frac {\text{speed}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.} 7047: 6916:

Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space

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As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis.

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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.

72:(such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to 5224:, 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. 11448: 7490:, 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. 6882:

adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),

4134:. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and 4551:

However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example,

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is a member of the group, having an inverse of L or 1/L. The operation of the group is multiplication, having the usual rules for handling exponents (

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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:

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has been studied since 1977. Implementations for Ada and C++ were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in

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In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like

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The origins of dimensional analysis have been disputed by historians. The first written application of dimensional analysis has been credited to

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specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that

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Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although

685:{\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.} 12714: 12428: 8211: 4524:) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for 4361:

There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not

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by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39). In the second edition of 1833, Poisson explicitly introduces the term

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In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example,

13406: 11296: 10208: 9284:. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form 13764: 10270: 7218: 5434:, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. 5131: 3674: 219:

can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally

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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.

5456:(but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit 10231: 10192: 3159: 3044: 2733:. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone. 2508:

denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression

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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

1878: 197:. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. 7397:{\displaystyle {\mathsf {L}}=\left({\mathsf {T}}^{-1}{\mathsf {L}}\right)^{a+b}\left({\mathsf {T}}^{-2}{\mathsf {L}}\right)^{c}} 6433: 2625:" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 574:

dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have

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Petty, G. W. (2001), "Automated computation and consistency checking of physical dimensions and units in scientific programs",

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Siano, Donald (1985), "Orientational Analysis, Tensor Analysis and The Group Properties of the SI Supplementary Units – II",

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adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),

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Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.

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associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a

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does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines

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Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

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On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:

4238: 4099:. Similarly, the dual space can be interpreted as having "negative" dimensions. This corresponds to the fact that under the 3668:

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

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In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a

123:. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on 9375:

is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written:

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Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator.

2218:, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s). The newton is defined as 13281: 13068: 8018: 481:

are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a

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Kennedy, A. (2010). "Types for Units-of-Measure: Theory and Practice". In Horváth, Z.; Plasmeijer, R.; Zsók, V. (eds.).

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There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:

11244: 3411: 2617:. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 212:

of nature. This may give insight into the fundamental properties of the system, as illustrated in the examples below.

13205: 11813: 11471: 10965: 10486: 4415: 3952:

The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an

11502: 4110:

The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The

13399: 13175: 13139: 13048: 12786: 12724: 7135:, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then 5640: 2141: 17: 12395:

Giancoli, Douglas C. (2014). "1. Introduction, Measurement, Estimating §1.8 Dimensions and Dimensional Analysis".

7874:

to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.

2207:, however, can be factored into the base units of length (m), thus they are considered derived or compound units. 13774: 13261: 13246: 13200: 13078: 13053: 13038: 12998: 12922: 12505: 12451: 6132: 5477:. In problems involving a gravitational field the latter limit should be taken such that the field stays finite. 11605: 13256: 13241: 13231: 13221: 13159: 13129: 13099: 13063: 12988: 12907: 12882: 12877: 12872: 12865: 12569: 12352:

Vignaux, GA (1992), "Dimensional Analysis in Data Modelling", in Erickson, Gary J.; Neudorfer, Paul O. (eds.),

12294:

Van Driest, E. R. (March 1946), "On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems",

6836: 6814: 12117:

Perry, J. H.; et al. (1944), "Standard System of Nomenclature for Chemical Engineering Unit Operations",

10741: 3658:{\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}} 3098: 13769: 13754: 13251: 13226: 13190: 13154: 13134: 13124: 13109: 13104: 13094: 13003: 12927: 12828: 12559: 12549: 11552: 10442:

JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM)

9205:

being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation

9042: 8898: 8754: 8578: 8548: 8518: 5945: 5494: 4394:, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the 3213:, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book 251: 10384:

Duff, M.J.; Okun, L.B.; Veneziano, G. (September 2002), "Trialogue on the number of fundamental constants",

13732: 13691: 13316: 13236: 13195: 13185: 13114: 13018: 12957: 12952: 12912: 12902: 12892: 12887: 12791: 12693: 12688: 12607: 12602: 12564: 12046:

Mendez, P.F.; Ordóñez, F. (September 2005), "Scaling Laws From Statistical Data and Dimensional Analysis",

10474: 6152: 2342: 12651: 10765: 2757:

has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid".

13392: 13144: 13119: 13058: 13008: 12993: 12983: 12947: 12838: 12801: 12770: 12641: 8314:

by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield

6826: 4511:

are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.

4111: 4037:

The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module,

2741:

In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the

11423: 6670: 5186:

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

2560:

of physical equations: the two sides of any equation must be commensurable or have the same dimensions.

2376: 13180: 13028: 12962: 12860: 12806: 12796: 12338: 6918: 3193:

is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were

2589: 8610: 8488: 7531: 7496: 6963:

This distinction is particularly important in the case of temperature, for which the numeric value of

5084:, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: 4664:{\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.} 1465:{\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.} 13589: 13073: 12612: 12443: 11580: 10449: 6578: 6470: 3835:. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. 11527: 11210: 4138:) together the measured quantities to produce something with the same unit as some derived quantity 3041:), important in high speed flows where the velocity approaches or exceeds the local speed of sound: 12479: 12068: 10203: 6319: 5898: 5407: 5195: 5183:

convert the quantities to the same unit so that their numerical values can be added or subtracted.

4454: 112: 9734:{\displaystyle \sin(\theta \,1_{\text{z}}+\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})} 9128: 9100: 9072: 9012: 8956: 8928: 8868: 8840: 8784: 8724: 8696: 8668: 8181: 8154: 7938: 7175: 7144: 7109: 7080: 6737: 6613: 5573: 5026: 13701: 12636: 12587: 12438: 11788: 6258: 5831: 5228: 4438: 3459: 2191: 2066: 1866: 11285: 10345: 10187: 7900: 4999: 3947: 3113: 2844: 166: 13725: 12063: 11205: 10534: 10511: 9569: 7808:. The increase in deductive power gained by the use of directed length dimensions is apparent. 6201: 5427: 5349: 4446: 4340: 4135: 4027: 3163: 3140: 2967: 2172: 2081: 2035: 1870: 485: 11924: 11245:"A typechecker plugin for units of measure: domain-specific constraint solving in GHC Haskell" 10751: 10649: 9595: 7450: 131:. It also serves as a guide and constraint in deriving equations that may describe a physical 100:

and have different dimensions, and can not be directly compared to each other, no matter what

13706: 12528: 11359:

Proceedings of the 11th ACM SIGPLAN International Conference on Software Language Engineering

10219: 10150:

are orientationally homogeneous using the above multiplication table, while expressions like

7410: 6487: 6378: 4051: 4026:, and all other vectors are called derived units. As in any module, one may choose different 4016: 3930: 2824:

is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because

2481:

only quantities of the same dimension can be added, subtracted, or compared. For example, if

2195: 2167:

Many parameters and measurements in the physical sciences and engineering are expressed as a

201: 12439:

A C++ implementation of compile-time dimensional analysis in the Boost open-source libraries

11892: 10526: 8289: 5987: 2653:

A simple application of dimensional analysis to mathematics is in computing the form of the

13759: 12334: 12228:

Siano, Donald (1985), "Orientational Analysis – A Supplement to Dimensional Analysis – I",

12207: 12055: 12006: 11825: 11754: 10403: 9748: 9156: 8984: 8812: 8640: 6950: 6868: 5368: 4458: 4442: 4348: 4322:

Knowing this restriction can be a powerful tool for obtaining new insight into the system.

3986: 3840: 3359: 3314:. From these we can form only one dimensionless product of powers of our chosen variables, 2152: 1889: 1874: 561: 482: 278: 236: 205: 194: 69: 12268:

Silberberg, I. H.; McKetta, J. J. Jr. (1953), "Learning How to Use Dimensional Analysis",

11176: 7021:

discussed below, namely Huntley's directed dimensions and Siano's orientational analysis.

4031: 3389:

is the only quantity that involves the dimension L. This implies that in this problem the

3139:

made the first credited important contributions based on the idea that physical laws like

2210:

Sometimes the names of units obscure the fact that they are derived units. For example, a

76:

from one dimensional unit to another, which can be used to evaluate scientific formulae.

8: 13616: 13301: 12821: 12752: 12139: 12132: 8323: 8315: 7878: 5955: 5886: 5289: 5210: 5081: 5011: 4411: 4395: 3568: 3355: 3242: 3154: 2184: 1944: 575: 282: 224: 101: 73: 12211: 12059: 12010: 11829: 11758: 11357:

Bennich-Björkman, O.; McKeever, S. (2018). "The next 700 unit of measurement checkers".

10415: 10407: 7074: 7073:

As an example of the usefulness of the first approach, suppose we wish to calculate the

6550: 5526: 2654: 247:, being based on only universal constants, may be thought of as being "less arbitrary". 13641: 12816: 12592: 12324: 12173: 11704: 11656: 11477: 11380: 11157: 11122: 11004: 10982: 10981:

Duff, Michael James (July 2004). "Comment on time-variation of fundamental constants".

10891: 10504: 10419: 10393: 7032: 7011: 6806: 6025: 2414: 1893: 286: 11855: 7811:

Huntley's concept of directed length dimensions however has some serious limitations:

2764:

also has the unit year (debt has a unit of currency, GDP has a unit of currency/year).

12467: 12410: 12400: 12375: 12357: 12261: 12241: 12177: 12143: 12033: 12025: 12018: 11976: 11956: 11932: 11907: 11796: 11738: 11712: 11481: 11467: 11370: 11223: 11182: 11091: 11059: 11035: 11011: 10961: 10655: 10629: 10601: 10571: 10538: 10527: 10482: 10361: 10317: 10246: 10241: 5809: 5214: 4399: 3979: 3975: 3535: 3121: 2767: 2761: 2609: 2602: 2428:

Only commensurable quantities (physical quantities having the same dimension) may be

2401: 2077: 216: 209: 45: 11745:

Bhaskar, R.; Nigam, Anil (1991), "Qualitative Explanations of Red Giant Formation",

11458:. Advances in Mathematics for Applied Sciences. World Scientific. pp. 331–345. 11384: 11161: 11126: 10895: 10423: 3863:

Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness

3395:

is irrelevant. Dimensional analysis can sometimes yield strong statements about the

13554: 12848: 12346: 12316: 12257: 12237: 12215: 12165: 12093: 12073: 12014: 11950: 11870: 11841: 11833: 11809: 11762: 11734: 11725:

Bhaskar, R.; Nigam, Anil (1990), "Qualitative Physics Using Dimensional Analysis",

11666: 11459: 11419: 11410: 11362: 11293:

28ièmes Journées Francophones des Langaeges Applicatifs, Jan 2017, Gourette, France

11259: 11215: 11149: 11114: 11087: 10883: 10699: 10597: 10567: 10411: 10353: 10308: 10283: 10236: 9186: 6222: 5819: 5799: 5615: 4434: 4050:. However, it is not possible to take arbitrary fractional powers of units, due to 3957: 3184: 3180: 2746: 2420: 2394: 2111: 489: 274: 190: 97: 85: 65: 8339: 6995: 4034:

whether the unit for charge is derived from the unit for current, or vice versa).

2621:. Since any quantity can be multiplied by 1 without changing it, the expression " 13013: 12455: 12185: 11984: 11814:"On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis" 11204:. Lecture Notes in Computer Science. Vol. 6299. Springer. pp. 268–305. 10732: 8422:

Siano has suggested that the directed dimensions of Huntley be replaced by using

6015: 5975: 5631: 5419: 4419: 4387: 4336: 4100: 3844: 2852: 2840: 2425:

The most basic rule of dimensional analysis is that of dimensional homogeneity.

2168: 2145: 1399: 1249: 818: 493: 220: 12488: 11219: 10704: 10357: 13363: 12937: 12554: 11920: 11670: 11463: 10868: 10268:

Bolster, Diogo; Hershberger, Robert E.; Donnelly, Russell E. (September 2011).

9779: 6840: 6724: 6538: 6111: 6101: 5411: 4391: 4096: 3526: 3151:

should be independent of the units employed to measure the physical variables.

3136: 3117: 2742: 2176: 147: 12461:

Units, quantities, and fundamental constants project dimensional analysis maps

12307:

Whitney, H. (1968), "The Mathematics of Physical Quantities, Parts I and II",

11846: 11644: 10954:

Guide for the Use of the International System of Units (SI): The Metric System

10887: 10558:

Macagno, Enzo O. (1971). "Historico-critical review of dimensional analysis".

5292:

if necessary. In contrast, a corresponding numerical-value equation would be:

5014:

within the dimension and a dimensionless numerical value or numerical factor,

3850: 3201:

equation for mass, results in charge having the same dimensions as mass, viz.

13748: 12811: 12683: 12623: 12449:

Quantity System calculator for units conversion based on dimensional approach

12414: 11980: 11444: 11186: 10952: 10197: 9257:, which is not unreasonable. Analogous reasoning forces the conclusion that 8141:{\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}} 7817: 6991: 6964: 4104: 4023: 3953: 3210: 2911: 2778: 2448: 2211: 244: 204:, which begins with dimensional analysis, and involves scaling quantities by 200:

A dimensional equation can have the dimensions reduced or eliminated through

49: 11943: 11366: 11263: 7208:

With these four quantities, we may conclude that the equation for the range

5354:

The dimensionless constants that arise in the results obtained, such as the

4483:, where the logarithm is taken in any base, holds for dimensionless numbers 13415: 13368: 12656: 12646: 12631: 11988: 11970: 11952:

Multidimensional Analysis: Algebras and Systems for Science and Engineering

11118: 10923: 6909: 6901: 6861: 6122: 2557: 2297: 2180: 696: 124: 12434:

Unicalc Live web calculator doing units conversion by dimensional analysis

11875: 11837: 11456:

Advanced Mathematical and Computational Tools in Metrology and Testing XII

11140:

Cmelik, R. F.; Gehani, N. H. (May 1988). "Dimensional analysis with C++".

8478:, the following multiplication table for the orientation symbols results: 6839:

to support Hart's matrices. McBride and Nordvall-Forsberg show how to use

3855: 119:

have the same dimensions on its left and right sides, a property known as

13631: 12967: 12698: 12597: 12497: 12475: 12282: 11329: 10987: 10924:"Square bracket notation for dimensions and units: usage and conventions" 8327: 6810: 5380: 4344: 3034: 1862: 1643: 128: 33: 11412:

A unit-aware matrix language and its application in control and auditing

10398: 10115:

must be an odd integer. In fact, the required function of theta will be

9785:

As an example, for the projectile problem, using orientational symbols,

12328: 11314: 10213: 8272:{\displaystyle C={\frac {p_{\mathrm {x} }\rho r^{4}}{\eta {\dot {m}}}}} 2745:. More generally, dimensional analysis is used in interpreting various 533: 297: 294: 12097: 12077: 10287: 4103:

between a vector space and its dual, the dimensions cancel, leaving a

3483:

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

13572: 13384: 12448: 12433: 12220: 10588:

Martins, Roberto De A. (1981). "The origin of dimensional analysis".

9201:

that lies in the z-plane. Form a right triangle in the z-plane with

4515: 4450: 4331: 3517: 3502: 2754: 2148: 1858: 12460: 12320: 12169: 11153: 11105:

Gehani, N. (June 1985). "Ada's derived types and units of measure".

10783:"Dimensional Analysis and Numerical Experiments for a Rotating Disc" 4232:

equation for the physics of the system can be rewritten in the form

4022:

A basis for such a module of dimensional symbols is called a set of

3236: 13544: 11767: 11661: 10614:

Martins, p. 403 in the Proceedings book containing his article

10316:(in English and French) (v. 1.08, 9th ed.). pp. 136–137. 9242:

we conclude that an angle in the xy-plane must have an orientation

8390:-axis. Conventional analysis will yield the dimensionless variable 7276:{\displaystyle R\propto v_{\text{x}}^{a}\,v_{\text{y}}^{b}\,g^{c}.} 6925:

Affine quantities cannot be added, but may be subtracted, yielding

6867:

define a position, this assumes an implicit point of reference: an

5725: 5715: 5564: 5172:{\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.} 3731:{\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,} 3102: 2668: 2261: 949: 512: 239:

used to express the amount of that physical quantity. For example,

158:"Dimension (physics)" redirects here. For physical dimensions, see 108: 93: 27:

Analysis of the relationships between different physical quantities

11929:

Proceedings of the Fifth SIAM Conference on Applied Linear Algebra

3964:, and the inverse of L is 1/L or L. L raised to any integer power 3859:

Dimensional analysis and numerical experiments for a rotating disc

3813:

to experiments to discover the form for the unknown function

2264:) has dimension TL—length from position, time due to the gradient; 13661: 13436: 13373: 12372:

Dimensional Analysis in the Identification of Mathematical Models

12194: 6949:

unit, one must not only choose a unit of measurement, but also a

6830: 6818: 6518: 6029: 4681: 3929:

Through the use of numerical experiments using, for example, the

1854: 1476: 250:

There are many possible choices of base physical dimensions. The

37: 5367:

order unity. This observation can allow one to sometimes make "

5335:

Generally, the use of numerical-value equations is discouraged.

4410:) is also defined as a base dimension, N. In the interaction of 13606: 13579: 13495: 13485: 13043: 11645:"Angles in the SI: a detailed proposal for solving the problem" 11503:"NondimensionalizationTransform—Wolfram Language Documentation" 10174: 6528: 6143: 3508: 2564: 2204: 2200: 2187:), powers (like m for square metres), or combinations thereof. 1104: 266: 132: 53: 4708: = 0.01 minutes. Then the first term would be 4418:, connected with the symmetry properties of the collisionless 13519: 11343: 8376:-axis, with the force of gravity directed along the negative 8344: 6822: 5554: 5237: 3106: 3025:{\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.} 2593:

height in feet, then they must be the same height in metres.

2408: 2315: 2215: 595: 169:

describes how every physically meaningful equation involving

12119:

Transactions of the American Institute of Chemical Engineers

12086:

Transactions of the American Society of Mechanical Engineers

11053: 8440:

to denote vector directions, and an orientationless symbol 1

8333: 2843:, dimensional analysis is performed to obtain dimensionless 13559: 13529: 13470: 11606:"QuantityVariableDimensions—Wolfram Language Documentation" 11078:

Gehani, N. (1977). "Units of measure as a data attribute".

10479:

Essential of Fluid Mechanics: Fundamentals and Applications

10304: 6422:{\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}} 5965: 5941: 5705: 4425: 3851:

A third example: demand versus capacity for a rotating disc

2818:

is a derivative. From the previous point, the dimension of

2613:. For example, kPa and bar are both units of pressure, and 2144:

the values of exponents in the main equation, and form the

270: 262: 159: 61: 57: 12370:

Kasprzak, Wacław; Lysik, Bertold; Rybaczuk, Marek (1990),

11032:

Physics for Scientists and Engineers – with Modern Physics

10267: 7815:

It does not deal well with vector equations involving the

6462:{\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}} 3158:

can be derived from length and time by assuming a form of

175:

variables can be equivalently rewritten as an equation of

11202:

Central European Functional Programming School. CEFP 2009

7205:

the downward acceleration of gravity, with dimension TL.

6971:−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F, 6933:

may then be added to each other or to an affine quantity.

5784:{\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I} 5209:, is an equation that remains valid independently of the 4674:

This is the height to which an object rises in time

4368:

Velocity, being expressible in terms of length and time (

2225: 566:. A quantity that has all exponents null is said to have 11553:"DimensionalCombinations—Wolfram Language Documentation" 11443: 11356: 10102:{\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1} 4380: 3254:

attached to an ideal linear spring with spring constant

2958:{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.} 2902:{\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.} 2577:, they are fundamentally different physical quantities. 300:

typeface. Mathematically, the dimension of the quantity

12084:

Moody, L. F. (1944), "Friction Factors for Pipe Flow",

10908: 10866: 8151:

If we distinguish between inertial mass with dimension

6967:

is not the origin 0 in some scales. For absolute zero,

4993: 4218:{\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.} 2859:), generally important in all types of fluid problems: 2462:

quantities (quantities with different dimensions), and

2065:

Express each of the quantities in the equation in some

488:– for instance, one could replace the dimension (I) of 44:

is the analysis of the relationships between different

12287:"A mathematical formalisation of dimensional analysis" 11709:

Scaling, Self-Similarity, and Intermediate Asymptotics

10920:

For a review of the different conventions in use see:

10867:

Berberan-Santos, Mário N.; Pogliani, Lionello (1999).

