Dimensional analysis

3890:(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 ( 6995:). 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. 13674: 7847:, 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? 3836:

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.)

1836: 4012:. 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 4711: 9987: 7779: 5407:, 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: 1634: 1654: 1095: 4419:, 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. 1390: 1240: 3486:, 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. 3225:, 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. 4980:{\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}}} 940: 9805: 9558: 6868:. 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). 7586: 9779:. 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: 809: 2401:

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

2168:—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 4427:

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

3930:, 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. 8202:, 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: 5380:

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,

1487: 1831:{\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}}}.} 687: 9186:

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

7399: 960: 1260: 1115: 6976:, 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. 3660: 10663:

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

9982:{\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}.\,} 10809: 829: 9378: 7774:{\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}} 4666: 1467: 7007:. (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 5387:) 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 4382:

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: 10797: 9740:. 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 5434:

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.

3816:. 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 9775:

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

1629:{\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}.} 7057:

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:

6424: 4336:. 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 6464: 5786: 10104: 4351:

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.

2960: 2904: 11482: 8327:, 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. 7289: 4220: 1090:{\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}}.} 4314: 5123: 3798: 1385:{\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.

6088: 1235:{\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.

9370: 7878:. 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: 579:; 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. 3443: 10818:, "However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units ...". 6957:

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.

6188: 3574: 2368: 12716: 6699: 935:{\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.} 9553:{\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),} 8631: 8509: 7552: 7517: 4428:

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

10806:, "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 ...". 4054:

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

2394:: 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). 6353: 5928: 4554: 4458:. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.) 2585:

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

6282: 5855: 3475: 7921: 3403:: 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: 9628: 9587: 12351:

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

9621: 7483: 5368:" 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. 4117:) 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, 7859:

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}} 10779: 11264: 6565: 5541: 5376:

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.

804:{\displaystyle \operatorname {dim} a={\frac {\text{speed}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.} 7044: 6913:

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 5221:, 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. 11445: 7487:, 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. 6879:

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),

4131:. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and 4548:

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

682:{\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.} 12711: 12425: 8208: 4521:) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for 4358:

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,

13403: 11293: 10205: 9281:. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form 13761: 10267: 7215: 5431:, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. 5128: 3671: 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.

5453:(but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit 10228: 10189: 3156: 3041: 2730:. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone. 2505:

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

1875: 197:. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. 7394:{\displaystyle {\mathsf {L}}=\left({\mathsf {T}}^{-1}{\mathsf {L}}\right)^{a+b}\left({\mathsf {T}}^{-2}{\mathsf {L}}\right)^{c}} 6430: 2622:" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 571:

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:

4235: 4096:. Similarly, the dual space can be interpreted as having "negative" dimensions. This corresponds to the fact that under the 3665:

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 9372:

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.

2215:, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s). The newton is defined as 13278: 13065: 8015: 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:

11241: 3408: 2614:. 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.

13202: 11810: 11468: 10962: 10483: 4412: 3949:

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

11499: 4107:

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

13396: 13172: 13136: 13045: 12783: 12721: 7132:, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then 5637: 2138: 12392:

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

7871:

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

2204:, however, can be factored into the base units of length (m), thus they are considered derived or compound units. 13771: 13258: 13243: 13197: 13075: 13050: 13035: 12995: 12919: 12502: 12448: 6129: 5474:. In problems involving a gravitational field the latter limit should be taken such that the field stays finite. 11602: 13253: 13238: 13228: 13218: 13156: 13126: 13096: 13060: 12985: 12904: 12879: 12874: 12869: 12862: 12566: 12349:

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

12291:

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

6833: 6811: 12114:

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

10738: 3655:{\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}} 3095: 13766: 13751: 13248: 13223: 13187: 13151: 13131: 13121: 13106: 13101: 13091: 13000: 12924: 12825: 12556: 12546: 11549: 10439:

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

9202:

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

9039: 8895: 8751: 8575: 8545: 8515: 5942: 5491: 4391:, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the 3210:, who was trying to understand why the sky is blue. Rayleigh first published the technique in his 1877 book 251: 10381:

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

13729: 13688: 13313: 13233