8407:, but offers no insight into the relationship between 6885:

subtracting two positions should yield a displacement,

6850: 4911: 4818: 4724: 4562: 4312:{\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.} 3426: 3296:. The four quantities have the following dimensions: 2974:), used in problems in which pressure is of interest: 2346: 2203:

and time are normally chosen as base units. Units for

2114:

these equations to obtain the values of the exponents

1905:

is a variable that depends upon independent variables

12429:

List of dimensions for variety of physical quantities

12369: 11344:"CamFort: Specify, verify, and refactor Fortran code" 10017: 9811: 9751: 9634: 9598: 9572: 9384: 9317: 9159: 9131: 9103: 9075: 9045: 9015: 8987: 8959: 8931: 8901: 8871: 8843: 8815: 8787: 8757: 8727: 8699: 8671: 8643: 8613: 8581: 8551: 8521: 8491: 8292: 8214: 8184: 8157: 8074: 8021: 7941: 7903: 7592: 7534: 7499: 7453: 7413: 7295: 7221: 7178: 7147: 7112: 7083: 7035: 7001: 6740: 6673: 6616: 6553: 6490: 6436: 6393: 6322: 6261: 6155: 6042: 5990: 5901: 5834: 5737: 5643: 5576: 5529: 5134: 5121:{\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} } 5093: 5029: 4717: 4560: 4241: 4153: 3796:{\displaystyle E=Asf\left({\frac {\ell }{A}}\right),} 3756: 3677: 3580: 3462: 3414: 3172: 3047: 2980: 2924: 2865: 2736: 2556:

is fine. Thus, dimensional analysis may be used as a

2345: 1660: 1493: 1416: 1266: 1121: 966: 835: 713: 612: 315: 231:

of a physical quantity is more fundamental than some

11346:. University of Cambridge; University of Kent. 2018. 7866:), and mass as a measure of the quantity of matter. 7493:

However, if we use directed length dimensions, then

6086:{\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu } 4330:

The dimension of physical quantities of interest in

3847:, which may be interpreted by dimensional analysis. 2331:

the integral of force with respect to the distance (

594:

As examples, the dimension of the physical quantity

11054:Martin, B.R.; Shaw, G.; Manchester Physics (2008), 10869:"Two alternative derivations of Bridgman's theorem" 10310:

SI Brochure: The International System of Units (SI)

8060:{\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}} 7832:

It also is often quite difficult to assign the L, L

254:selects the following dimensions and corresponding 13599: 12131: 11003: 10651:The Mathematics of Measurement: A Critical History 10503: 10269: 10101: 9984: 9764: 9733: 9618: 9584: 9555: 9368:{\displaystyle \sin(\theta +\pi /2)=\cos(\theta )} 9367: 9172: 9144: 9116: 9088: 9060: 9028: 9000: 8972: 8944: 8916: 8884: 8856: 8828: 8800: 8772: 8740: 8712: 8684: 8656: 8628: 8596: 8566: 8536: 8506: 8306: 8286:is an undetermined constant (found to be equal to 8271: 8197: 8170: 8140: 8059: 7954: 7918: 7776: 7549: 7514: 7480: 7437: 7396: 7275: 7191: 7160: 7125: 7096: 7041: 6761: 6696: 6639: 6562: 6502: 6461: 6421: 6350: 6279: 6185: 6085: 5999: 5925: 5852: 5783: 5689: 5599: 5538: 5171: 5120: 5068: 4982: 4663: 4311: 4217: 3795: 3730: 3657: 3503:A more complex example: energy of a vibrating wire 3472: 3440: 3074: 3024: 2957: 2901: 2365: 2175:, e.g. 60 km/h. Other relations can involve 2162: 1833: 1631: 1464: 1387: 1237: 1092: 937: 806: 684: 428: 12267: 11449:"Type systems for programs respecting dimensions" 10767:A Treatise on Electricity and Magnetism, volume 1 10383: 9995:Dimensional homogeneity will now correctly yield 9791:, being in the xy-plane will thus have dimension 6922:an affine space), yielding a new affine quantity. 4030:, which yields different systems of units (e.g., 3237:A simple example: period of a harmonic oscillator 2808:is the continuously compounded interest rate and 2477:The rule implies that in a physically meaningful 293:The symbols are by convention usually written in 146:, and of dimensional analysis, was introduced by 13746: 11775:Boucher; Alves (1960), "Dimensionless Numbers", 11581:"UnityDimensions—Wolfram Language Documentation" 11315:"Units of Measure in Rust with Refinement Types" 10170:are not, and are (correctly) deemed unphysical. 6896:This illustrates the subtle distinction between 3441:{\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}} 3272:of some dimensionless equation in the variables 2524:is meaningful, but the heterogeneous expression 11528:"UnitDimensions—Wolfram Language Documentation" 11181:(Phd). Vol. 391. University of Cambridge. 10583: 10581: 10299: 10297: 10271:"Dynamic similarity, the dimensionless science" 10225: 8456:with L specifying the dimension of length, and 3989:over the integers, with the dimensional symbol 12106:Bulletin of the Virginia Polytechnic Institute 12104:Murphy, N. F. (1949), "Dimensional Analysis", 11034:(6th ed.), San Francisco: W. H. Freeman, 10698:, Clarendon Press series, Oxford, p. 45, 10127:which is a series consisting of odd powers of 10009:, and orientational homogeneity requires that 7077:when fired with a vertical velocity component 6941:length, while displacements have dimension of 3747:is some unknown function, or, equivalently as 510:(with all other exponents zero) is known as a 135:in the absence of a more rigorous derivation. 13400: 12513: 12045: 11711:, Cambridge, UK: Cambridge University Press, 10472: 6986:1 K = 1 °C ≠ 1 °F = 1 °R. 6843:to extend type systems for units of measure. 5690:{\displaystyle mv^{2}\equiv pv\equiv p^{2}/m} 5406:It has been argued by some physicists, e.g., 5004:The value of a dimensional physical quantity 3075:{\displaystyle \mathrm {Ma} ={\frac {u}{c}},} 2667:dimensions), or the area of its surface, the 2250:derivative of position with respect to time ( 11856:"On the foundations of dimensional analysis" 11774: 11744: 11724: 11424:11245.1/fd7be191-700f-4468-a329-4c8ecd9007ba 11139: 10731:Rayleigh, Baron John William Strutt (1877), 10578: 10501: 10294: 9185:The orientational symbols form a group (the 7139:, the distance travelled, with dimension L, 3135:In 1822, the important Napoleonic scientist 2073: 11995: 11853: 11029: 6937:Properly then, positions have dimension of 6835:Griffioen's 2019 thesis extended Kennedy's 5480: 3480:from the original dimensionless equation). 2749:, economics ratios, and accounting ratios. 2409:Dimensional homogeneity (commensurability) 13407: 13393: 12527: 12520: 12506: 12486: 12293: 11808: 11703: 11283: 11006:The Cambridge Handbook of Physics Formulas 10882:: 255–261, See §5 General Results p. 259. 10502:de Jong, Frits J.; Quade, Wilhelm (1967). 10307:(2019). "2.3.3 Dimensions of quantities". 6957:unit only requires a unit of measurement. 4414:with strong laser pulses, a dimensionless 3941: 2830:is in the "denominator" of the derivative. 2682:-dimensional figure, the volume scales as 12219: 12067: 12030:Dimensional Analysis and Theory of Models 11874: 11845: 11766: 11660: 11408: 11209: 10986: 10914: 10703: 10524: 10397: 10263: 10261: 10209:Rayleigh's method of dimensional analysis 9981: 9960: 9893: 9862: 9839: 9828: 9717: 9678: 9647: 9534: 9508: 9475: 9449: 9416: 9399: 9212:and the side opposite has an orientation 8386:, at which point the mass returns to the 8334:Siano's extension: orientational analysis 7870:is defined by Huntley as a quantity only 7259: 7243: 5338: 5156: 5147: 5113: 5098: 4305: 4211: 3974:). Physically, 1/L can be interpreted as 3507:Consider the case of a vibrating wire of 2945: 2883: 1884:The method involves the following steps: 12394: 12247: 12024: 11901: 11787: 11688: 11047: 11030:Mosca, Gene; Tipler, Paul Allen (2007), 10950: 10854: 10757: 10730: 10687: 10672: 10628:, New York: Collier Books, p. 169, 10346:"Principles of the Theory of Dimensions" 10232:Covariance and contravariance of vectors 10216:– an application of dimensional analysis 10193:Dimensionless numbers in fluid mechanics 9311:are real scalars. An expression such as 7877:For example, consider the derivation of 6929:quantities which are vectors, and these 6800: 5360:in the Poiseuille's Law problem and the 4426:Polynomials and transcendental functions 4398:(the number of molecules divided by the 3854: 2393:In economics, one distinguishes between 492:of the SI basis with a dimension (Q) of 12490:An introduction to dimensional analysis 12351: 12306: 12227: 12198:(1915), "The Principle of Similitude", 11968: 11890: 11684: 11642: 11630: 11199: 11174: 10946: 10944: 10921: 10763: 10747: 10696:A Treatise on Electricity and Magnetism 10693: 10681:A Treatise on Electricity and Magnetism 10678: 10641: 10587: 10557: 9223:to indicate orientational equivalence) 9061:{\displaystyle \mathbf {1_{\text{z}}} } 8917:{\displaystyle \mathbf {1_{\text{y}}} } 8773:{\displaystyle \mathbf {1_{\text{x}}} } 8597:{\displaystyle \mathbf {1_{\text{z}}} } 8567:{\displaystyle \mathbf {1_{\text{y}}} } 8537:{\displaystyle \mathbf {1_{\text{x}}} } 8380:-axis. It is desired to find the range 7024: 3543:(LM/T), and we want to know the energy 2802:is the value of a bond (or portfolio), 1642:The dimension of the physical quantity 1475:The dimension of the physical quantity 1398:The dimension of the physical quantity 1248:The dimension of the physical quantity 1103:The dimension of the physical quantity 948:The dimension of the physical quantity 817:The dimension of the physical quantity 695:The dimension of the physical quantity 14: 13747: 13414: 12465: 12183: 12103: 11433:from the original on 21 February 2020. 11327: 11312: 11302:from the original on 10 November 2020. 11242: 11104: 11077: 10379: 10377: 10258: 9967: 9944: 9939: 9908: 9888: 9857: 8178:and quantity of matter with dimension 7749: 7730: 7695: 7676: 7641: 7622: 7596: 7378: 7362: 7331: 7315: 7298: 7068: 3881:(M/L), rotates at an angular velocity 2918:), modeling flow with a free surface: 2585:to convert 35 yards to 32.004 m. 2226:Percentages, derivatives and integrals 1821: 1817: 1807: 1804: 1800: 1790: 1787: 1783: 1773: 1769: 1746: 1738: 1725: 1708: 1698: 1691: 1612: 1604: 1591: 1581: 1578: 1574: 1562: 1555: 1542: 1525: 1454: 1447: 1377: 1364: 1347: 1335: 1328: 1315: 1298: 1227: 1214: 1197: 1186: 1176: 1169: 1153: 1082: 1066: 1049: 1030: 1021: 1014: 998: 927: 920: 904: 893: 877: 866: 796: 780: 768: 761: 745: 674: 658: 646: 641: 415: 401: 387: 373: 359: 345: 331: 13388: 12501: 12397:Physics: Principles with Applications 12155: 12129: 12116: 12083: 11575: 11573: 11447:; Nordvall-Forsberg, Fredrik (2022). 10718: 10647: 10623: 10473:Cimbala, John; Çengel, Yunus (2006). 10343: 10134:It is seen that the Taylor series of 7857: 6186:{\displaystyle pE\equiv mB\equiv IAB} 5227:For example, a quantity equation for 5189: 4381:Other fields of physics and chemistry 4229: 4144:can be expressed in the general form 3160:Newton's law of universal gravitation 3120:also treated the same problem of the 2366:{\displaystyle \textstyle \int F\ ds} 1991:Write the above equation in the form 11948: 11925:"The theory of dimensioned matrices" 11919: 11886:from the original on 16 January 2004 11397: 11273:from the original on 10 August 2017. 11178:Programming languages and dimensions 11001: 10980: 10941: 10435: 10303: 10200:– used to teach dimensional analysis 7583:. The dimensional equation becomes: 7104:and a horizontal velocity component 6855: 6217:= area (bounded by a current loop), 4994:Combining units and numerical values 4688:and the initial upward speed is 500 4534:makes sense (as an area), while for 3985:An abelian group is equivalent to a 2743:distinction between stocks and flows 2596: 1844: 84:physical quantities are of the same 12281: 11418:(Thesis). University of Amsterdam. 10909:Berberan-Santos & Pogliani 1999 10857:, 2. Dimensional Formulas pp. 17–27 10830: 10818: 10806: 10506:Dimensional analysis for economists 10374: 9052: 8908: 8764: 8588: 8558: 8528: 7824:nor does it handle well the use of 6851:Geometry: position vs. displacement 6805:Dimensional correctness as part of 5000:Physical quantity § Components 3916:thickness/radius or aspect ratio = 24: 13765:Conversion of units of measurement 13697:International System of Quantities 12550:International System of Units (SI) 12388: 11570: 10951:Thompson, Ambler (November 2009). 10780: 9869: 8230: 8097: 7756: 7702: 7648: 7603: 7541: 7506: 7002:Orientation and frame of reference 6697:{\displaystyle T\delta S/\delta r} 5160: 5157: 5148: 5114: 5102: 5099: 4969: 4895: 4877: 4869: 4866: 4863: 4792: 4789: 4786: 4760: 4751: 4645: 4637: 4598: 4589: 4514:Similarly, while one can evaluate 3978:, and 1/T as reciprocal time (see 3571:of the variables chosen, given by 3450:, for some dimensionless constant 3266:? That period is the solution for 3052: 3049: 2995: 2985: 2982: 2929: 2926: 2870: 2867: 2834: 2737:Finance, economics, and accounting 2328:(mass multiplied by acceleration); 2155:the variables with like exponents. 2069:in which the solution is required. 538:. A quantity that has only all of 25: 13791: 12422: 12250:Journal of the Franklin Institute 12230:Journal of the Franklin Institute 12196:J. W. Strutt (3rd Baron Rayleigh) 12158:Software: Practice and Experience 11491:from the original on 17 May 2022. 11175:Kennedy, Andrew J. (April 1996). 11107:Software: Practice and Experience 10876:Journal of Mathematical Chemistry 10590:Journal of the Franklin Institute 10560:Journal of the Franklin Institute 10533:. New York: McGraw-Hill. p. 9197:. For angles, consider an angle 8340:Angle § Dimensional analysis 6945:length. To assign a number to an 6829:, Python, and a code checker for 5403:is the dimension of the lattice. 4416:relativistic similarity parameter 3567:be two dimensionless products of 3260:suspended in gravity of strength 3220:The original meaning of the word 2627:5 bar × 100 kPa / 1 bar = 500 kPa 1892:that are likely to influence the 517:. A quantity that has only both 13721: 13720: 13675: 11897:(in French), Paris: Firmin Didot 11894:Theorie analytique de la chaleur 10922:Pisanty, E (17 September 2013). 10525:Waite, Lee; Fine, Jerry (2007). 9798:and the range of the projectile 9048: 8904: 8760: 8629:{\displaystyle \mathbf {1_{0}} } 8620: 8616: 8584: 8554: 8524: 8507:{\displaystyle \mathbf {1_{0}} } 8498: 8494: 7964:pressure gradient along the pipe 7550:{\displaystyle v_{\mathrm {y} }} 7515:{\displaystyle v_{\mathrm {x} }} 6953:, while to assign a number to a 6817:. There are implementations for 4372:), is redundant (the set is not 3347:for some dimensionless constant 3197:, which, after substituting his 2708:-ball in terms of the radius is 2688:, while the surface area, being 2447:However, the dimensions form an 185:dimensionless parameters, where 11677: 11636: 11623: 11598: 11545: 11520: 11495: 11437: 11402: 11391: 11350: 11336: 11321: 11306: 11277: 11236: 11193: 11168: 11133: 11098: 11071: 11023: 10995: 10974: 10902: 10860: 10848: 10824: 10774: 10724: 10711: 10617: 10608: 7787:and we may solve completely as 7018: 6900:quantities (ones modeled by an 5288:may be expressed in any units, 5010:is written as the product of a 3887:(T) and this leads to a stress 2640: 2567:and energy share the dimension 2247:) has the dimension L (length); 2163:Concrete numbers and base units 589: 578:that relate them. For example, 12278:, (5): 147, (6): 101, (7): 129 11313:Teller, David (January 2020). 11243:Gundry, Adam (December 2015). 11010:, Cambridge University Press, 10624:Mason, Stephen Finney (1962), 10551: 10518: 10495: 10475:"§7-2 Dimensional homogeneity" 10466: 10448:(3rd ed.), archived from 10429: 10386:Journal of High Energy Physics 10337: 10066: 10033: 9728: 9711: 9689: 9675: 9661: 9641: 9528: 9519: 9469: 9460: 9362: 9356: 9344: 9324: 7407:from which we may deduce that 7201:, both dimensioned as TL, and 7052:He introduced two approaches: 6908:quantities (ones modeled by a 5320:when expressed in seconds and 5245:multiplied by time difference 5063: 5057: 5048: 5042: 4882: 4859: 4797: 4776: 4770: 4738: 4649: 4627: 4608: 4576: 4296: 4245: 4195: 4181: 2648: 2633:, and bar/bar cancels out, so 153: 13: 1: 12560:US customary units (USCS/USC) 12309:American Mathematical Monthly 11777:Chemical Engineering Progress 11697: 11284:Garrigue, J.; Ly, D. (2017). 10764:Maxwell, James Clerk (1873), 10694:Maxwell, James Clerk (1873), 10679:Maxwell, James Clerk (1873), 10481:. McGraw-Hill. p. 203–. 10416:10.1088/1126-6708/2002/03/023 7075:distance a cannonball travels 6982:For temperature differences, 6351:{\displaystyle S/r\equiv L/r} 5926:{\displaystyle AIt\equiv ASt} 5495:International System of Units 5374: 4228:Consequently, every possible 4114:describes some number (e.g., 3936: 3875:(L). The disc has a density 1853:is a conceptual tool used in 1679:electric potential difference 13692:History of the metric system 12296:Journal of Applied Mechanics 12262:10.1016/0016-0032(85)90032-8 12242:10.1016/0016-0032(85)90031-6 12048:Journal of Applied Mechanics 12019:10.1016/0009-2509(55)80004-8 11999:Chemical Engineering Science 11739:10.1016/0004-3702(90)90038-2 11092:10.1016/0096-0551(77)90010-8 10602:10.1016/0016-0032(81)90475-0 10572:10.1016/0016-0032(71)90160-8 10226:Related areas of mathematics 9145:{\displaystyle 1_{\text{x}}} 9117:{\displaystyle 1_{\text{y}}} 9089:{\displaystyle 1_{\text{z}}} 9029:{\displaystyle 1_{\text{x}}} 8973:{\displaystyle 1_{\text{z}}} 8945:{\displaystyle 1_{\text{y}}} 8885:{\displaystyle 1_{\text{y}}} 8857:{\displaystyle 1_{\text{z}}} 8801:{\displaystyle 1_{\text{x}}} 8741:{\displaystyle 1_{\text{z}}} 8713:{\displaystyle 1_{\text{y}}} 8685:{\displaystyle 1_{\text{x}}} 8322:substance, the SI dimension 8198:{\displaystyle M_{\text{m}}} 8171:{\displaystyle M_{\text{i}}} 7955:{\displaystyle p_{\text{x}}} 7192:{\displaystyle v_{\text{y}}} 7161:{\displaystyle v_{\text{x}}} 7126:{\displaystyle v_{\text{x}}} 7097:{\displaystyle v_{\text{y}}} 6762:{\displaystyle Eq\equiv Bqv} 6640:{\displaystyle ma\equiv p/t} 5600:{\displaystyle S/t\equiv Pt} 5343: 5069:{\displaystyle Z=n\times =n} 4686:metres per second per second 4325: 4069:, one has the vector spaces 3088:is the local speed of sound. 2337:) the object has travelled ( 7: 11927:, in Lewis, John G. (ed.), 11295:(in French). hal-01503084. 11286:"Des unités dans le typeur" 11220:10.1007/978-3-642-17685-2_8 10833:"Similarly, one can define 10358:10.1007/978-1-349-00245-0_1 10181: 7049:of the dimensional matrix. 6280:{\displaystyle mv\equiv Ft} 5853:{\displaystyle pV\equiv NT} 5488: 4692:. It is not necessary for 3999:corresponding to the tuple 3473:{\displaystyle {\sqrt {C}}} 3231: 3105:, in a 1799 article at the 2583:1 yard = 0.9144 m 10: 13796: 12487:Dureisseix, David (2019). 12184:Porter, Alfred W. (1933), 11931:, SIAM, pp. 186–190, 11464:10.1142/9789811242380_0020 11328:Grecco, Hernan E. (2022). 10529:Applied Biofluid Mechanics 10350:Theory of Hydraulic Models 8337: 7919:{\displaystyle {\dot {m}}} 6859: 6837:Hindley–Milner type system 5492: 5347: 5332:when expressed in metres. 5205:, also sometimes called a 5193: 4997: 3945: 3897:) non-dimensional groups: 3092: 2600: 2451:under multiplication, so: 2418: 2412: 157: 107:Any physically meaningful 13780:Environmental engineering 13715: 13684: 13673: 13590:thermodynamic temperature 13435: 13430: 13422: 13356: 13330: 13274: 13214: 13168: 13087: 12976: 12779: 12745: 12738: 12707: 12676: 12669: 12622: 12613:English Engineering Units 12580: 12542: 12535: 12399:(7th ed.). Pearson. 11795:, Yale University Press, 11747:The Astrophysical Journal 10787:Ramsay Maunder Associates 10654:, Springer, p. 203, 10626:A history of the sciences 10510:. North Holland. p. 9585:{\displaystyle a=\theta } 9039: 8895: 8751: 8607: 7522:will be dimensioned as TL 6975:where the symbol ≘ means 6912:, such as displacement). 6602: 6599: 6588: 6579:magnetic vector potential 6471:root mean square velocity 6252: 6247: 6244: 6233: 5981: 5520: 5515: 5512: 5501: 5213:used when expressing the 4455:inhomogeneous polynomials 3549:(LM/T) in the wire. Let 3358:. They are often called 2702:. Thus the volume of the 2540:is meaningless. However, 1849:In dimensional analysis, 503:A quantity that has only 12480:University of Nottingham 12454:24 December 2017 at the 12187:The Method of Dimensions 11949:Hart, George W. (1995), 11891:Fourier, Joseph (1822), 11854:Drobot, S. (1953–1954), 11671:10.1088/1681-7575/ac023f 11330:"Pint: makes units easy" 10344:Yalin, M. Selim (1971). 10252: 10204:Numerical-value equation 9619:{\displaystyle b=\pi /2} 7481:{\displaystyle a+b+2c=0} 6904:, such as position) and 5481:Dimensional equivalences 5326:is the numeric value of 5314:is the numeric value of 5222:numerical-value equation 5196:Quantity theory of money 5194:Not to be confused with 4702:. For example, suppose 4459:dimensionless quantities 4439:transcendental functions 4052:representation-theoretic 3179:. By assuming a form of 2696:-dimensional, scales as 2084:involving the exponents 2062:are arbitrary exponents. 12652:Quantum chromodynamical 12444:Buckingham's pi-theorem 12315:(2): 115–138, 227–256, 12190:(3rd ed.), Methuen 12005:(3): 130–140, 167–177, 11969:Huntley, H. E. (1967), 11902:Gibbings, J.C. (2011), 11727:Artificial Intelligence 11367:10.1145/3276604.3276613 11264:10.1145/2887747.2804305 11058:(2nd ed.), Wiley, 10888:10.1023/A:1019102415633 10111:. In other words, that 9852: which means 7990:dynamic fluid viscosity 7438:{\displaystyle a+b+c=1} 6990:(Here °R refers to the 6503:{\displaystyle \rho Vv} 6477:= mass (of a molecule) 4680:if the acceleration of 4546:(3 m) + 3 m = 9 m + 3 m 3942:Mathematical properties 2777:In financial analysis, 2590:Newton's laws of motion 2267:the second derivative ( 2074:dimensional homogeneity 1867:functional relationship 121:dimensional homogeneity 13775:Mechanical engineering 13707:Systems of measurement 12765:centimetre–gram–second 12529:Systems of measurement 12468:"Dimensional Analysis" 12466:Bowley, Roger (2009). 12339:"Theory of Dimensions" 12138:, MIT Press, pp. 11643:Quincey, Paul (2021). 11409:Griffioen, P. (2019). 11119:10.1002/spe.4380150604 10928:Physics Stack Exchange 10648:Roche, John J (1998), 10103: 9986: 9766: 9735: 9620: 9586: 9557: 9369: 9174: 9146: 9118: 9090: 9062: 9030: 9002: 8974: 8946: 8918: 8886: 8858: 8830: 8802: 8774: 8742: 8714: 8686: 8658: 8630: 8598: 8568: 8538: 8508: 8308: 8307:{\displaystyle \pi /8} 8273: 8199: 8172: 8142: 8061: 7956: 7920: 7828:as physical variables. 7778: 7551: 7516: 7482: 7439: 7398: 7277: 7193: 7162: 7127: 7098: 7043: 6763: 6698: 6641: 6564: 6504: 6463: 6423: 6352: 6281: 6202:electric dipole moment 6187: 6087: 6001: 6000:{\displaystyle q\phi } 5927: 5854: 5785: 5691: 5601: 5540: 5350:Dimensionless quantity 5339:Dimensionless concepts 5173: 5122: 5070: 4984: 4665: 4313: 4219: 4180: 3860: 3797: 3732: 3659: 3516:(L) vibrating with an 3474: 3442: 3241:What is the period of 3164:gravitational constant 3128:instead of the Daviet 3076: 3026: 2959: 2903: 2367: 2082:simultaneous equations 2036:dimensionless constant 1835: 1633: 1466: 1389: 1239: 1094: 939: 808: 686: 430: 13287:Biblical and Talmudic 12753:metre–kilogram–second 12493:(lecture). INSA Lyon. 12130:Pesic, Peter (2005), 11876:10.4064/sm-14-1-84-99 11838:10.1103/PhysRev.4.345 11610:reference.wolfram.com 11585:reference.wolfram.com 11557:reference.wolfram.com 11532:reference.wolfram.com 11507:reference.wolfram.com 10839:as the dual space to 10220:System of measurement 10104: 9987: 9802:will be of the form: 9767: 9765:{\displaystyle 1_{0}} 9736: 9621: 9587: 9558: 9370: 9175: 9173:{\displaystyle 1_{0}} 9147: 9119: 9091: 9063: 9031: 9003: 9001:{\displaystyle 1_{0}} 8975: 8947: 8919: 8887: 8859: 8831: 8829:{\displaystyle 1_{0}} 8803: 8775: 8743: 8715: 8687: 8659: 8657:{\displaystyle 1_{0}} 8631: 8599: 8569: 8539: 8509: 8424:orientational symbols 8309: 8274: 8200: 8173: 8143: 8062: 7957: 7921: 7779: 7552: 7517: 7483: 7440: 7399: 7278: 7194: 7163: 7128: 7099: 7044: 6860:Further information: 6801:Programming languages 6764: 6699: 6642: 6565: 6505: 6464: 6424: 6353: 6282: 6188: 6088: 6028:(for changes this is 6002: 5928: 5855: 5786: 5692: 5602: 5541: 5174: 5123: 5071: 4985: 4666: 4548:does not make sense. 4314: 4220: 4160: 4017:scalar multiplication 3946:Further information: 3931:finite element method 3858: 3798: 3733: 3660: 3525:(L). The wire has a 3475: 3443: 3360:dimensionless numbers 3226:Theorie de la Chaleur 3077: 3027: 2960: 2904: 2419:Further information: 2368: 2196:system of measurement 1890:independent variables 1877:. It was named after 1836: 1634: 1467: 1390: 1240: 1095: 940: 809: 687: 431: 202:nondimensionalization 48:by identifying their 13770:Chemical engineering 13755:Dimensional analysis 12374:, World Scientific, 11972:Dimensional Analysis 11904:Dimensional Analysis 11793:Dimensional Analysis 11361:. pp. 121–132. 10960:. DIANE Publishing. 10705:2027/uc1.l0065867749 10455:on 23 September 2015 10188:Buckingham π theorem 10015: 9809: 9749: 9632: 9596: 9570: 9382: 9315: 9233: + ... ~ 1 9157: 9129: 9101: 9073: 9043: 9013: 8985: 8957: 8929: 8899: 8869: 8841: 8813: 8785: 8755: 8725: 8697: 8669: 8641: 8611: 8579: 8549: 8519: 8489: 8290: 8212: 8182: 8155: 8072: 8019: 7939: 7901: 7590: 7532: 7497: 7451: 7411: 7293: 7219: 7176: 7145: 7110: 7081: 7033: 7025:Huntley's extensions 6931:relative differences 6738: 6723:= displacement (see 6671: 6614: 6551: 6488: 6434: 6391: 6371:= angular momentum, 6320: 6259: 6153: 6040: 5988: 5899: 5832: 5735: 5641: 5574: 5527: 5369:back of the envelope 5276:= 5 m/s, where 5132: 5091: 5027: 4715: 4558: 4374:linearly independent 4349:linearly independent 4239: 4151: 3948:Buckingham π theorem 3841:dimensionless number 3825:, and so infer that 3754: 3675: 3578: 3460: 3412: 3114:Buckingham π theorem 3109:Academy of Science. 3045: 2978: 2922: 2863: 2722:, for some constant 2343: 2179:(often shown with a 1875:exponential equation 1658: 1491: 1414: 1264: 1119: 964: 833: 711: 610: 483:linearly independent 313: 279:absolute temperature 206:characteristic units 167:Buckingham π theorem 70:units of measurement 42:dimensional analysis 13617:amount of substance 12356:, Kluwer Academic, 12212:1915Natur..95...66R 12060:2005JAM....72..648M 12011:1955ChEnS...4..130K 11955:, Springer-Verlag, 11830:1914PhRv....4..345B 11759:1991ApJ...372..592B 10734:The Theory of Sound 10408:2002JHEP...03..023D 10092: 10065: 10050: 9977: 8444:. Thus, Huntley's L 8324:amount of substance 7258: 7242: 7069:Directed dimensions 6210:= magnetic moment, 5887:amount of substance 5215:physical quantities 5211:unit of measurement 4412:relativistic plasma 4396:amount of substance 3534:(M/L) and is under 3215:The Theory of Sound 3175:, thereby defining 3155:James Clerk Maxwell 2663:(the solid ball in 2619:100 kPa / 1 bar = 1 1945:functional equation 283:amount of substance 215:The dimension of a 193:of the dimensional 74:conversion of units 46:physical quantities 13642:luminous intensity 13416:SI base quantities 12759:metre–tonne–second 12555:UK imperial system 12026:Langhaar, Henry L. 11863:Studia Mathematica 11847:10338.dmlcz/101743 10099: 10072: 10051: 10036: 9982: 9961: 9762: 9731: 9616: 9582: 9553: 9365: 9170: 9142: 9114: 9086: 9058: 9026: 8998: 8970: 8942: 8914: 8882: 8854: 8826: 8798: 8770: 8738: 8710: 8682: 8654: 8626: 8594: 8564: 8534: 8504: 8304: 8269: 8195: 8168: 8138: 8057: 8003:radius of the pipe 7952: 7916: 7868:Quantity of matter 7858:Quantity of matter 7774: 7547: 7512: 7478: 7435: 7394: 7273: 7244: 7228: 7189: 7158: 7123: 7094: 7039: 7017:This leads to the 7012:frame of reference 6951:point of reference 6892:add two positions. 6782:= magnetic field, 6776:= electric field, 6759: 6694: 6637: 6563:{\displaystyle qA} 6560: 6500: 6459: 6419: 6348: 6277: 6183: 6083: 6026:electric potential 5997: 5923: 5850: 5781: 5687: 5597: 5539:{\displaystyle Fd} 5536: 5220:In contrast, in a 5190:Quantity equations 5169: 5118: 5066: 4980: 4978: 4920: 4827: 4733: 4661: 4571: 4347:the space, and be 4309: 4215: 3901:demand/capacity = 3861: 3793: 3728: 3655: 3653: 3470: 3438: 3435: 3072: 3022: 2955: 2899: 2781:can be defined as 2415:Apples and oranges 2402:debt-to-GDP ratios 2363: 2362: 2318:has the dimension 1947:can be written as 1894:dependent variable 1873:in the form of an 1831: 1629: 1462: 1385: 1235: 1090: 935: 804: 682: 576:conversion factors 426: 287:luminous intensity 210:physical constants 144:quantity dimension 140:physical dimension 13742: 13741: 13670: 13669: 13382: 13381: 13270: 13269: 12734: 12733: 12725:Foot–pound–second 12665: 12664: 12642:Heaviside–Lorentz 12406:978-0-321-62592-2 12381:978-981-02-0304-7 12363:978-0-7923-2031-9 12270:Petroleum Refiner 12149:978-0-262-16234-0 12098:10.1115/1.4018140 12078:10.1115/1.1943434 12039:978-0-88275-682-0 11962:978-0-387-94417-3 11938:978-0-89871-336-7 11913:978-1-84996-316-9 11810:Buckingham, Edgar 11802:978-0-548-91029-0 11718:978-0-521-43522-2 11705:Barenblatt, G. I. 11376:978-1-4503-6029-6 11229:978-3-642-17684-5 11189:. UCAM-CL-TR-391. 11065:978-0-470-03294-7 11041:978-0-7167-8964-2 11017:978-0-521-57507-2 11002:Woan, G. (2010), 10661:978-0-387-91581-4 10635:978-0-02-093400-4 10544:978-0-07-147217-3 10367:978-1-349-00247-4 10352:. pp. 1–34. 10323:978-92-822-2272-0 10288:10.1063/PT.3.1258 10247:Quantity calculus 10242:Geometric algebra 9948: 9919: 9901: 9853: 9725: 9702: 9686: 9655: 9542: 9516: 9483: 9457: 9424: 9407: 9280:has orientation 1 9183: 9182: 9139: 9111: 9083: 9054: 9023: 8967: 8939: 8910: 8879: 8851: 8795: 8766: 8735: 8707: 8679: 8590: 8560: 8530: 8267: 8263: 8192: 8165: 8136: 8127: 8055: 8045: 8010: 8009: 7949: 7913: 7286:Or dimensionally 7251: 7235: 7186: 7155: 7120: 7091: 7042:{\displaystyle m} 6856:Affine quantities 6798: 6797: 6584: 6583: 6457: 6417: 6229: 6228: 5810:moment of inertia 5207:complete equation 5203:quantity equation 5164: 5082:conversion factor 4919: 4826: 4785: 4750: 4732: 4690:metres per second 4636: 4588: 4570: 4544:, the expression 4530:, the expression 4453:functions, or to 4400:Avogadro constant 4081:, and can define 3980:reciprocal second 3976:reciprocal length 3960:is written as 1; 3784: 3712: 3699: 3646: 3615: 3468: 3436: 3434: 3122:parallelogram law 3067: 3017: 2950: 2949: 2894: 2768:Velocity of money 2762:debt-to-GDP ratio 2753:For example, the 2631:5 × 100 / 1 = 500 2610:conversion factor 2603:Conversion factor 2597:Conversion factor 2355: 2214:(N) is a unit of 1865:. It expresses a 1851:Rayleigh's method 1845:Rayleigh's method 1761: 1681: 1680: 1677: 1566: 1514: 1513: 1510: 1440: 1432: 1339: 1287: 1286: 1283: 1145: 1137: 1041: 987: 986: 983: 859: 851: 772: 734: 733: 730: 650: 633: 632: 629: 256:dimension symbols 217:physical quantity 96:are of different 16:(Redirected from 13787: 13724: 13723: 13679: 13678: 13653: 13623: 13601: 13596: 13574: 13569: 13565: 13563: 13555:electric current 13536: 13510: 13506: 13502: 13477: 13428: 13427: 13409: 13402: 13395: 13386: 13385: 12849:Mesures usuelles 12743: 12742: 12674: 12673: 12540: 12539: 12522: 12515: 12508: 12499: 12498: 12494: 12483: 12418: 12384: 12366: 12347:Internet Archive 12341:, chapter XI of 12335:Wilson, Edwin B. 12331: 12303: 12290: 12277: 12264: 12244: 12224: 12223: 12221:10.1038/095066c0 12191: 12180: 12152: 12137: 12126: 12113: 12100: 12092:(671): 671–678, 12080: 12071: 12042: 12021: 11992: 11965: 11941: 11916: 11898: 11887: 11885: 11878: 11860: 11850: 11849: 11805: 11784: 11771: 11770: 11741: 11721: 11692: 11681: 11675: 11674: 11664: 11640: 11634: 11627: 11621: 11620: 11618: 11616: 11602: 11596: 11595: 11593: 11591: 11577: 11568: 11567: 11565: 11563: 11549: 11543: 11542: 11540: 11538: 11524: 11518: 11517: 11515: 11513: 11499: 11493: 11492: 11490: 11453: 11441: 11435: 11434: 11432: 11417: 11406: 11400: 11395: 11389: 11388: 11354: 11348: 11347: 11340: 11334: 11333: 11325: 11319: 11318: 11310: 11304: 11303: 11301: 11290: 11281: 11275: 11274: 11272: 11249: 11240: 11234: 11233: 11213: 11197: 11191: 11190: 11172: 11166: 11165: 11137: 11131: 11130: 11102: 11096: 11095: 11075: 11069: 11068: 11056:Particle Physics 11051: 11045: 11044: 11027: 11021: 11020: 11009: 10999: 10993: 10992: 10990: 10988:hep-th/0208093v3 10978: 10972: 10971: 10959: 10948: 10939: 10938: 10936: 10934: 10918: 10912: 10906: 10900: 10899: 10873: 10864: 10858: 10852: 10846: 10844: 10838: 10828: 10822: 10816: 10810: 10804: 10798: 10797: 10795: 10793: 10778: 10772: 10771: 10761: 10755: 10745: 10739: 10738: 10728: 10722: 10715: 10709: 10708: 10707: 10691: 10685: 10684: 10676: 10670: 10669: 10645: 10639: 10638: 10621: 10615: 10612: 10606: 10605: 10585: 10576: 10575: 10555: 10549: 10548: 10532: 10522: 10516: 10515: 10509: 10499: 10493: 10492: 10470: 10464: 10463: 10462: 10460: 10454: 10447: 10433: 10427: 10426: 10401: 10381: 10372: 10371: 10341: 10335: 10334: 10332: 10330: 10315: 10301: 10292: 10291: 10273: 10265: 10237:Exterior algebra 10169: 10161: 10149: 10141: 10130: 10126: 10114: 10110: 10108: 10106: 10105: 10100: 10091: 10080: 10064: 10059: 10049: 10044: 10032: 10027: 10026: 10008: 10001: 9991: 9989: 9988: 9983: 9976: 9971: 9970: 9959: 9958: 9953: 9949: 9947: 9942: 9937: 9930: 9929: 9924: 9920: 9918: 9917: 9912: 9911: 9904: 9903: 9902: 9899: 9892: 9891: 9884: 9874: 9873: 9872: 9861: 9860: 9854: 9851: 9849: 9848: 9838: 9837: 9827: 9826: 9801: 9797: 9790: 9773: 9771: 9769: 9768: 9763: 9761: 9760: 9742: 9740: 9738: 9737: 9732: 9727: 9726: 9723: 9704: 9703: 9700: 9688: 9687: 9684: 9671: 9657: 9656: 9653: 9625: 9623: 9622: 9617: 9612: 9591: 9589: 9588: 9583: 9562: 9560: 9559: 9554: 9549: 9545: 9544: 9543: 9540: 9518: 9517: 9514: 9490: 9486: 9485: 9484: 9481: 9459: 9458: 9455: 9431: 9427: 9426: 9425: 9422: 9409: 9408: 9405: 9374: 9372: 9371: 9366: 9340: 9310: 9306: 9302: 9279: 9271: 9265:has orientation 9264: 9256: 9241: 9222: 9219:. Since (using 9218: 9211: 9204: 9200: 9196: 9187:Klein four-group 9179: 9177: 9176: 9171: 9169: 9168: 9151: 9149: 9148: 9143: 9141: 9140: 9137: 9123: 9121: 9120: 9115: 9113: 9112: 9109: 9095: 9093: 9092: 9087: 9085: 9084: 9081: 9067: 9065: 9064: 9059: 9057: 9056: 9055: 9035: 9033: 9032: 9027: 9025: 9024: 9021: 9007: 9005: 9004: 8999: 8997: 8996: 8979: 8977: 8976: 8971: 8969: 8968: 8965: 8951: 8949: 8948: 8943: 8941: 8940: 8937: 8923: 8921: 8920: 8915: 8913: 8912: 8911: 8891: 8889: 8888: 8883: 8881: 8880: 8877: 8863: 8861: 8860: 8855: 8853: 8852: 8849: 8835: 8833: 8832: 8827: 8825: 8824: 8807: 8805: 8804: 8799: 8797: 8796: 8793: 8779: 8777: 8776: 8771: 8769: 8768: 8767: 8747: 8745: 8744: 8739: 8737: 8736: 8733: 8719: 8717: 8716: 8711: 8709: 8708: 8705: 8691: 8689: 8688: 8683: 8681: 8680: 8677: 8663: 8661: 8660: 8655: 8653: 8652: 8635: 8633: 8632: 8627: 8625: 8624: 8623: 8603: 8601: 8600: 8595: 8593: 8592: 8591: 8573: 8571: 8570: 8565: 8563: 8562: 8561: 8543: 8541: 8540: 8535: 8533: 8532: 8531: 8513: 8511: 8510: 8505: 8503: 8502: 8501: 8481: 8480: 8477: 8462: 8454: 8448: 8439: 8418: 8412: 8406: 8385: 8371: 8365: 8359: 8316:Poiseuille's law 8313: 8311: 8310: 8305: 8300: 8285: 8278: 8276: 8275: 8270: 8268: 8266: 8265: 8264: 8256: 8249: 8248: 8247: 8235: 8234: 8233: 8222: 8204: 8202: 8201: 8196: 8194: 8193: 8190: 8177: 8175: 8174: 8169: 8167: 8166: 8163: 8147: 8145: 8144: 8139: 8137: 8135: 8134: 8129: 8128: 8120: 8116: 8115: 8114: 8102: 8101: 8100: 8089: 8084: 8083: 8066: 8064: 8063: 8058: 8056: 8054: 8046: 8038: 8036: 8031: 8030: 8000: 7987: 7974: 7961: 7959: 7958: 7953: 7951: 7950: 7947: 7925: 7923: 7922: 7917: 7915: 7914: 7906: 7884: 7883: 7879:Poiseuille's Law 7848: 7842: 7836: 7807: 7800: 7793: 7783: 7781: 7780: 7775: 7773: 7772: 7767: 7763: 7762: 7761: 7760: 7759: 7753: 7752: 7744: 7743: 7742: 7734: 7733: 7719: 7718: 7713: 7709: 7708: 7707: 7706: 7705: 7699: 7698: 7690: 7689: 7688: 7680: 7679: 7665: 7664: 7659: 7655: 7654: 7653: 7652: 7651: 7645: 7644: 7636: 7635: 7634: 7626: 7625: 7608: 7607: 7606: 7600: 7599: 7581: 7576: 7571: 7566: 7561: 7556: 7554: 7553: 7548: 7546: 7545: 7544: 7526: 7521: 7519: 7518: 7513: 7511: 7510: 7509: 7489: 7487: 7485: 7484: 7479: 7444: 7442: 7441: 7436: 7403: 7401: 7400: 7395: 7393: 7392: 7387: 7383: 7382: 7381: 7375: 7374: 7366: 7365: 7352: 7351: 7340: 7336: 7335: 7334: 7328: 7327: 7319: 7318: 7302: 7301: 7282: 7280: 7279: 7274: 7269: 7268: 7257: 7252: 7249: 7241: 7236: 7233: 7212:may be written: 7211: 7204: 7200: 7198: 7196: 7195: 7190: 7188: 7187: 7184: 7169: 7167: 7165: 7164: 7159: 7157: 7156: 7153: 7138: 7134: 7132: 7130: 7129: 7124: 7122: 7121: 7118: 7103: 7101: 7100: 7095: 7093: 7092: 7089: 7048: 7046: 7045: 7040: 6793: 6787: 6781: 6775: 6768: 6766: 6765: 6760: 6732:Electromagnetic 6722: 6716: 6710: 6703: 6701: 6700: 6695: 6687: 6659: 6653: 6646: 6644: 6643: 6638: 6633: 6594: 6586: 6585: 6576: 6569: 6567: 6566: 6561: 6545:Electromagnetic 6536: 6526: 6516: 6509: 6507: 6506: 6501: 6468: 6466: 6465: 6460: 6458: 6456: 6452: 6451: 6438: 6428: 6426: 6425: 6420: 6418: 6416: 6412: 6411: 6398: 6376: 6370: 6364: 6357: 6355: 6354: 6349: 6344: 6330: 6311: 6305: 6299: 6293: 6286: 6284: 6283: 6278: 6239: 6231: 6230: 6223:electric current 6216: 6209: 6199: 6192: 6190: 6189: 6184: 6141: 6130: 6120: 6109: 6099: 6092: 6090: 6089: 6084: 6079: 6071: 6070: 6055: 6054: 6023: 6013: 6006: 6004: 6003: 5998: 5982:Electromagnetic 5973: 5963: 5953: 5939: 5932: 5930: 5929: 5924: 5884: 5878: 5872: 5866: 5859: 5857: 5856: 5851: 5820:angular velocity 5817: 5807: 5800:angular momentum 5797: 5790: 5788: 5787: 5782: 5777: 5772: 5771: 5750: 5749: 5723: 5713: 5703: 5696: 5694: 5693: 5688: 5683: 5678: 5677: 5656: 5655: 5629: 5623: 5613: 5606: 5604: 5603: 5598: 5584: 5562: 5552: 5545: 5543: 5542: 5537: 5507: 5499: 5498: 5476: 5469: 5462: 5455: 5449: 5443: 5432: 5424: 5416: 5402: 5396: 5389: 5365: 5359: 5331: 5325: 5319: 5313: 5304: 5287: 5281: 5275: 5266: 5250: 5244: 5235: 5178: 5176: 5175: 5170: 5165: 5163: 5151: 5142: 5128:is identical to 5127: 5125: 5124: 5119: 5117: 5105: 5075: 5073: 5072: 5067: 5019: 5009: 4989: 4987: 4986: 4981: 4979: 4972: 4964: 4963: 4951: 4947: 4946: 4921: 4912: 4907: 4898: 4890: 4889: 4880: 4876: 4858: 4854: 4853: 4828: 4819: 4814: 4805: 4804: 4795: 4783: 4769: 4768: 4767: 4758: 4748: 4734: 4725: 4721: 4707: 4697: 4691: 4687: 4679: 4670: 4668: 4667: 4662: 4648: 4644: 4634: 4623: 4622: 4607: 4606: 4605: 4596: 4586: 4572: 4563: 4547: 4543: 4533: 4529: 4523: 4510: 4504: 4494: 4488: 4482: 4409: 4407: 4371: 4318: 4316: 4315: 4310: 4295: 4294: 4270: 4269: 4257: 4256: 4224: 4222: 4221: 4216: 4210: 4209: 4208: 4207: 4193: 4192: 4179: 4174: 4143: 4133: 4119: 4094: 4080: 4074: 4068: 4062: 4049: 4040: 4014: 3998: 3997: 3994: 3973: 3969: 3963: 3925: 3913: 3896: 3892: 3886: 3880: 3874: 3868: 3834: 3824: 3818: 3811: 3802: 3800: 3799: 3794: 3789: 3785: 3777: 3746: 3737: 3735: 3734: 3729: 3718: 3714: 3713: 3705: 3700: 3698: 3687: 3664: 3662: 3661: 3656: 3654: 3647: 3639: 3630: 3629: 3616: 3614: 3603: 3594: 3593: 3566: 3557: 3548: 3542: 3533: 3524: 3515: 3498: 3488: 3479: 3477: 3476: 3471: 3469: 3464: 3455: 3449: 3447: 3445: 3444: 3439: 3437: 3427: 3425: 3405: 3394: 3388: 3384: 3380: 3376: 3372: 3368: 3352: 3346: 3333: 3313: 3307: 3303: 3299: 3295: 3289: 3283: 3277: 3271: 3265: 3259: 3253: 3249: 3204: 3200: 3196: 3185:Coulomb constant 3178: 3170: 3149: 3087: 3081: 3079: 3078: 3073: 3068: 3060: 3055: 3040: 3031: 3029: 3028: 3023: 3018: 3016: 3015: 3014: 3001: 2993: 2988: 2973: 2964: 2962: 2961: 2956: 2951: 2941: 2937: 2932: 2917: 2908: 2906: 2905: 2900: 2895: 2890: 2878: 2873: 2858: 2829: 2823: 2817: 2807: 2801: 2795: 2747:financial ratios 2732: 2721: 2707: 2701: 2695: 2687: 2681: 2673: 2660: 2636: 2632: 2628: 2624: 2620: 2616: 2584: 2576: 2573: 2570: 2555: 2539: 2523: 2507: 2498: 2489: 2421:Kind of quantity 2395:stocks and flows 2388: 2385: 2382: 2379:) has dimension 2374: 2372: 2370: 2369: 2364: 2353: 2336: 2327: 2324: 2321: 2306: 2303: 2300:) has dimension 2295: 2259: 2246: 2233: 2221: 2137: 2131: 2125: 2119: 2107: 2101: 2095: 2089: 2061: 2055: 2049: 2043: 2033: 2027: 1987: 1942: 1931: 1922: 1913: 1904: 1840: 1838: 1837: 1832: 1827: 1826: 1825: 1824: 1813: 1812: 1811: 1810: 1796: 1795: 1794: 1793: 1779: 1778: 1777: 1776: 1762: 1760: 1759: 1758: 1750: 1749: 1742: 1741: 1735: 1734: 1729: 1728: 1721: 1720: 1712: 1711: 1703: 1702: 1701: 1695: 1694: 1687: 1682: 1678: 1675: 1674: 1650: 1638: 1636: 1635: 1630: 1625: 1624: 1616: 1615: 1608: 1607: 1601: 1600: 1595: 1594: 1587: 1586: 1585: 1584: 1567: 1565: 1560: 1559: 1558: 1552: 1551: 1546: 1545: 1538: 1537: 1529: 1528: 1520: 1515: 1511: 1508: 1507: 1483: 1471: 1469: 1468: 1463: 1458: 1457: 1451: 1450: 1441: 1438: 1433: 1430: 1406: 1394: 1392: 1391: 1386: 1381: 1380: 1374: 1373: 1368: 1367: 1360: 1359: 1351: 1350: 1340: 1338: 1333: 1332: 1331: 1325: 1324: 1319: 1318: 1311: 1310: 1302: 1301: 1293: 1288: 1284: 1281: 1280: 1256: 1244: 1242: 1241: 1236: 1231: 1230: 1224: 1223: 1218: 1217: 1210: 1209: 1201: 1200: 1190: 1189: 1180: 1179: 1173: 1172: 1166: 1165: 1157: 1156: 1146: 1143: 1138: 1135: 1111: 1099: 1097: 1096: 1091: 1086: 1085: 1079: 1078: 1070: 1069: 1062: 1061: 1053: 1052: 1042: 1040: 1039: 1034: 1033: 1026: 1025: 1024: 1018: 1017: 1011: 1010: 1002: 1001: 993: 988: 984: 981: 980: 956: 944: 942: 941: 936: 931: 930: 924: 923: 917: 916: 908: 907: 897: 896: 890: 889: 881: 880: 870: 869: 860: 857: 852: 849: 825: 813: 811: 810: 805: 800: 799: 793: 792: 784: 783: 773: 771: 766: 765: 764: 758: 757: 749: 748: 740: 735: 731: 728: 727: 703: 691: 689: 688: 683: 678: 677: 671: 670: 662: 661: 651: 649: 644: 639: 634: 630: 627: 626: 602: 581: 558: 551: 544: 530: 523: 509: 499: 490:electric current 480: 474: 468: 462: 456: 450: 444: 435: 433: 432: 427: 425: 424: 419: 418: 411: 410: 405: 404: 397: 396: 391: 390: 383: 382: 377: 376: 369: 368: 363: 362: 355: 354: 349: 348: 341: 340: 335: 334: 305: 275:electric current 184: 174: 66:electric current 21: 18:Dimension symbol 13795: 13794: 13790: 13789: 13788: 13786: 13785: 13784: 13745: 13744: 13743: 13738: 13711: 13680: 13676: 13671: 13652: 13646: 13621: 13594: 13573: I 13567: 13561: 13534: 13508: 13504: 13500: 13475: 13464: 13459: 13451: 13423:Base quantities 13418: 13413: 13383: 13378: 13352: 13326: 13266: 13210: 13164: 13083: 12972: 12775: 12730: 12703: 12661: 12618: 12576: 12531: 12526: 12456:Wayback Machine 12425: 12407: 12391: 12389:Further reading 12382: 12364: 12321:10.2307/2315883 12170:10.1002/spe.401 12164:(11): 1067–76, 12150: 12134:Sky in a Bottle 12040: 11963: 11939: 11921:Hart, George W. 11914: 11883: 11858: 11818:Physical Review 11803: 11789:Bridgman, P. W. 11733:(1–2): 73–111, 11719: 11700: 11695: 11682: 11678: 11641: 11637: 11628: 11624: 11614: 11612: 11604: 11603: 11599: 11589: 11587: 11579: 11578: 11571: 11561: 11559: 11551: 11550: 11546: 11536: 11534: 11526: 11525: 11521: 11511: 11509: 11501: 11500: 11496: 11488: 11474: 11451: 11442: 11438: 11430: 11415: 11407: 11403: 11396: 11392: 11377: 11355: 11351: 11342: 11341: 11337: 11326: 11322: 11311: 11307: 11299: 11288: 11282: 11278: 11270: 11252:SIGPLAN Notices 11247: 11241: 11237: 11230: 11211:10.1.1.174.6901 11198: 11194: 11173: 11169: 11154:10.1109/52.2021 11138: 11134: 11103: 11099: 11076: 11072: 11066: 11052: 11048: 11042: 11028: 11024: 11018: 11000: 10996: 10979: 10975: 10968: 10957: 10949: 10942: 10932: 10930: 10919: 10915: 10907: 10903: 10871: 10865: 10861: 10853: 10849: 10840: 10834: 10829: 10825: 10817: 10813: 10805: 10801: 10791: 10789: 10781:Ramsay, Angus. 10779: 10775: 10762: 10758: 10746: 10742: 10729: 10725: 10716: 10712: 10692: 10688: 10677: 10673: 10662: 10646: 10642: 10636: 10622: 10618: 10613: 10609: 10586: 10579: 10556: 10552: 10545: 10523: 10519: 10500: 10496: 10489: 10471: 10467: 10458: 10456: 10452: 10445: 10434: 10430: 10399:physics/0110060 10382: 10375: 10368: 10342: 10338: 10328: 10326: 10324: 10313: 10302: 10295: 10266: 10259: 10255: 10228: 10184: 10163: 10151: 10143: 10135: 10128: 10116: 10112: 10081: 10076: 10060: 10055: 10045: 10040: 10028: 10022: 10018: 10016: 10013: 10012: 10010: 10003: 9996: 9972: 9966: 9965: 9954: 9943: 9938: 9936: 9932: 9931: 9925: 9913: 9907: 9906: 9905: 9898: 9894: 9887: 9886: 9885: 9883: 9879: 9878: 9868: 9867: 9863: 9856: 9855: 9850: 9844: 9840: 9833: 9829: 9822: 9818: 9810: 9807: 9806: 9799: 9796: 9792: 9786: 9756: 9752: 9750: 9747: 9746: 9744: 9722: 9718: 9699: 9695: 9683: 9679: 9667: 9652: 9648: 9633: 9630: 9629: 9627: 9608: 9597: 9594: 9593: 9571: 9568: 9567: 9539: 9535: 9513: 9509: 9504: 9500: 9480: 9476: 9454: 9450: 9445: 9441: 9421: 9417: 9404: 9400: 9395: 9391: 9383: 9380: 9379: 9336: 9316: 9313: 9312: 9308: 9304: 9285: 9283: 9273: 9270: 9266: 9258: 9255: 9251: 9247: 9243: 9240: 9236: 9224: 9220: 9217: 9213: 9210: 9206: 9202: 9198: 9195: 9191: 9164: 9160: 9158: 9155: 9154: 9136: 9132: 9130: 9127: 9126: 9108: 9104: 9102: 9099: 9098: 9080: 9076: 9074: 9071: 9070: 9051: 9047: 9046: 9044: 9041: 9040: 9020: 9016: 9014: 9011: 9010: 8992: 8988: 8986: 8983: 8982: 8964: 8960: 8958: 8955: 8954: 8936: 8932: 8930: 8927: 8926: 8907: 8903: 8902: 8900: 8897: 8896: 8876: 8872: 8870: 8867: 8866: 8848: 8844: 8842: 8839: 8838: 8820: 8816: 8814: 8811: 8810: 8792: 8788: 8786: 8783: 8782: 8763: 8759: 8758: 8756: 8753: 8752: 8732: 8728: 8726: 8723: 8722: 8704: 8700: 8698: 8695: 8694: 8676: 8672: 8670: 8667: 8666: 8648: 8644: 8642: 8639: 8638: 8619: 8615: 8614: 8612: 8609: 8608: 8587: 8583: 8582: 8580: 8577: 8576: 8557: 8553: 8552: 8550: 8547: 8546: 8527: 8523: 8522: 8520: 8517: 8516: 8497: 8493: 8492: 8490: 8487: 8486: 8476: 8470: 8464: 8461: 8457: 8455: 8452: 8449: 8446: 8443: 8438: 8434: 8430: 8426: 8414: 8408: 8391: 8381: 8367: 8361: 8349: 8342: 8336: 8296: 8291: 8288: 8287: 8283: 8282:where now only 8255: 8254: 8250: 8243: 8239: 8229: 8228: 8224: 8223: 8221: 8213: 8210: 8209: 8189: 8185: 8183: 8180: 8179: 8162: 8158: 8156: 8153: 8152: 8130: 8119: 8118: 8117: 8110: 8106: 8096: 8095: 8091: 8090: 8088: 8079: 8075: 8073: 8070: 8069: 8047: 8037: 8035: 8026: 8022: 8020: 8017: 8016: 7998: 7985: 7972: 7946: 7942: 7940: 7937: 7936: 7905: 7904: 7902: 7899: 7898: 7860: 7849: 7846: 7843: 7840: 7837: 7834: 7802: 7795: 7788: 7768: 7755: 7754: 7748: 7747: 7746: 7745: 7735: 7729: 7728: 7727: 7726: 7725: 7721: 7720: 7714: 7701: 7700: 7694: 7693: 7692: 7691: 7681: 7675: 7674: 7673: 7672: 7671: 7667: 7666: 7660: 7647: 7646: 7640: 7639: 7638: 7637: 7627: 7621: 7620: 7619: 7618: 7617: 7613: 7612: 7602: 7601: 7595: 7594: 7593: 7591: 7588: 7587: 7582: 7579: 7574: 7572: 7569: 7564: 7562: 7559: 7540: 7539: 7535: 7533: 7530: 7529: 7527: 7524: 7505: 7504: 7500: 7498: 7495: 7494: 7452: 7449: 7448: 7446: 7412: 7409: 7408: 7388: 7377: 7376: 7367: 7361: 7360: 7359: 7358: 7354: 7353: 7341: 7330: 7329: 7320: 7314: 7313: 7312: 7311: 7307: 7306: 7297: 7296: 7294: 7291: 7290: 7264: 7260: 7253: 7248: 7237: 7232: 7220: 7217: 7216: 7209: 7202: 7183: 7179: 7177: 7174: 7173: 7171: 7152: 7148: 7146: 7143: 7142: 7140: 7136: 7117: 7113: 7111: 7108: 7107: 7105: 7088: 7084: 7082: 7079: 7078: 7071: 7059: 7034: 7031: 7030: 7027: 7004: 6864: 6858: 6853: 6841:dependent types 6834: 6813:, and later in 6803: 6789: 6783: 6777: 6771: 6739: 6736: 6735: 6718: 6717:= temperature, 6712: 6706: 6683: 6672: 6669: 6668: 6660:= acceleration 6655: 6649: 6629: 6615: 6612: 6611: 6590: 6572: 6552: 6549: 6548: 6532: 6522: 6512: 6489: 6486: 6485: 6447: 6443: 6439: 6437: 6435: 6432: 6431: 6407: 6403: 6399: 6397: 6392: 6389: 6388: 6372: 6366: 6360: 6340: 6326: 6321: 6318: 6317: 6307: 6301: 6295: 6289: 6260: 6257: 6256: 6235: 6212: 6211: 6205: 6195: 6154: 6151: 6150: 6137: 6136: 6126: 6116: 6115: 6105: 6095: 6075: 6066: 6062: 6050: 6046: 6041: 6038: 6037: 6019: 6016:electric charge 6009: 5989: 5986: 5985: 5976:Poynting vector 5969: 5959: 5949: 5935: 5900: 5897: 5896: 5880: 5879:= temperature, 5874: 5868: 5862: 5833: 5830: 5829: 5813: 5803: 5793: 5773: 5767: 5763: 5745: 5741: 5736: 5733: 5732: 5719: 5709: 5699: 5679: 5673: 5669: 5651: 5647: 5642: 5639: 5638: 5625: 5619: 5609: 5580: 5575: 5572: 5571: 5558: 5548: 5528: 5525: 5524: 5503: 5497: 5491: 5483: 5471: 5464: 5457: 5451: 5445: 5439: 5428: 5420: 5412: 5408:Michael J. Duff 5398: 5391: 5385: 5377: 5361: 5355: 5352: 5346: 5341: 5327: 5321: 5315: 5309: 5296: 5283: 5277: 5271: 5255: 5246: 5240: 5231: 5199: 5192: 5152: 5143: 5141: 5133: 5130: 5129: 5109: 5094: 5092: 5089: 5088: 5028: 5025: 5024: 5015: 5005: 5002: 4996: 4977: 4976: 4968: 4959: 4955: 4942: 4938: 4934: 4910: 4908: 4906: 4900: 4899: 4894: 4885: 4881: 4872: 4862: 4849: 4845: 4841: 4817: 4815: 4813: 4807: 4806: 4800: 4796: 4779: 4763: 4759: 4754: 4741: 4723: 4718: 4716: 4713: 4712: 4703: 4693: 4689: 4685: 4675: 4640: 4630: 4618: 4614: 4601: 4597: 4592: 4579: 4561: 4559: 4556: 4555: 4545: 4535: 4531: 4525: 4519: 4506: 4500: 4490: 4484: 4465: 4428: 4420:Vlasov equation 4405: 4403: 4388:electric charge 4383: 4369: 4337:change of basis 4328: 4290: 4286: 4265: 4261: 4252: 4248: 4240: 4237: 4236: 4203: 4199: 4198: 4194: 4188: 4184: 4175: 4164: 4152: 4149: 4148: 4139: 4131: 4125: 4121: 4115: 4101:natural pairing 4082: 4076: 4070: 4064: 4058: 4045: 4038: 4024:base quantities 4019:in the module. 4000: 3995: 3992: 3990: 3971: 3965: 3961: 3950: 3944: 3939: 3917: 3902: 3894: 3888: 3882: 3876: 3870: 3869:(L) and radius 3864: 3853: 3845:Reynolds number 3826: 3820: 3814: 3807: 3776: 3772: 3755: 3752: 3751: 3742: 3704: 3691: 3686: 3685: 3681: 3676: 3673: 3672: 3652: 3651: 3638: 3631: 3625: 3621: 3618: 3617: 3607: 3602: 3595: 3589: 3585: 3581: 3579: 3576: 3575: 3565: 3559: 3556: 3550: 3544: 3538: 3529: 3520: 3511: 3505: 3494: 3484: 3463: 3461: 3458: 3457: 3451: 3424: 3413: 3410: 3409: 3407: 3401: 3390: 3386: 3382: 3378: 3374: 3370: 3366: 3348: 3341: 3335: 3321: 3315: 3309: 3305: 3301: 3297: 3291: 3285: 3279: 3273: 3267: 3261: 3255: 3251: 3245: 3239: 3234: 3224:, in Fourier's 3202: 3198: 3194: 3192: 3176: 3166: 3141: 3101:, a student of 3099:François Daviet 3095: 3083: 3059: 3048: 3046: 3043: 3042: 3038: 3010: 3006: 3002: 2994: 2992: 2981: 2979: 2976: 2975: 2971: 2936: 2925: 2923: 2920: 2919: 2915: 2879: 2877: 2866: 2864: 2861: 2860: 2856: 2853:Reynolds number 2841:fluid mechanics 2837: 2835:Fluid mechanics 2825: 2819: 2809: 2803: 2797: 2782: 2739: 2731: 2723: 2717: 2709: 2703: 2697: 2689: 2683: 2677: 2669: 2656: 2651: 2643: 2635:5 bar = 500 kPa 2634: 2630: 2626: 2623:100 kPa / 1 bar 2622: 2618: 2615:100 kPa = 1 bar 2614: 2605: 2599: 2582: 2574: 2571: 2568: 2554: 2547: 2541: 2538: 2531: 2525: 2522: 2515: 2509: 2506: 2500: 2497: 2491: 2488: 2482: 2471: 2460:incommensurable 2445: 2423: 2417: 2411: 2386: 2383: 2380: 2344: 2341: 2340: 2338: 2332: 2325: 2322: 2319: 2304: 2301: 2268: 2251: 2242: 2231: 2228: 2219: 2169:concrete number 2165: 2146:non-dimensional 2133: 2127: 2121: 2115: 2103: 2097: 2091: 2085: 2057: 2051: 2045: 2039: 2029: 2026: 2017: 2011: 2005: 1992: 1985: 1976: 1969: 1962: 1948: 1941: 1933: 1930: 1924: 1921: 1915: 1912: 1906: 1900: 1888:Gather all the 1847: 1820: 1816: 1815: 1814: 1803: 1799: 1798: 1797: 1786: 1782: 1781: 1780: 1772: 1768: 1767: 1766: 1751: 1745: 1744: 1743: 1737: 1736: 1730: 1724: 1723: 1722: 1713: 1707: 1706: 1705: 1704: 1697: 1696: 1690: 1689: 1688: 1686: 1676:electric charge 1673: 1659: 1656: 1655: 1646: 1617: 1611: 1610: 1609: 1603: 1602: 1596: 1590: 1589: 1588: 1577: 1573: 1572: 1571: 1561: 1554: 1553: 1547: 1541: 1540: 1539: 1530: 1524: 1523: 1522: 1521: 1519: 1506: 1492: 1489: 1488: 1479: 1453: 1452: 1446: 1445: 1437: 1429: 1415: 1412: 1411: 1402: 1400:electric charge 1376: 1375: 1369: 1363: 1362: 1361: 1352: 1346: 1345: 1344: 1334: 1327: 1326: 1320: 1314: 1313: 1312: 1303: 1297: 1296: 1295: 1294: 1292: 1279: 1265: 1262: 1261: 1252: 1226: 1225: 1219: 1213: 1212: 1211: 1202: 1196: 1195: 1194: 1185: 1184: 1175: 1174: 1168: 1167: 1158: 1152: 1151: 1150: 1142: 1134: 1120: 1117: 1116: 1107: 1081: 1080: 1071: 1065: 1064: 1063: 1054: 1048: 1047: 1046: 1035: 1029: 1028: 1027: 1020: 1019: 1013: 1012: 1003: 997: 996: 995: 994: 992: 979: 965: 962: 961: 952: 926: 925: 919: 918: 909: 903: 902: 901: 892: 891: 882: 876: 875: 874: 865: 864: 856: 848: 834: 831: 830: 821: 795: 794: 785: 779: 778: 777: 767: 760: 759: 750: 744: 743: 742: 741: 739: 726: 712: 709: 708: 699: 673: 672: 663: 657: 656: 655: 645: 640: 638: 625: 611: 608: 607: 598: 592: 579: 553: 546: 539: 525: 518: 504: 497: 494:electric charge 476: 470: 464: 458: 452: 446: 440: 420: 414: 413: 412: 406: 400: 399: 398: 392: 386: 385: 384: 378: 372: 371: 370: 364: 358: 357: 356: 350: 344: 343: 342: 336: 330: 329: 328: 314: 311: 310: 301: 208:of a system or 176: 170: 163: 156: 138:The concept of 90:Incommensurable 50:base quantities 28: 23: 22: 15: 12: 11: 5: 13793: 13783: 13782: 13777: 13772: 13767: 13762: 13757: 13740: 13739: 13737: 13736: 13729: 13716: 13713: 13712: 13710: 13709: 13704: 13699: 13694: 13688: 13686: 13682: 13681: 13674: 13672: 13668: 13667: 13664: 13659: 13657: 13654: 13650: 13644: 13638: 13637: 13634: 13629: 13627: 13624: 13619: 13613: 13612: 13609: 13604: 13602: 13597: 13592: 13586: 13585: 13582: 13577: 13575: 13570: 13557: 13551: 13550: 13547: 13542: 13540: 13537: 13532: 13526: 13525: 13522: 13517: 13515: 13512: 13498: 13492: 13491: 13488: 13483: 13481: 13478: 13473: 13471:time, duration 13467: 13466: 13461: 13456: 13454: 13447: 13444: 13440: 13439: 13434: 13432: 13426: 13424: 13420: 13419: 13412: 13411: 13404: 13397: 13389: 13380: 13379: 13377: 13376: 13371: 13366: 13364:Absolute scale 13360: 13358: 13354: 13353: 13351: 13350: 13345: 13340: 13334: 13332: 13328: 13327: 13325: 13324: 13319: 13314: 13309: 13304: 13299: 13294: 13289: 13284: 13278: 13276: 13272: 13271: 13268: 13267: 13265: 13264: 13259: 13254: 13249: 13244: 13239: 13234: 13229: 13224: 13218: 13216: 13212: 13211: 13209: 13208: 13203: 13198: 13193: 13188: 13183: 13178: 13172: 13170: 13166: 13165: 13163: 13162: 13157: 13152: 13147: 13142: 13137: 13132: 13127: 13122: 13117: 13112: 13107: 13102: 13097: 13091: 13089: 13085: 13084: 13082: 13081: 13076: 13071: 13066: 13061: 13056: 13051: 13046: 13041: 13036: 13031: 13026: 13021: 13016: 13011: 13006: 13001: 12996: 12991: 12986: 12980: 12978: 12974: 12973: 12971: 12970: 12965: 12960: 12955: 12950: 12945: 12940: 12935: 12930: 12925: 12920: 12915: 12910: 12905: 12900: 12895: 12890: 12885: 12880: 12875: 12870: 12869: 12868: 12858: 12853: 12852: 12851: 12846: 12836: 12831: 12826: 12825: 12824: 12819: 12809: 12804: 12799: 12794: 12789: 12783: 12781: 12777: 12776: 12774: 12773: 12768: 12762: 12756: 12749: 12747: 12740: 12736: 12735: 12732: 12731: 12729: 12728: 12722: 12717: 12711: 12709: 12705: 12704: 12702: 12701: 12696: 12691: 12686: 12680: 12678: 12671: 12667: 12666: 12663: 12662: 12660: 12659: 12654: 12649: 12644: 12639: 12634: 12628: 12626: 12620: 12619: 12617: 12616: 12610: 12605: 12600: 12595: 12590: 12584: 12582: 12578: 12577: 12575: 12574: 12573: 12572: 12562: 12557: 12552: 12546: 12544: 12537: 12533: 12532: 12525: 12524: 12517: 12510: 12502: 12496: 12495: 12484: 12463: 12458: 12446: 12441: 12436: 12431: 12424: 12423:External links 12421: 12420: 12419: 12405: 12390: 12387: 12386: 12385: 12380: 12367: 12362: 12349: 12332: 12304: 12291: 12279: 12265: 12256:(6): 285–302, 12245: 12236:(6): 267–283, 12225: 12206:(2368): 66–8, 12192: 12181: 12153: 12148: 12127: 12114: 12101: 12081: 12069:10.1.1.422.610 12054:(5): 648–657, 12043: 12038: 12022: 11993: 11991:, LOC 67-17978 11966: 11961: 11946: 11937: 11917: 11912: 11899: 11888: 11851: 11824:(4): 345–376, 11806: 11801: 11785: 11772: 11768:10.1086/170003 11742: 11722: 11717: 11699: 11696: 11694: 11693: 11676: 11635: 11622: 11597: 11569: 11544: 11519: 11494: 11472: 11445:McBride, Conor 11436: 11401: 11390: 11375: 11349: 11335: 11320: 11305: 11276: 11235: 11228: 11192: 11167: 11132: 11113:(6): 555–569. 11097: 11070: 11064: 11046: 11040: 11022: 11016: 10994: 10973: 10966: 10940: 10913: 10901: 10859: 10847: 10823: 10811: 10799: 10773: 10756: 10748:Fourier (1822) 10740: 10723: 10710: 10686: 10671: 10660: 10640: 10634: 10616: 10607: 10596:(5): 331–337. 10577: 10566:(6): 391–340. 10550: 10543: 10517: 10494: 10487: 10465: 10428: 10373: 10366: 10336: 10322: 10293: 10256: 10254: 10251: 10250: 10249: 10244: 10239: 10234: 10227: 10224: 10223: 10222: 10217: 10211: 10206: 10201: 10198:Fermi estimate 10195: 10190: 10183: 10180: 10098: 10095: 10090: 10087: 10084: 10079: 10075: 10071: 10068: 10063: 10058: 10054: 10048: 10043: 10039: 10035: 10031: 10025: 10021: 9993: 9992: 9980: 9975: 9969: 9964: 9957: 9952: 9946: 9941: 9935: 9928: 9923: 9916: 9910: 9897: 9890: 9882: 9877: 9871: 9866: 9859: 9847: 9843: 9836: 9832: 9825: 9821: 9817: 9814: 9794: 9759: 9755: 9730: 9721: 9716: 9713: 9710: 9707: 9698: 9694: 9691: 9682: 9677: 9674: 9670: 9666: 9663: 9660: 9651: 9646: 9643: 9640: 9637: 9615: 9611: 9607: 9604: 9601: 9581: 9578: 9575: 9564: 9563: 9552: 9548: 9538: 9533: 9530: 9527: 9524: 9521: 9512: 9507: 9503: 9499: 9496: 9493: 9489: 9479: 9474: 9471: 9468: 9465: 9462: 9453: 9448: 9444: 9440: 9437: 9434: 9430: 9420: 9415: 9412: 9403: 9398: 9394: 9390: 9387: 9364: 9361: 9358: 9355: 9352: 9349: 9346: 9343: 9339: 9335: 9332: 9329: 9326: 9323: 9320: 9281: 9268: 9253: 9249: 9245: 9238: 9234: 9215: 9208: 9193: 9181: 9180: 9167: 9163: 9152: 9135: 9124: 9107: 9096: 9079: 9068: 9050: 9037: 9036: 9019: 9008: 8995: 8991: 8980: 8963: 8952: 8935: 8924: 8906: 8893: 8892: 8875: 8864: 8847: 8836: 8823: 8819: 8808: 8791: 8780: 8762: 8749: 8748: 8731: 8720: 8703: 8692: 8675: 8664: 8651: 8647: 8636: 8622: 8618: 8605: 8604: 8586: 8574: 8556: 8544: 8526: 8514: 8500: 8496: 8484: 8472: 8466: 8459: 8451: 8445: 8441: 8436: 8432: 8428: 8335: 8332: 8303: 8299: 8295: 8280: 8279: 8262: 8259: 8253: 8246: 8242: 8238: 8232: 8227: 8220: 8217: 8188: 8161: 8149: 8148: 8133: 8126: 8123: 8113: 8109: 8105: 8099: 8094: 8087: 8082: 8078: 8067: 8053: 8050: 8044: 8041: 8034: 8029: 8025: 8008: 8007: 8004: 8001: 7995: 7994: 7991: 7988: 7982: 7981: 7978: 7975: 7969: 7968: 7965: 7962: 7945: 7933: 7932: 7929: 7928:mass flow rate 7926: 7912: 7909: 7895: 7894: 7891: 7888: 7873: 7859: 7856: 7845: 7839: 7833: 7830: 7829: 7822: 7785: 7784: 7771: 7766: 7758: 7751: 7741: 7738: 7732: 7724: 7717: 7712: 7704: 7697: 7687: 7684: 7678: 7670: 7663: 7658: 7650: 7643: 7633: 7630: 7624: 7616: 7611: 7605: 7598: 7578: 7568: 7558: 7543: 7538: 7523: 7508: 7503: 7477: 7474: 7471: 7468: 7465: 7462: 7459: 7456: 7434: 7431: 7428: 7425: 7422: 7419: 7416: 7405: 7404: 7391: 7386: 7380: 7373: 7370: 7364: 7357: 7350: 7347: 7344: 7339: 7333: 7326: 7323: 7317: 7310: 7305: 7300: 7284: 7283: 7272: 7267: 7263: 7256: 7247: 7240: 7231: 7227: 7224: 7182: 7151: 7116: 7087: 7070: 7067: 7066: 7065: 7062: 7057: 7038: 7026: 7023: 7003: 7000: 6988: 6987: 6977:corresponds to 6973: 6972: 6935: 6934: 6923: 6894: 6893: 6886: 6883: 6880: 6857: 6854: 6852: 6849: 6802: 6799: 6796: 6795: 6769: 6758: 6755: 6752: 6749: 6746: 6743: 6733: 6729: 6728: 6725:entropic force 6704: 6693: 6690: 6686: 6682: 6679: 6676: 6666: 6662: 6661: 6647: 6636: 6632: 6628: 6625: 6622: 6619: 6609: 6605: 6604: 6601: 6598: 6582: 6581: 6570: 6559: 6556: 6546: 6542: 6541: 6539:phase velocity 6510: 6499: 6496: 6493: 6483: 6479: 6478: 6455: 6450: 6446: 6442: 6429: 6415: 6410: 6406: 6402: 6396: 6386: 6382: 6381: 6358: 6347: 6343: 6339: 6336: 6333: 6329: 6325: 6314: 6313: 6287: 6276: 6273: 6270: 6267: 6264: 6254: 6250: 6249: 6246: 6243: 6227: 6226: 6193: 6182: 6179: 6176: 6173: 6170: 6167: 6164: 6161: 6158: 6147: 6146: 6112:magnetic field 6102:electric field 6093: 6082: 6078: 6074: 6069: 6065: 6061: 6058: 6053: 6049: 6045: 6034: 6033: 6007: 5996: 5993: 5983: 5979: 5978: 5933: 5922: 5919: 5916: 5913: 5910: 5907: 5904: 5894: 5890: 5889: 5860: 5849: 5846: 5843: 5840: 5837: 5827: 5823: 5822: 5791: 5780: 5776: 5770: 5766: 5762: 5759: 5756: 5753: 5748: 5744: 5740: 5729: 5728: 5697: 5686: 5682: 5676: 5672: 5668: 5665: 5662: 5659: 5654: 5650: 5646: 5635: 5634: 5607: 5596: 5593: 5590: 5587: 5583: 5579: 5568: 5567: 5546: 5535: 5532: 5522: 5518: 5517: 5514: 5511: 5493:Main article: 5490: 5487: 5482: 5479: 5376: 5373: 5348:Main article: 5345: 5342: 5340: 5337: 5306: 5305: 5268: 5267: 5191: 5188: 5180: 5179: 5168: 5162: 5159: 5155: 5150: 5146: 5140: 5137: 5116: 5112: 5108: 5104: 5101: 5097: 5077: 5076: 5065: 5062: 5059: 5056: 5053: 5050: 5047: 5044: 5041: 5038: 5035: 5032: 4998:Main article: 4995: 4992: 4991: 4990: 4975: 4971: 4967: 4962: 4958: 4954: 4950: 4945: 4941: 4937: 4933: 4930: 4927: 4924: 4918: 4915: 4909: 4905: 4902: 4901: 4897: 4893: 4888: 4884: 4879: 4875: 4871: 4868: 4865: 4861: 4857: 4852: 4848: 4844: 4840: 4837: 4834: 4831: 4825: 4822: 4816: 4812: 4809: 4808: 4803: 4799: 4794: 4791: 4788: 4782: 4778: 4775: 4772: 4766: 4762: 4757: 4753: 4747: 4744: 4740: 4737: 4731: 4728: 4722: 4720: 4672: 4671: 4660: 4657: 4654: 4651: 4647: 4643: 4639: 4633: 4629: 4626: 4621: 4617: 4613: 4610: 4604: 4600: 4595: 4591: 4585: 4582: 4578: 4575: 4569: 4566: 4495:, but it does 4474:) = log 4427: 4424: 4392:thermodynamics 4382: 4379: 4378: 4377: 4366: 4363:span the space 4327: 4324: 4320: 4319: 4308: 4304: 4301: 4298: 4293: 4289: 4285: 4282: 4279: 4276: 4273: 4268: 4264: 4260: 4255: 4251: 4247: 4244: 4226: 4225: 4214: 4206: 4202: 4197: 4191: 4187: 4183: 4178: 4173: 4170: 4167: 4163: 4159: 4156: 4136:exponentiating 4127: 4123: 4097:tensor product 4054:obstructions. 3943: 3940: 3938: 3935: 3927: 3926: 3914: 3852: 3849: 3804: 3803: 3792: 3788: 3783: 3780: 3775: 3771: 3768: 3765: 3762: 3759: 3739: 3738: 3727: 3724: 3721: 3717: 3711: 3708: 3703: 3697: 3694: 3690: 3684: 3680: 3666: 3665: 3650: 3645: 3642: 3637: 3634: 3632: 3628: 3624: 3620: 3619: 3613: 3610: 3606: 3601: 3598: 3596: 3592: 3588: 3584: 3583: 3563: 3554: 3527:linear density 3504: 3501: 3467: 3433: 3430: 3423: 3420: 3417: 3339: 3334:, and putting 3319: 3238: 3235: 3233: 3230: 3190: 3137:Joseph Fourier 3118:Simeon Poisson 3094: 3091: 3090: 3089: 3071: 3066: 3063: 3058: 3054: 3051: 3032: 3021: 3013: 3009: 3005: 3000: 2997: 2991: 2987: 2984: 2965: 2954: 2948: 2944: 2940: 2935: 2931: 2928: 2909: 2898: 2893: 2889: 2886: 2882: 2876: 2872: 2869: 2836: 2833: 2832: 2831: 2775: 2771: 2765: 2760:In economics, 2758: 2738: 2735: 2727: 2713: 2650: 2647: 2642: 2639: 2601:Main article: 2598: 2595: 2552: 2545: 2536: 2529: 2520: 2513: 2504: 2495: 2486: 2453: 2427: 2410: 2407: 2391: 2390: 2361: 2358: 2352: 2349: 2329: 2309: 2308: 2265: 2248: 2227: 2224: 2220:1 N = 1 kg⋅m⋅s 2177:multiplication 2164: 2161: 2157: 2156: 2139: 2109: 2070: 2063: 2022: 2015: 2009: 2003: 1989: 1981: 1974: 1967: 1960: 1937: 1928: 1919: 1910: 1897: 1846: 1843: 1842: 1841: 1830: 1823: 1819: 1809: 1806: 1802: 1792: 1789: 1785: 1775: 1771: 1765: 1757: 1754: 1748: 1740: 1733: 1727: 1719: 1716: 1710: 1700: 1693: 1685: 1672: 1669: 1666: 1663: 1640: 1639: 1628: 1623: 1620: 1614: 1606: 1599: 1593: 1583: 1580: 1576: 1570: 1564: 1557: 1550: 1544: 1536: 1533: 1527: 1518: 1505: 1502: 1499: 1496: 1473: 1472: 1461: 1456: 1449: 1444: 1436: 1428: 1425: 1422: 1419: 1396: 1395: 1384: 1379: 1372: 1366: 1358: 1355: 1349: 1343: 1337: 1330: 1323: 1317: 1309: 1306: 1300: 1291: 1278: 1275: 1272: 1269: 1246: 1245: 1234: 1229: 1222: 1216: 1208: 1205: 1199: 1193: 1188: 1183: 1178: 1171: 1164: 1161: 1155: 1149: 1141: 1133: 1130: 1127: 1124: 1101: 1100: 1089: 1084: 1077: 1074: 1068: 1060: 1057: 1051: 1045: 1038: 1032: 1023: 1016: 1009: 1006: 1000: 991: 978: 975: 972: 969: 946: 945: 934: 929: 922: 915: 912: 906: 900: 895: 888: 885: 879: 873: 868: 863: 855: 847: 844: 841: 838: 815: 814: 803: 798: 791: 788: 782: 776: 770: 763: 756: 753: 747: 738: 725: 722: 719: 716: 693: 692: 681: 676: 669: 666: 660: 654: 648: 643: 637: 624: 621: 618: 615: 591: 588: 580:1 in = 2.54 cm 559:is known as a 531:is known as a 437: 436: 423: 417: 409: 403: 395: 389: 381: 375: 367: 361: 353: 347: 339: 333: 327: 324: 321: 318: 291: 290: 155: 152: 148:Joseph Fourier 127:equations and 26: 9: 6: 4: 3: 2: 13792: 13781: 13778: 13776: 13773: 13771: 13768: 13766: 13763: 13761: 13758: 13756: 13753: 13752: 13750: 13735: 13734: 13730: 13728: 13727: 13718: 13717: 13714: 13708: 13705: 13703: 13702:2019 revision 13700: 13698: 13695: 13693: 13690: 13689: 13687: 13683: 13665: 13663: 13660: 13658: 13655: 13649: 13645: 13643: 13640: 13639: 13635: 13633: 13630: 13628: 13625: 13620: 13618: 13615: 13614: 13610: 13608: 13605: 13603: 13598: 13593: 13591: 13588: 13587: 13583: 13581: 13578: 13576: 13571: 13558: 13556: 13553: 13552: 13548: 13546: 13543: 13541: 13538: 13533: 13531: 13528: 13527: 13523: 13521: 13518: 13516: 13513: 13499: 13497: 13494: 13493: 13489: 13487: 13484: 13482: 13479: 13474: 13472: 13469: 13468: 13462: 13457: 13455: 13453: 13448: 13445: 13442: 13441: 13438: 13433: 13429: 13425: 13421: 13417: 13410: 13405: 13403: 13398: 13396: 13391: 13390: 13387: 13375: 13372: 13370: 13367: 13365: 13362: 13361: 13359: 13355: 13349: 13346: 13344: 13341: 13339: 13336: 13335: 13333: 13331:List articles 13329: 13323: 13320: 13318: 13315: 13313: 13310: 13308: 13305: 13303: 13300: 13298: 13295: 13293: 13290: 13288: 13285: 13283: 13280: 13279: 13277: 13273: 13263: 13260: 13258: 13255: 13253: 13250: 13248: 13245: 13243: 13240: 13238: 13235: 13233: 13230: 13228: 13225: 13223: 13220: 13219: 13217: 13215:South America 13213: 13207: 13204: 13202: 13199: 13197: 13194: 13192: 13189: 13187: 13184: 13182: 13179: 13177: 13174: 13173: 13171: 13169:North America 13167: 13161: 13158: 13156: 13153: 13151: 13150:South African 13148: 13146: 13143: 13141: 13138: 13136: 13133: 13131: 13128: 13126: 13123: 13121: 13118: 13116: 13113: 13111: 13108: 13106: 13103: 13101: 13098: 13096: 13093: 13092: 13090: 13086: 13080: 13077: 13075: 13072: 13070: 13067: 13065: 13062: 13060: 13057: 13055: 13052: 13050: 13047: 13045: 13042: 13040: 13037: 13035: 13032: 13030: 13027: 13025: 13022: 13020: 13017: 13015: 13012: 13010: 13007: 13005: 13002: 13000: 12997: 12995: 12992: 12990: 12987: 12985: 12982: 12981: 12979: 12975: 12969: 12966: 12964: 12961: 12959: 12956: 12954: 12951: 12949: 12946: 12944: 12941: 12939: 12936: 12934: 12931: 12929: 12926: 12924: 12921: 12919: 12916: 12914: 12911: 12909: 12906: 12904: 12901: 12899: 12898:Luxembourgian 12896: 12894: 12891: 12889: 12886: 12884: 12881: 12879: 12876: 12874: 12871: 12867: 12864: 12863: 12862: 12859: 12857: 12854: 12850: 12847: 12845: 12842: 12841: 12840: 12837: 12835: 12832: 12830: 12827: 12823: 12820: 12818: 12815: 12814: 12813: 12810: 12808: 12805: 12803: 12800: 12798: 12795: 12793: 12790: 12788: 12785: 12784: 12782: 12778: 12772: 12771:gravitational 12769: 12766: 12763: 12760: 12757: 12754: 12751: 12750: 12748: 12744: 12741: 12737: 12726: 12723: 12721: 12718: 12716: 12713: 12712: 12710: 12706: 12700: 12697: 12695: 12692: 12690: 12687: 12685: 12682: 12681: 12679: 12675: 12672: 12668: 12658: 12655: 12653: 12650: 12648: 12645: 12643: 12640: 12638: 12635: 12633: 12630: 12629: 12627: 12625: 12621: 12614: 12611: 12609: 12606: 12604: 12601: 12599: 12596: 12594: 12591: 12589: 12588:Apothecaries' 12586: 12585: 12583: 12579: 12571: 12568: 12567: 12566: 12563: 12561: 12558: 12556: 12553: 12551: 12548: 12547: 12545: 12541: 12538: 12534: 12530: 12523: 12518: 12516: 12511: 12509: 12504: 12503: 12500: 12492: 12491: 12485: 12481: 12477: 12473: 12472:Sixty Symbols 12469: 12464: 12462: 12459: 12457: 12453: 12450: 12447: 12445: 12442: 12440: 12437: 12435: 12432: 12430: 12427: 12426: 12416: 12412: 12408: 12402: 12398: 12393: 12392: 12383: 12377: 12373: 12368: 12365: 12359: 12355: 12350: 12348: 12344: 12340: 12336: 12333: 12330: 12326: 12322: 12318: 12314: 12310: 12305: 12301: 12297: 12292: 12288: 12284: 12280: 12275: 12271: 12266: 12263: 12259: 12255: 12251: 12246: 12243: 12239: 12235: 12231: 12226: 12222: 12217: 12213: 12209: 12205: 12201: 12197: 12193: 12189: 12188: 12182: 12179: 12175: 12171: 12167: 12163: 12159: 12154: 12151: 12145: 12141: 12136: 12135: 12128: 12124: 12120: 12115: 12111: 12107: 12102: 12099: 12095: 12091: 12087: 12082: 12079: 12075: 12070: 12065: 12061: 12057: 12053: 12049: 12044: 12041: 12035: 12031: 12027: 12023: 12020: 12016: 12012: 12008: 12004: 12000: 11994: 11990: 11986: 11982: 11978: 11974: 11973: 11967: 11964: 11958: 11954: 11953: 11947: 11945: 11940: 11934: 11930: 11926: 11922: 11918: 11915: 11909: 11905: 11900: 11896: 11895: 11889: 11882: 11877: 11872: 11868: 11864: 11857: 11852: 11848: 11843: 11839: 11835: 11831: 11827: 11823: 11819: 11815: 11811: 11807: 11804: 11798: 11794: 11790: 11786: 11782: 11778: 11773: 11769: 11764: 11760: 11756: 11752: 11748: 11743: 11740: 11736: 11732: 11728: 11723: 11720: 11714: 11710: 11706: 11702: 11701: 11690: 11686: 11680: 11672: 11668: 11663: 11658: 11655:(5): 053002. 11654: 11650: 11646: 11639: 11632: 11626: 11611: 11607: 11601: 11586: 11582: 11576: 11574: 11558: 11554: 11548: 11533: 11529: 11523: 11508: 11504: 11498: 11487: 11483: 11479: 11475: 11473:9789811242380 11469: 11465: 11461: 11457: 11450: 11446: 11440: 11429: 11425: 11421: 11414: 11413: 11405: 11399: 11394: 11386: 11382: 11378: 11372: 11368: 11364: 11360: 11353: 11345: 11339: 11331: 11324: 11316: 11309: 11298: 11294: 11287: 11280: 11269: 11265: 11261: 11258:(12): 11–22. 11257: 11253: 11246: 11239: 11231: 11225: 11221: 11217: 11212: 11207: 11203: 11196: 11188: 11184: 11180: 11179: 11171: 11163: 11159: 11155: 11151: 11147: 11143: 11142:IEEE Software 11136: 11128: 11124: 11120: 11116: 11112: 11108: 11101: 11093: 11089: 11086:(3): 93–111. 11085: 11081: 11074: 11067: 11061: 11057: 11050: 11043: 11037: 11033: 11026: 11019: 11013: 11008: 11007: 10998: 10989: 10984: 10977: 10969: 10967:9781437915594 10963: 10956: 10955: 10947: 10945: 10929: 10925: 10917: 10911:, p. 256 10910: 10905: 10897: 10893: 10889: 10885: 10881: 10877: 10870: 10863: 10856: 10855:Bridgman 1922 10851: 10843: 10837: 10832: 10827: 10820: 10815: 10808: 10803: 10788: 10784: 10777: 10769: 10768: 10760: 10753: 10749: 10744: 10736: 10735: 10727: 10720: 10714: 10706: 10701: 10697: 10690: 10682: 10675: 10668: 10663: 10657: 10653: 10652: 10644: 10637: 10631: 10627: 10620: 10611: 10603: 10599: 10595: 10591: 10584: 10582: 10573: 10569: 10565: 10561: 10554: 10546: 10540: 10536: 10531: 10530: 10521: 10513: 10508: 10507: 10498: 10490: 10488:9780073138350 10484: 10480: 10476: 10469: 10451: 10444: 10443: 10438: 10432: 10425: 10421: 10417: 10413: 10409: 10405: 10400: 10395: 10391: 10387: 10380: 10378: 10369: 10363: 10359: 10355: 10351: 10347: 10340: 10325: 10319: 10312: 10311: 10306: 10300: 10298: 10289: 10285: 10281: 10277: 10276:Physics Today 10272: 10264: 10262: 10257: 10248: 10245: 10243: 10240: 10238: 10235: 10233: 10230: 10229: 10221: 10218: 10215: 10212: 10210: 10207: 10205: 10202: 10199: 10196: 10194: 10191: 10189: 10186: 10185: 10179: 10176: 10171: 10167: 10159: 10155: 10147: 10139: 10132: 10124: 10120: 10096: 10093: 10088: 10085: 10082: 10077: 10073: 10069: 10061: 10056: 10052: 10046: 10041: 10037: 10029: 10023: 10019: 10006: 9999: 9978: 9973: 9962: 9955: 9950: 9933: 9926: 9921: 9914: 9895: 9880: 9875: 9864: 9845: 9841: 9834: 9830: 9823: 9819: 9815: 9812: 9805: 9804: 9803: 9789: 9783: 9781: 9775: 9757: 9753: 9719: 9714: 9708: 9705: 9696: 9692: 9680: 9672: 9668: 9664: 9658: 9649: 9644: 9638: 9635: 9613: 9609: 9605: 9602: 9599: 9579: 9576: 9573: 9550: 9546: 9536: 9531: 9525: 9522: 9510: 9505: 9501: 9497: 9494: 9491: 9487: 9477: 9472: 9466: 9463: 9451: 9446: 9442: 9438: 9435: 9432: 9428: 9418: 9413: 9410: 9401: 9396: 9392: 9388: 9385: 9378: 9377: 9376: 9359: 9353: 9350: 9347: 9341: 9337: 9333: 9330: 9327: 9321: 9318: 9300: 9296: 9292: 9288: 9277: 9262: 9232: 9228: 9188: 9165: 9161: 9153: 9133: 9125: 9105: 9097: 9077: 9069: 9038: 9017: 9009: 8993: 8989: 8981: 8961: 8953: 8933: 8925: 8894: 8873: 8865: 8845: 8837: 8821: 8817: 8809: 8789: 8781: 8750: 8729: 8721: 8701: 8693: 8673: 8665: 8649: 8645: 8637: 8606: 8575: 8545: 8515: 8485: 8483: 8482: 8479: 8475: 8469: 8425: 8420: 8417: 8411: 8405: 8401: 8398: 8394: 8389: 8384: 8379: 8375: 8370: 8364: 8357: 8353: 8346: 8341: 8331: 8329: 8325: 8319: 8317: 8301: 8297: 8293: 8260: 8257: 8251: 8244: 8240: 8236: 8225: 8218: 8215: 8208: 8207: 8206: 8186: 8159: 8131: 8124: 8121: 8111: 8107: 8103: 8092: 8085: 8080: 8076: 8068: 8051: 8048: 8042: 8039: 8032: 8027: 8023: 8015: 8014: 8013: 8005: 8002: 7997: 7996: 7992: 7989: 7984: 7983: 7979: 7976: 7971: 7970: 7966: 7963: 7943: 7935: 7934: 7930: 7927: 7910: 7907: 7897: 7896: 7892: 7889: 7886: 7885: 7882: 7880: 7875: 7871: 7869: 7865: 7864:inertial mass 7855: 7851: 7827: 7823: 7820: 7819: 7818:cross product 7814: 7813: 7812: 7809: 7805: 7798: 7791: 7769: 7764: 7739: 7736: 7722: 7715: 7710: 7685: 7682: 7668: 7661: 7656: 7631: 7628: 7614: 7609: 7586: 7585: 7584: 7536: 7501: 7491: 7475: 7472: 7469: 7466: 7463: 7460: 7457: 7454: 7432: 7429: 7426: 7423: 7420: 7417: 7414: 7389: 7384: 7371: 7368: 7355: 7348: 7345: 7342: 7337: 7324: 7321: 7308: 7303: 7289: 7288: 7287: 7270: 7265: 7261: 7254: 7245: 7238: 7229: 7225: 7222: 7215: 7214: 7213: 7206: 7180: 7149: 7114: 7085: 7076: 7063: 7055: 7054: 7053: 7050: 7036: 7022: 7020: 7015: 7013: 7009: 6999: 6997: 6996:Réaumur scale 6993: 6992:Rankine scale 6985: 6984: 6983: 6980: 6978: 6970: 6969: 6968: 6966: 6965:absolute zero 6961: 6958: 6956: 6952: 6948: 6944: 6940: 6932: 6928: 6924: 6921: 6920: 6915: 6914: 6913: 6911: 6907: 6903: 6899: 6891: 6887: 6884: 6881: 6878: 6877: 6876: 6872: 6870: 6863: 6848: 6844: 6842: 6838: 6832: 6828: 6824: 6820: 6816: 6812: 6808: 6807:type checking 6792: 6786: 6780: 6774: 6770: 6756: 6753: 6750: 6747: 6744: 6741: 6734: 6731: 6730: 6726: 6721: 6715: 6709: 6705: 6691: 6688: 6684: 6680: 6677: 6674: 6667: 6664: 6663: 6658: 6652: 6648: 6634: 6630: 6626: 6623: 6620: 6617: 6610: 6607: 6606: 6603:Nomenclature 6597: 6593: 6587: 6580: 6575: 6571: 6557: 6554: 6547: 6544: 6543: 6540: 6535: 6530: 6525: 6520: 6515: 6511: 6497: 6494: 6491: 6484: 6481: 6480: 6476: 6472: 6453: 6448: 6444: 6440: 6430: 6413: 6408: 6404: 6400: 6394: 6387: 6384: 6383: 6380: 6375: 6369: 6363: 6359: 6345: 6341: 6337: 6334: 6331: 6327: 6323: 6316: 6315: 6310: 6304: 6298: 6292: 6288: 6274: 6271: 6268: 6265: 6262: 6255: 6251: 6248:Nomenclature 6242: 6238: 6232: 6224: 6220: 6215: 6208: 6203: 6198: 6194: 6180: 6177: 6174: 6171: 6168: 6165: 6162: 6159: 6156: 6149: 6148: 6145: 6140: 6134: 6129: 6124: 6119: 6113: 6108: 6103: 6098: 6094: 6080: 6076: 6072: 6067: 6063: 6059: 6056: 6051: 6047: 6043: 6036: 6035: 6031: 6027: 6022: 6017: 6012: 6008: 5994: 5991: 5984: 5980: 5977: 5972: 5967: 5962: 5957: 5952: 5947: 5943: 5938: 5934: 5920: 5917: 5914: 5911: 5908: 5905: 5902: 5895: 5892: 5891: 5888: 5883: 5877: 5871: 5865: 5861: 5847: 5844: 5841: 5838: 5835: 5828: 5825: 5824: 5821: 5816: 5811: 5806: 5801: 5796: 5792: 5778: 5774: 5768: 5764: 5760: 5757: 5754: 5751: 5746: 5742: 5738: 5731: 5730: 5727: 5722: 5717: 5712: 5707: 5702: 5698: 5684: 5680: 5674: 5670: 5666: 5663: 5660: 5657: 5652: 5648: 5644: 5637: 5636: 5633: 5628: 5622: 5617: 5612: 5608: 5594: 5591: 5588: 5585: 5581: 5577: 5570: 5569: 5566: 5561: 5556: 5551: 5547: 5533: 5530: 5523: 5519: 5516:Nomenclature 5510: 5506: 5500: 5496: 5486: 5478: 5474: 5467: 5460: 5454: 5448: 5442: 5435: 5433: 5431: 5425: 5423: 5417: 5415: 5409: 5404: 5401: 5395: 5388: 5382: 5372: 5370: 5364: 5358: 5351: 5336: 5333: 5330: 5324: 5318: 5312: 5303: 5299: 5295: 5294: 5293: 5291: 5286: 5280: 5274: 5265: 5262: 5258: 5254: 5253: 5252: 5249: 5243: 5239: 5234: 5230: 5225: 5223: 5218: 5216: 5212: 5208: 5204: 5197: 5187: 5184: 5166: 5153: 5144: 5138: 5135: 5110: 5106: 5095: 5087: 5086: 5085: 5083: 5060: 5054: 5051: 5045: 5039: 5036: 5033: 5030: 5023: 5022: 5021: 5018: 5013: 5008: 5001: 4973: 4965: 4960: 4956: 4952: 4948: 4943: 4939: 4935: 4931: 4928: 4925: 4922: 4916: 4913: 4903: 4891: 4886: 4873: 4855: 4850: 4846: 4842: 4838: 4835: 4832: 4829: 4823: 4820: 4810: 4801: 4780: 4773: 4764: 4755: 4745: 4742: 4735: 4729: 4726: 4711: 4710: 4709: 4706: 4701: 4696: 4683: 4678: 4658: 4655: 4652: 4641: 4631: 4624: 4619: 4615: 4611: 4602: 4593: 4583: 4580: 4573: 4567: 4564: 4554: 4553: 4552: 4549: 4542: 4538: 4528: 4522: 4517: 4512: 4509: 4503: 4498: 4493: 4487: 4481: 4477: 4473: 4469: 4462: 4460: 4456: 4452: 4448: 4447:trigonometric 4444: 4440: 4437:arguments to 4436: 4432: 4423: 4421: 4417: 4413: 4401: 4397: 4393: 4389: 4375: 4367: 4364: 4360: 4359: 4358: 4355: 4352: 4350: 4346: 4342: 4338: 4333: 4323: 4306: 4302: 4299: 4291: 4287: 4283: 4280: 4277: 4274: 4271: 4266: 4262: 4258: 4253: 4249: 4242: 4235: 4234: 4233: 4231: 4212: 4204: 4200: 4189: 4185: 4176: 4171: 4168: 4165: 4161: 4157: 4154: 4147: 4146: 4145: 4142: 4137: 4130: 4118: 4113: 4108: 4106: 4105:dimensionless 4102: 4098: 4093: 4089: 4085: 4079: 4073: 4067: 4061: 4055: 4053: 4048: 4042: 4035: 4033: 4029: 4025: 4020: 4018: 4012: 4008: 4004: 3988: 3983: 3981: 3977: 3968: 3959: 3955: 3954:abelian group 3949: 3934: 3932: 3924: 3920: 3915: 3912: 3908: 3905: 3900: 3899: 3898: 3891: 3885: 3879: 3873: 3867: 3857: 3848: 3846: 3842: 3836: 3833: 3829: 3823: 3817: 3810: 3790: 3786: 3781: 3778: 3773: 3769: 3766: 3763: 3760: 3757: 3750: 3749: 3748: 3745: 3725: 3722: 3719: 3715: 3709: 3706: 3701: 3695: 3692: 3688: 3682: 3678: 3671: 3670: 3669: 3648: 3643: 3640: 3635: 3633: 3626: 3622: 3611: 3608: 3604: 3599: 3597: 3590: 3586: 3574: 3573: 3572: 3570: 3562: 3553: 3547: 3541: 3537: 3532: 3528: 3523: 3519: 3514: 3510: 3500: 3497: 3490: 3487: 3481: 3465: 3454: 3431: 3428: 3421: 3418: 3415: 3404: 3398: 3393: 3365:The variable 3363: 3361: 3357: 3351: 3345: 3338: 3332: 3328: 3325: 3318: 3312: 3294: 3288: 3282: 3276: 3270: 3264: 3258: 3248: 3244: 3229: 3227: 3223: 3218: 3216: 3212: 3211:Lord Rayleigh 3206: 3189: 3186: 3183:in which the 3182: 3181:Coulomb's law 3174: 3169: 3165: 3162:in which the 3161: 3156: 3152: 3150: 3148: 3144: 3138: 3133: 3131: 3127: 3123: 3119: 3115: 3110: 3108: 3104: 3100: 3086: 3069: 3064: 3061: 3056: 3036: 3033: 3019: 3011: 3007: 3003: 2998: 2989: 2969: 2966: 2952: 2946: 2942: 2938: 2933: 2913: 2912:Froude number 2910: 2896: 2891: 2887: 2884: 2880: 2874: 2854: 2851: 2850: 2849: 2846: 2842: 2828: 2822: 2816: 2812: 2806: 2800: 2794: 2790: 2786: 2780: 2779:bond duration 2776: 2772: 2769: 2766: 2763: 2759: 2756: 2752: 2751: 2750: 2748: 2744: 2734: 2730: 2726: 2720: 2716: 2712: 2706: 2700: 2693: 2686: 2680: 2675: 2672: 2666: 2662: 2659: 2655:volume of an 2646: 2638: 2612: 2611: 2604: 2594: 2591: 2586: 2578: 2566: 2561: 2559: 2551: 2544: 2535: 2528: 2519: 2512: 2503: 2494: 2485: 2480: 2475: 2469: 2465: 2461: 2457: 2454:One may take 2452: 2450: 2449:abelian group 2443: 2439: 2435: 2431: 2426: 2422: 2416: 2406: 2403: 2398: 2396: 2378: 2359: 2356: 2350: 2347: 2335: 2330: 2317: 2314: 2313: 2312: 2299: 2294: 2290: 2286: 2282: 2278: 2274: 2271: 2266: 2263: 2258: 2254: 2249: 2245: 2240: 2239: 2238: 2235: 2223: 2217: 2213: 2208: 2206: 2202: 2197: 2193: 2188: 2186: 2185:juxtaposition 2182: 2178: 2174: 2170: 2160: 2154: 2150: 2147: 2143: 2140: 2136: 2130: 2124: 2118: 2113: 2110: 2106: 2100: 2094: 2088: 2083: 2079: 2075: 2071: 2068: 2064: 2060: 2054: 2048: 2042: 2037: 2032: 2025: 2021: 2014: 2008: 2002: 1999: 1995: 1990: 1984: 1980: 1973: 1966: 1959: 1955: 1951: 1946: 1940: 1936: 1927: 1918: 1909: 1903: 1898: 1895: 1891: 1887: 1886: 1885: 1882: 1880: 1879:Lord Rayleigh 1876: 1872: 1868: 1864: 1860: 1856: 1852: 1828: 1763: 1755: 1752: 1731: 1717: 1714: 1683: 1670: 1667: 1664: 1661: 1654: 1653: 1652: 1649: 1645: 1626: 1621: 1618: 1597: 1568: 1548: 1534: 1531: 1516: 1503: 1500: 1497: 1494: 1487: 1486: 1485: 1482: 1478: 1459: 1442: 1434: 1426: 1423: 1420: 1417: 1410: 1409: 1408: 1405: 1401: 1382: 1370: 1356: 1353: 1341: 1321: 1307: 1304: 1289: 1276: 1273: 1270: 1267: 1260: 1259: 1258: 1255: 1251: 1232: 1220: 1206: 1203: 1191: 1181: 1162: 1159: 1147: 1139: 1131: 1128: 1125: 1122: 1115: 1114: 1113: 1110: 1106: 1087: 1075: 1072: 1058: 1055: 1043: 1036: 1007: 1004: 989: 976: 973: 970: 967: 960: 959: 958: 955: 951: 932: 913: 910: 898: 886: 883: 871: 861: 853: 845: 842: 839: 836: 829: 828: 827: 824: 820: 801: 789: 786: 774: 754: 751: 736: 723: 720: 717: 714: 707: 706: 705: 702: 698: 679: 667: 664: 652: 635: 622: 619: 616: 613: 606: 605: 604: 601: 597: 587: 583: 577: 571: 569: 568:dimension one 565: 563: 556: 549: 542: 537: 535: 528: 521: 516: 514: 507: 501: 495: 491: 487: 484: 479: 473: 467: 461: 455: 449: 443: 421: 407: 393: 379: 365: 351: 337: 325: 322: 319: 316: 309: 308: 307: 306:is given by 304: 299: 296: 288: 284: 280: 276: 272: 268: 264: 261: 260: 259: 257: 253: 248: 246: 245:Natural units 242: 238: 234: 230: 226: 222: 218: 213: 211: 207: 203: 198: 196: 192: 188: 183: 179: 173: 168: 161: 151: 149: 145: 141: 136: 134: 130: 126: 122: 118: 114: 110: 105: 103: 99: 95: 91: 87: 83: 82: 81:Commensurable 77: 75: 71: 67: 63: 59: 55: 51: 47: 43: 39: 35: 30: 19: 13731: 13719: 13647: 13449: 13312:Mesopotamian 13206:Puerto Rican 12603:Astronomical 12489: 12471: 12396: 12371: 12353: 12342: 12312: 12308: 12299: 12295: 12283:Tao, Terence 12273: 12269: 12253: 12249: 12233: 12229: 12203: 12199: 12186: 12161: 12157: 12133: 12122: 12118: 12109: 12105: 12089: 12085: 12051: 12047: 12029: 12002: 11998: 11971: 11951: 11928: 11906:, Springer, 11903: 11893: 11866: 11862: 11821: 11817: 11792: 11780: 11776: 11750: 11746: 11730: 11726: 11708: 11683:Siano ( 11679: 11652: 11648: 11638: 11631:Huntley 1967 11625: 11613:. Retrieved 11609: 11600: 11588:. Retrieved 11584: 11560:. Retrieved 11556: 11547: 11535:. Retrieved 11531: 11522: 11510:. Retrieved 11506: 11497: 11455: 11439: 11411: 11404: 11393: 11358: 11352: 11338: 11323: 11308: 11292: 11279: 11255: 11251: 11238: 11201: 11195: 11177: 11170: 11148:(3): 21–27. 11145: 11141: 11135: 11110: 11106: 11100: 11083: 11080:Comput. Lang 11079: 11073: 11055: 11049: 11031: 11025: 11005: 10997: 10976: 10953: 10931:. Retrieved 10927: 10916: 10904: 10879: 10875: 10862: 10850: 10841: 10835: 10826: 10814: 10802: 10790:. Retrieved 10786: 10776: 10766: 10759: 10743: 10733: 10726: 10713: 10695: 10689: 10680: 10674: 10665: 10650: 10643: 10625: 10619: 10610: 10593: 10589: 10563: 10559: 10553: 10528: 10520: 10505: 10497: 10478: 10468: 10457:, retrieved 10450:the original 10441: 10431: 10389: 10385: 10349: 10339: 10327:. Retrieved 10309: 10282:(9): 42–47. 10279: 10275: 10172: 10165: 10157: 10153: 10145: 10137: 10133: 10122: 10118: 10004: 9997: 9994: 9787: 9784: 9776: 9565: 9298: 9294: 9290: 9286: 9275: 9260: 9230: 9226: 9184: 8473: 8467: 8423: 8421: 8415: 8409: 8403: 8399: 8396: 8392: 8387: 8382: 8377: 8373: 8368: 8362: 8355: 8351: 8343: 8326:, with unit 8320: 8281: 8150: 8011: 7876: 7872:proportional 7867: 7863: 7861: 7852: 7831: 7825: 7816: 7810: 7803: 7796: 7789: 7786: 7492: 7406: 7285: 7207: 7072: 7051: 7028: 7016: 7007: 7005: 6989: 6981: 6976: 6974: 6962: 6959: 6954: 6946: 6942: 6938: 6936: 6930: 6926: 6917: 6910:vector space 6905: 6902:affine space 6897: 6895: 6889: 6888:but one may 6873: 6865: 6862:Affine space 6845: 6804: 6790: 6788:= velocity, 6784: 6778: 6772: 6719: 6713: 6707: 6656: 6650: 6595: 6591: 6573: 6533: 6523: 6513: 6474: 6379:displacement 6373: 6367: 6361: 6308: 6302: 6300:= velocity, 6296: 6290: 6240: 6236: 6218: 6213: 6206: 6196: 6138: 6133:permeability 6127: 6123:permittivity 6117: 6106: 6096: 6020: 6010: 5970: 5960: 5950: 5936: 5881: 5875: 5869: 5867:= pressure, 5863: 5826:Ideal gases 5814: 5804: 5794: 5720: 5710: 5700: 5626: 5620: 5610: 5559: 5549: 5508: 5504: 5484: 5472: 5465: 5458: 5452: 5446: 5440: 5436: 5429: 5421: 5413: 5405: 5399: 5393: 5386: 5378: 5362: 5356: 5353: 5334: 5328: 5322: 5316: 5310: 5307: 5301: 5297: 5284: 5278: 5272: 5269: 5263: 5260: 5256: 5247: 5241: 5232: 5229:displacement 5226: 5221: 5219: 5206: 5202: 5200: 5185: 5181: 5078: 5016: 5006: 5003: 4704: 4699: 4694: 4676: 4673: 4550: 4540: 4536: 4526: 4520: 4513: 4507: 4501: 4496: 4491: 4485: 4479: 4478:− log 4475: 4471: 4467: 4463: 4433: 4429: 4384: 4373: 4362: 4356: 4353: 4343:: they must 4329: 4321: 4230:commensurate 4227: 4140: 4128: 4116: 4109: 4091: 4087: 4083: 4077: 4071: 4065: 4059: 4056: 4046: 4043: 4036: 4021: 4010: 4006: 4002: 3984: 3966: 3951: 3928: 3922: 3918: 3910: 3906: 3903: 3889: 3883: 3877: 3871: 3865: 3862: 3843:such as the 3837: 3831: 3827: 3821: 3815: 3808: 3805: 3743: 3740: 3667: 3560: 3551: 3545: 3539: 3530: 3521: 3512: 3506: 3495: 3491: 3485: 3482: 3452: 3402: 3396: 3391: 3364: 3349: 3343: 3336: 3330: 3326: 3323: 3316: 3310: 3292: 3286: 3280: 3274: 3268: 3262: 3256: 3246: 3240: 3225: 3221: 3219: 3214: 3207: 3187: 3171:is taken as 3167: 3153: 3146: 3142: 3134: 3129: 3125: 3111: 3096: 3084: 2968:Euler number 2838: 2826: 2820: 2814: 2810: 2804: 2798: 2792: 2788: 2784: 2740: 2728: 2724: 2718: 2714: 2710: 2704: 2698: 2691: 2684: 2678: 2670: 2664: 2657: 2652: 2644: 2641:Applications 2608: 2606: 2587: 2579: 2562: 2558:sanity check 2549: 2542: 2533: 2526: 2517: 2510: 2501: 2492: 2483: 2478: 2476: 2472: 2467: 2463: 2459: 2455: 2446: 2441: 2437: 2433: 2429: 2424: 2399: 2392: 2333: 2310: 2298:acceleration 2292: 2288: 2284: 2280: 2276: 2272: 2269: 2256: 2252: 2243: 2236: 2229: 2209: 2189: 2181:centered dot 2166: 2158: 2134: 2128: 2122: 2116: 2104: 2098: 2092: 2086: 2058: 2052: 2046: 2040: 2030: 2023: 2019: 2012: 2006: 2000: 1997: 1993: 1982: 1978: 1971: 1964: 1957: 1953: 1949: 1938: 1934: 1925: 1916: 1907: 1901: 1883: 1850: 1848: 1647: 1641: 1480: 1474: 1403: 1397: 1253: 1247: 1144:displacement 1108: 1102: 953: 947: 858:acceleration 822: 816: 700: 697:acceleration 694: 599: 593: 590:Simple cases 584: 572: 567: 560: 554: 547: 540: 532: 526: 519: 511: 505: 502: 477: 471: 465: 459: 453: 447: 441: 438: 302: 292: 255: 249: 240: 232: 228: 214: 199: 186: 181: 177: 171: 164: 143: 139: 137: 129:computations 120: 116: 106: 89: 80: 79: 78: 41: 31: 29: 13760:Measurement 13176:Costa Rican 13140:Seychellois 13049:Singaporean 12844:Traditional 12699:Metrication 12637:Geometrised 12593:Avoirdupois 12476:Brady Haran 12343:Aeronautics 10770:, p. 5 10737:, Macmillan 10683:, p. 4 10329:1 September 9780:normal form 8360:at a speed 7061:consistent. 6811:Standard ML 6711:= entropy, 6608:Mechanical 6600:Expression 6253:Mechanical 6245:Expression 5521:Mechanical 5513:Expression 5381:Ising model 4532:(3 m) = 9 m 4451:logarithmic 4443:exponential 4408:10 mol 3397:irrelevance 3243:oscillation 3130:homogeneity 3035:Mach number 2676:: being an 2649:Mathematics 2076:, obtain a 1943:, then the 1863:engineering 1644:capacitance 252:SI standard 154:Formulation 34:engineering 13749:Categories 13262:Venezuelan 13247:Paraguayan 13201:Nicaraguan 13079:Vietnamese 13054:Sri Lankan 13039:Philippine 12999:Indonesian 12923:Portuguese 12817:Winchester 12720:Comparison 12670:Background 12608:Electrical 11944:postscript 11698:References 11662:2108.05704 11649:Metrologia 10750:, p. 10719:Pesic 2005 10392:(3): 023, 10214:Similitude 9566:which for 8450:becomes L1 8372:above the 8366:and angle 8358:) = (0, 0) 8338:See also: 7893:Dimension 7019:extensions 6994:, not the 6365:= action, 6234:Momentum, 5946:wave front 5873:= volume, 5375:Formalisms 5251:would be: 4457:, must be 3937:Properties 3456:(equal to 3385:, because 3250:of a mass 2479:expression 2442:subtracted 2413:See also: 2241:position ( 2232:1% = 1/100 2192:base units 2149:parameters 2142:Substitute 2067:base units 298:sans serif 113:inequality 94:quantities 13450:Dimension 13431:Quantity 13257:Uruguayan 13242:Colombian 13232:Brazilian 13222:Argentine 13160:Tanzanian 13130:Mauritian 13100:Ethiopian 13064:Taiwanese 13034:Pakistani 13014:Mongolian 12989:Cambodian 12908:Norwegian 12878:Icelandic 12873:Hungarian 12866:Byzantine 12822:Exchequer 12570:Hong Kong 12415:853154197 12178:206506776 12064:CiteSeerX 12032:, Wiley, 11981:682090763 11975:, Dover, 11869:: 84–99, 11753:: 592–6, 11482:243831207 11398:Hart 1995 11206:CiteSeerX 11187:1476-2986 9876:∼ 9842:θ 9715:θ 9709:⁡ 9665:π 9645:θ 9639:⁡ 9606:π 9580:θ 9526:⁡ 9498:⁡ 9467:⁡ 9439:⁡ 9389:⁡ 9360:θ 9354:⁡ 9334:π 9328:θ 9322:⁡ 8294:π 8261:˙ 8252:η 8237:ρ 8125:˙ 8104:ρ 8077:π 8049:η 8043:˙ 8024:π 7911:˙ 7737:− 7683:− 7629:− 7369:− 7322:− 7226:∝ 7008:direction 6794:= charge 6748:≡ 6689:δ 6678:δ 6624:≡ 6492:ρ 6335:≡ 6306:= force, 6269:≡ 6172:≡ 6163:≡ 6081:μ 6060:≡ 6044:ε 5995:ϕ 5956:intensity 5912:≡ 5842:≡ 5761:≡ 5758:ω 5752:≡ 5743:ω 5667:≡ 5658:≡ 5589:≡ 5344:Constants 5290:converted 5040:× 4966:⋅ 4953:⋅ 4932:⋅ 4926:− 4923:⋅ 4892:⋅ 4839:⋅ 4833:− 4830:⋅ 4774:⋅ 4743:− 4736:⋅ 4698:to be in 4653:⋅ 4612:⋅ 4581:− 4574:⋅ 4516:monomials 4332:mechanics 4326:Mechanics 4288:π 4263:π 4250:π 4186:π 4162:∏ 4086: := 4039:(0, 0, 0) 3972:L × L = L 3895:5 − 3 = 2 3779:ℓ 3707:ℓ 3641:ℓ 3623:π 3587:π 3518:amplitude 3422:κ 3362:as well. 3222:dimension 3126:dimension 3004:ρ 2996:Δ 2892:μ 2881:ρ 2755:P/E ratio 2348:∫ 2190:A set of 2072:By using 1871:variables 1859:chemistry 1805:− 1788:− 1753:− 1715:− 1665:⁡ 1619:− 1579:− 1532:− 1498:⁡ 1435:× 1421:⁡ 1354:− 1305:− 1271:⁡ 1204:− 1182:× 1160:− 1140:× 1126:⁡ 1073:− 1056:− 1005:− 971:⁡ 911:− 884:− 872:× 854:× 840:⁡ 787:− 752:− 718:⁡ 665:− 617:⁡ 534:kinematic 513:geometric 388:Θ 320:⁡ 229:dimension 150:in 1822. 92:physical 52:(such as 13726:Category 13685:See also 13545:kilogram 13343:Obsolete 13338:Humorous 13292:Egyptian 13252:Peruvian 13227:Bolivian 13191:Honduran 13155:Tunisian 13135:Moroccan 13125:Malagasy 13110:Eritrean 13105:Egyptian 13095:Algerian 13024:Nepalese 13004:Japanese 12938:Scottish 12928:Romanian 12829:Estonian 12739:Historic 12715:Overview 12684:Overview 12581:Specific 12478:for the 12452:Archived 12285:(2012). 12028:(1951), 11989:6128830M 11923:(1994), 11881:archived 11812:(1914), 11791:(1922), 11707:(1996), 11615:19 April 11590:19 April 11562:19 April 11537:19 April 11512:19 April 11486:Archived 11428:Archived 11385:53089559 11297:Archived 11268:Archived 11162:22450087 11127:40558757 10896:14833238 10831:Tao 2012 10819:Tao 2012 10807:Tao 2012 10792:15 April 10439:(2012), 10424:15806354 10182:See also 10156:) + sin( 9303:, where 7890:Variable 6927:relative 6665:Thermal 6654:= mass, 6454:⟩ 6441:⟨ 6414:⟩ 6401:⟨ 6385:Thermal 6294:= mass, 6225:in loop 5726:momentum 5716:velocity 5624:= time, 5565:distance 5502:Energy, 5489:SI units 5397:, where 4499:hold if 4441:such as 4126:, ..., π 4107:scalar. 4032:choosing 3958:identity 3232:Examples 3103:Lagrange 2845:pi terms 2796:, where 2629:because 2464:multiply 2430:compared 2262:velocity 2173:division 2153:grouping 2028:, where 1869:of some 950:pressure 564:quantity 536:quantity 515:quantity 496:, since 285:(N) and 221:rational 109:equation 13733:Outline 13662:candela 13564: 13560: 13511:, etc. 13465:symbol 13446:Symbol 13437:SI unit 13374:Modulor 13348:Unusual 13317:Persian 13275:Ancient 13237:Chilean 13196:Mexican 13186:Haitian 13115:Guinean 13019:Myanmar 12958:Swedish 12953:Spanish 12943:Serbian 12933:Russian 12913:Ottoman 12903:Maltese 12893:Latvian 12888:Italian 12834:Finnish 12812:English 12792:Cypriot 12787:Cornish 12694:History 12689:Outline 12624:Natural 12565:Chinese 12543:General 12536:Current 12337:(1920) 12329:2315883 12208:Bibcode 12056:Bibcode 12007:Bibcode 11826:Bibcode 11783:: 55–64 11755:Bibcode 11689:1985-II 10933:15 July 10404:Bibcode 10109:⁠ 10011:⁠ 9772:⁠ 9745:⁠ 9741:⁠ 9628:⁠ 9626:yields 8435: 1 8431: 1 7977:density 7488:⁠ 7447:⁠ 7199:⁠ 7172:⁠ 7168:⁠ 7141:⁠ 7133:⁠ 7106:⁠ 6919:acts on 6831:Fortran 6819:Haskell 6589:Force, 6519:density 6312:= time 6030:voltage 5954:= wave 4700:seconds 4684:is 9.8 4682:gravity 4370:V = L/T 4112:nullity 4095:as the 3536:tension 3448:⁠ 3408:⁠ 3308:; and 3195:Q = TLM 3093:History 2674:-sphere 2434:equated 2373:⁠ 2339:⁠ 2132:, ..., 2102:, ..., 2056:, ..., 1977:, ..., 1932:, ..., 1855:physics 1512:current 1477:voltage 1431:current 562:dynamic 189:is the 125:derived 38:science 13607:kelvin 13600:Θ 13580:ampere 13496:length 13486:second 13452:symbol 13369:N-body 13307:Indian 13282:Arabic 13145:Somali 13120:Libyan 13088:Africa 13059:Syrian 13009:Korean 12994:Indian 12984:Afghan 12948:Slovak 12918:Polish 12856:German 12839:French 12802:Danish 12780:Europe 12746:Metric 12677:Metric 12657:Stoney 12647:Planck 12632:Atomic 12413: 12403: 12378: 12360: 12345:, via 12327: 12302:(A–34) 12276:(4): 5 12200:Nature 12176: 12146: 12066: 12036: 11987: 11979: 11959: 11935: 11910: 11799: 11715: 11685:1985-I 11480: 11470: 11383: 11373: 11226: 11208: 11185: 11160: 11125: 11062: 11038: 11014: 10964: 10894: 10658: 10632: 10541: 10485: 10459:2 June 10422: 10364: 10320: 10175:radian 9272:while 8345:Angles 7887:Symbol 7826:angles 6955:vector 6947:affine 6943:vector 6939:affine 6906:vector 6898:affine 6869:origin 6825:, and 6529:volume 6482:Waves 6144:volume 5893:Waves 5616:action 5450:, and 5426:, and 5308:where 5145:0.3048 5111:0.3048 4784: 4749: 4635: 4587: 4435:Scalar 4390:. In 3987:module 3956:: The 3806:where 3741:where 3569:powers 3509:length 3381:, and 3290:, and 3203:Q = TL 3199:M = TL 3177:M = TL 3082:where 2565:torque 2468:divide 2456:ratios 2354: 2212:newton 2205:volume 2201:length 2194:for a 1861:, and 1282:energy 1105:energy 628:length 552:, and 498:Q = TI 439:where 267:length 227:. The 195:matrix 133:system 68:) and 64:, and 54:length 13520:metre 13460:name 13443:Name 13357:Other 13322:Roman 13302:Hindu 13297:Greek 13181:Cuban 13069:Tatar 13029:Omani 12968:Welsh 12963:Swiss 12883:Irish 12861:Greek 12807:Dutch 12797:Czech 12767:(CGS) 12761:(MTS) 12755:(MKS) 12727:(FPS) 12708:UK/US 12325:JSTOR 12174:S2CID 12140:227–8 12125:(251) 11884:(PDF) 11859:(PDF) 11657:arXiv 11489:(PDF) 11478:S2CID 11452:(PDF) 11431:(PDF) 11416:(PDF) 11381:S2CID 11300:(PDF) 11289:(PDF) 11271:(PDF) 11248:(PDF) 11158:S2CID 11123:S2CID 10983:arXiv 10958:(PDF) 10892:S2CID 10872:(PDF) 10667:well. 10453:(PDF) 10446:(PDF) 10420:S2CID 10394:arXiv 10314:(PDF) 10253:Notes 10121:)cos( 7577:as TL 7557:as TL 6823:OCaml 6142:= 3d 5632:power 5555:force 5238:speed 4341:basis 4028:bases 3962:L = 1 3373:with 3356:group 3173:unity 3107:Turin 2661:-ball 2470:them. 2440:, or 2438:added 2316:force 2216:force 2112:Solve 2034:is a 1509:power 1250:power 1136:force 982:force 819:force 729:speed 596:speed 486:basis 295:roman 281:(Θ), 277:(I), 273:(M), 269:(L), 265:(T), 233:scale 225:power 111:, or 102:units 98:kinds 13636:mol 13632:mole 13530:mass 13463:Unit 13458:Unit 13074:Thai 13044:Pegu 12977:Asia 12615:(US) 12598:Troy 12411:OCLC 12401:ISBN 12376:ISBN 12358:ISBN 12144:ISBN 12034:ISBN 11977:OCLC 11957:ISBN 11933:ISBN 11908:ISBN 11797:ISBN 11713:ISBN 11617:2023 11592:2023 11564:2023 11539:2023 11514:2023 11468:ISBN 11371:ISBN 11224:ISBN 11183:ISSN 11060:ISBN 11036:ISBN 11012:ISBN 10962:ISBN 10935:2014 10845:..." 10794:2017 10656:ISBN 10630:ISBN 10539:ISBN 10483:ISBN 10461:2015 10437:JCGM 10390:2002 10362:ISBN 10331:2021 10318:ISBN 10305:BIPM 10164:exp( 10162:and 10152:cos( 10144:cos( 10142:and 10136:sin( 10117:sin( 10002:and 10000:= −1 9592:and 9307:and 9297:sin( 9293:) + 9289:cos( 9274:cos( 9259:sin( 9229:) = 9225:tan( 8413:and 8328:mole 7993:TLM 7967:TLM 7806:= −1 7801:and 7573:and 7567:as L 7445:and 6827:Rust 6596:TLM 6241:TLM 5966:time 5942:area 5706:mass 5509:TLM 5470:and 5392:~ 1/ 5300:= 5 5282:and 5270:for 5012:unit 4940:0.01 4847:0.01 4781:0.01 4505:and 4489:and 4466:log( 4449:and 4404:6.02 4402:, ≈ 4345:span 4075:and 4063:and 3558:and 2694:− 1) 2499:and 2377:work 2291:) / 2038:and 2018:... 1439:time 1285:time 985:area 957:is 850:mass 732:time 631:time 524:and 289:(J). 271:mass 263:time 241:mass 237:unit 191:rank 165:The 160:Size 117:must 86:kind 62:time 58:mass 36:and 13666:cd 13549:kg 12317:doi 12258:doi 12254:320 12238:doi 12234:320 12216:doi 12166:doi 12112:(6) 12094:doi 12074:doi 12015:doi 11942:As 11871:doi 11842:hdl 11834:doi 11763:doi 11751:372 11735:doi 11667:doi 11460:doi 11420:hdl 11363:doi 11260:doi 11216:doi 11150:doi 11115:doi 11088:doi 10884:doi 10752:156 10700:hdl 10598:doi 10594:311 10568:doi 10564:292 10535:260 10412:doi 10354:doi 10284:doi 10007:= 2 9706:cos 9636:sin 9523:cos 9495:sin 9464:cos 9436:sin 9386:sin 9351:cos 9319:sin 9252:= 1 8471:= 1 7980:LM 7931:TM 7844:, L 7838:, L 7799:= 1 7792:= 1 6890:not 5944:of 5475:→ 0 5468:→ 0 5461:→ ∞ 5236:as 4929:9.8 4836:9.8 4746:9.8 4632:500 4584:9.8 4497:not 3982:). 3300:; 2839:In 2553:man 2546:man 2537:man 2530:man 2521:rat 2514:man 2505:man 2496:rat 2487:man 2466:or 2458:of 2183:or 2151:by 2080:of 2078:set 1899:If 1662:dim 1651:is 1495:dim 1484:is 1418:dim 1407:is 1268:dim 1257:is 1123:dim 1112:is 968:dim 837:dim 826:is 715:dim 704:is 614:dim 603:is 557:≠ 0 550:≠ 0 543:≠ 0 529:≠ 0 522:≠ 0 508:≠ 0 317:dim 258:: 235:or 142:or 32:In 13751:: 13656:J 13626:N 13611:K 13584:A 13566:, 13539:M 13524:m 13514:L 13507:, 13503:, 13490:s 13480:T 12474:. 12470:. 12409:. 12323:, 12313:75 12311:, 12300:68 12298:, 12274:32 12272:, 12252:, 12232:, 12214:, 12204:95 12202:, 12172:, 12162:31 12160:, 12142:, 12123:40 12121:, 12110:42 12108:, 12090:66 12088:, 12072:, 12062:, 12052:72 12050:, 12013:, 12001:, 11985:OL 11983:, 11879:, 11867:14 11865:, 11861:, 11840:, 11832:, 11820:, 11816:, 11781:55 11779:, 11761:, 11749:, 11731:45 11729:, 11687:, 11665:. 11653:58 11651:. 11647:. 11608:. 11583:. 11572:^ 11555:. 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3377:, 3342:= 3322:= 3304:; 3284:, 3278:, 3217:. 3205:. 3147:ma 3145:= 3132:. 3116:. 3039:Ma 2972:Eu 2916:Fr 2857:Re 2827:dr 2815:dr 2811:dV 2791:)/ 2789:dr 2785:dV 2637:. 2532:+ 2516:+ 2490:, 2436:, 2432:, 2375:, 2296:, 2293:dt 2289:dt 2285:dx 2279:= 2277:dt 2260:, 2257:dt 2253:dx 2234:. 2222:. 2126:, 2120:, 2096:, 2090:, 2050:, 2044:, 1996:= 1970:, 1963:, 1952:= 1923:, 1914:, 1881:. 1857:, 570:. 545:, 500:. 475:, 469:, 463:, 457:, 451:, 445:, 223:) 180:− 115:, 60:, 56:, 40:, 13651:v 13648:I 13622:n 13595:T 13568:i 13562:I 13535:m 13509:r 13505:x 13501:l 13476:t 13408:e 13401:t 13394:v 12521:e 12514:t 12507:v 12482:. 12417:. 12319:: 12289:. 12260:: 12240:: 12218:: 12210:: 12168:: 12096:: 12076:: 12058:: 12017:: 12009:: 12003:4 11873:: 11844:: 11836:: 11828:: 11822:4 11765:: 11757:: 11737:: 11691:) 11673:. 11669:: 11659:: 11633:) 11629:( 11619:. 11594:. 11566:. 11541:. 11516:. 11462:: 11422:: 11387:. 11365:: 11332:. 11317:. 11262:: 11232:. 11218:: 11164:. 11152:: 11146:5 11129:. 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9506:b 9502:( 9492:+ 9488:) 9482:z 9478:1 9473:b 9470:( 9461:) 9456:z 9452:1 9447:a 9443:( 9433:= 9429:) 9423:z 9419:1 9414:b 9411:+ 9406:z 9402:1 9397:a 9393:( 9363:) 9357:( 9348:= 9345:) 9342:2 9338:/ 9331:+ 9325:( 9309:b 9305:a 9301:) 9299:θ 9295:b 9291:θ 9287:a 9282:0 9278:) 9276:θ 9269:z 9267:1 9263:) 9261:θ 9254:z 9250:x 9246:y 9244:1 9239:x 9235:y 9231:θ 9227:θ 9221:~ 9216:y 9214:1 9209:x 9207:1 9203:θ 9199:θ 9194:z 9192:1 9166:0 9162:1 9138:x 9134:1 9110:y 9106:1 9082:z 9078:1 9053:z 9049:1 9022:x 9018:1 8994:0 8990:1 8966:z 8962:1 8938:y 8934:1 8909:y 8905:1 8878:y 8874:1 8850:z 8846:1 8822:0 8818:1 8794:x 8790:1 8765:x 8761:1 8734:z 8730:1 8706:y 8702:1 8678:x 8674:1 8650:0 8646:1 8621:0 8617:1 8589:z 8585:1 8559:y 8555:1 8529:x 8525:1 8499:0 8495:1 8474:i 8468:i 8465:1 8460:x 8458:1 8453:x 8447:x 8442:0 8437:z 8433:y 8429:x 8427:1 8416:θ 8410:R 8404:v 8402:/ 8400:g 8397:R 8393:π 8388:x 8383:R 8378:y 8374:x 8369:θ 8363:v 8356:y 8352:x 8350:( 8302:8 8298:/ 8284:C 8258:m 8245:4 8241:r 8231:x 8226:p 8219:= 8216:C 8191:m 8187:M 8164:i 8160:M 8132:2 8122:m 8112:5 8108:r 8098:x 8093:p 8086:= 8081:2 8052:r 8040:m 8033:= 8028:1 7999:r 7986:η 7973:ρ 7948:x 7944:p 7908:m 7847:z 7841:y 7835:x 7821:, 7804:c 7797:b 7790:a 7770:c 7765:) 7757:y 7750:L 7740:2 7731:T 7723:( 7716:b 7711:) 7703:y 7696:L 7686:1 7677:T 7669:( 7662:a 7657:) 7649:x 7642:L 7632:1 7623:T 7615:( 7610:= 7604:x 7597:L 7580:y 7575:g 7570:x 7565:R 7560:y 7542:y 7537:v 7525:x 7507:x 7502:v 7476:0 7473:= 7470:c 7467:2 7464:+ 7461:b 7458:+ 7455:a 7433:1 7430:= 7427:c 7424:+ 7421:b 7418:+ 7415:a 7390:c 7385:) 7379:L 7372:2 7363:T 7356:( 7349:b 7346:+ 7343:a 7338:) 7332:L 7325:1 7316:T 7309:( 7304:= 7299:L 7271:. 7266:c 7262:g 7255:b 7250:y 7246:v 7239:a 7234:x 7230:v 7223:R 7210:R 7203:g 7185:y 7181:v 7154:x 7150:v 7137:R 7119:x 7115:v 7090:y 7086:v 7058:x 7037:m 6833:. 6791:q 6785:v 6779:B 6773:E 6757:v 6754:q 6751:B 6745:q 6742:E 6720:r 6714:T 6708:S 6692:r 6685:/ 6681:S 6675:T 6657:a 6651:m 6635:t 6631:/ 6627:p 6621:a 6618:m 6592:F 6574:A 6558:A 6555:q 6534:v 6524:V 6514:ρ 6498:v 6495:V 6475:m 6449:2 6445:v 6409:2 6405:v 6395:m 6374:r 6368:L 6362:S 6346:r 6342:/ 6338:L 6332:r 6328:/ 6324:S 6309:t 6303:F 6297:v 6291:m 6275:t 6272:F 6266:v 6263:m 6237:p 6219:I 6214:A 6207:m 6197:p 6181:B 6178:A 6175:I 6169:B 6166:m 6160:E 6157:p 6139:V 6128:μ 6118:ε 6107:B 6097:E 6077:/ 6073:V 6068:2 6064:B 6057:V 6052:2 6048:E 6021:ϕ 6011:q 5992:q 5971:S 5961:t 5951:I 5937:A 5921:t 5918:S 5915:A 5909:t 5906:I 5903:A 5882:N 5876:T 5870:V 5864:p 5848:T 5845:N 5839:V 5836:p 5815:ω 5805:I 5795:L 5779:I 5775:/ 5769:2 5765:L 5755:L 5747:2 5739:I 5721:p 5711:v 5701:m 5685:m 5681:/ 5675:2 5671:p 5664:v 5661:p 5653:2 5649:v 5645:m 5627:P 5621:t 5611:S 5595:t 5592:P 5586:t 5582:/ 5578:S 5560:d 5550:F 5534:d 5531:F 5505:E 5473:G 5466:ħ 5459:c 5453:G 5447:c 5441:ħ 5430:G 5422:ħ 5414:c 5400:d 5394:χ 5387:χ 5363:κ 5357:C 5329:d 5323:D 5317:t 5311:T 5302:T 5298:D 5285:d 5279:t 5273:s 5264:t 5261:s 5257:d 5248:t 5242:s 5233:d 5198:. 5167:. 5161:t 5158:f 5154:1 5149:m 5139:= 5136:1 5115:m 5107:= 5103:t 5100:f 5096:1 5064:] 5061:Z 5058:[ 5055:n 5052:= 5049:] 5046:Z 5043:[ 5037:n 5034:= 5031:Z 5017:n 5007:Z 4974:. 4970:m 4961:2 4949:) 4944:2 4936:( 4917:2 4914:1 4904:= 4896:m 4887:2 4883:) 4878:s 4874:/ 4870:n 4867:i 4864:m 4860:( 4856:) 4851:2 4843:( 4824:2 4821:1 4811:= 4802:2 4798:) 4793:n 4790:i 4787:m 4777:( 4771:) 4765:2 4761:s 4756:/ 4752:m 4739:( 4730:2 4727:1 4705:t 4695:t 4677:t 4659:. 4656:t 4650:) 4646:s 4642:/ 4638:m 4628:( 4625:+ 4620:2 4616:t 4609:) 4603:2 4599:s 4594:/ 4590:m 4577:( 4568:2 4565:1 4541:x 4537:x 4527:x 4521:x 4518:( 4508:b 4502:a 4492:b 4486:a 4480:b 4476:a 4472:b 4470:/ 4468:a 4406:× 4307:. 4303:0 4300:= 4297:) 4292:m 4284:, 4281:. 4278:. 4275:. 4272:, 4267:2 4259:, 4254:1 4246:( 4243:f 4213:. 4205:i 4201:k 4196:) 4190:i 4182:( 4177:m 4172:1 4169:= 4166:i 4158:= 4155:X 4141:X 4132:} 4129:m 4124:1 4117:m 4092:V 4088:V 4084:V 4078:V 4072:V 4066:L 4060:M 4047:V 4013:) 4011:k 4007:j 4003:i 4001:( 3996:M 3993:L 3991:T 3967:p 3923:R 3921:/ 3919:t 3911:S 3909:/ 3907:ω 3890:S 3884:ω 3878:ρ 3872:R 3866:t 3828:E 3822:ℓ 3816:f 3809:f 3791:, 3787:) 3782:A 3774:( 3770:f 3767:s 3764:A 3761:= 3758:E 3744:F 3726:, 3723:0 3720:= 3716:) 3710:A 3702:, 3696:s 3693:A 3689:E 3683:( 3679:F 3649:. 3644:A 3636:= 3627:2 3612:s 3609:A 3605:E 3600:= 3591:1 3564:2 3561:π 3555:1 3552:π 3546:E 3540:s 3531:ρ 3522:A 3513:ℓ 3496:κ 3486:g 3466:C 3453:κ 3432:k 3429:m 3419:= 3416:T 3403:g 3392:g 3387:g 3383:T 3379:m 3375:k 3371:g 3367:g 3350:C 3344:C 3340:1 3337:G 3331:m 3329:/ 3327:k 3324:T 3320:1 3317:G 3311:g 3306:k 3302:m 3298:T 3293:g 3287:k 3281:m 3275:T 3269:T 3263:g 3257:k 3252:m 3247:T 3191:e 3188:k 3168:G 3143:F 3085:c 3070:, 3065:c 3062:u 3057:= 3053:a 3050:M 3037:( 3020:. 3012:2 3008:u 2999:p 2990:= 2986:u 2983:E 2970:( 2953:. 2947:L 2943:g 2939:u 2934:= 2930:r 2927:F 2914:( 2897:. 2888:d 2885:u 2875:= 2871:e 2868:R 2855:( 2821:r 2813:/ 2805:r 2799:V 2793:V 2787:/ 2783:( 2729:n 2725:C 2719:r 2715:n 2711:C 2705:n 2699:x 2692:n 2690:( 2685:x 2679:n 2671:n 2665:n 2658:n 2575:M 2572:L 2569:T 2550:L 2548:/ 2543:m 2534:L 2527:m 2518:m 2511:m 2502:L 2493:m 2484:m 2444:. 2389:. 2387:M 2384:L 2381:T 2360:s 2357:d 2351:F 2334:s 2326:M 2323:L 2320:T 2307:. 2305:L 2302:T 2287:/ 2283:( 2281:d 2275:/ 2273:x 2270:d 2255:/ 2244:x 2138:. 2135:m 2129:c 2123:b 2117:a 2108:. 2105:m 2099:c 2093:b 2087:a 2059:m 2053:c 2047:b 2041:a 2031:C 2024:n 2020:R 2016:3 2013:R 2010:2 2007:R 2004:1 2001:R 1998:C 1994:R 1988:. 1986:) 1983:n 1979:R 1975:3 1972:R 1968:2 1965:R 1961:1 1958:R 1956:( 1954:F 1950:R 1939:n 1935:R 1929:3 1926:R 1920:2 1917:R 1911:1 1908:R 1902:R 1896:. 1829:. 1822:2 1818:I 1808:1 1801:M 1791:2 1784:L 1774:4 1770:T 1764:= 1756:1 1747:I 1739:M 1732:2 1726:L 1718:3 1709:T 1699:I 1692:T 1684:= 1671:= 1668:C 1648:C 1627:. 1622:1 1613:I 1605:M 1598:2 1592:L 1582:3 1575:T 1569:= 1563:I 1556:M 1549:2 1543:L 1535:3 1526:T 1517:= 1504:= 1501:V 1481:V 1460:. 1455:I 1448:T 1443:= 1427:= 1424:Q 1404:Q 1383:. 1378:M 1371:2 1365:L 1357:3 1348:T 1342:= 1336:T 1329:M 1322:2 1316:L 1308:2 1299:T 1290:= 1277:= 1274:P 1254:P 1233:. 1228:M 1221:2 1215:L 1207:2 1198:T 1192:= 1187:L 1177:M 1170:L 1163:2 1154:T 1148:= 1132:= 1129:E 1109:E 1088:. 1083:M 1076:1 1067:L 1059:2 1050:T 1044:= 1037:2 1031:L 1022:M 1015:L 1008:2 999:T 990:= 977:= 974:P 954:P 933:. 928:M 921:L 914:2 905:T 899:= 894:L 887:2 878:T 867:M 862:= 846:= 843:F 823:F 802:. 797:L 790:2 781:T 775:= 769:T 762:L 755:1 746:T 737:= 724:= 721:a 701:a 680:. 675:L 668:1 659:T 653:= 647:T 642:L 636:= 623:= 620:v 600:v 555:c 548:b 541:a 527:b 520:a 506:b 478:g 472:f 466:e 460:d 454:c 448:b 442:a 422:g 416:J 408:f 402:N 394:e 380:d 374:I 366:c 360:M 352:b 346:L 338:a 332:T 326:= 323:Q 303:Q 187:m 182:m 178:n 172:n 162:. 20:)

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Knowledge

Dimensional analysis

Index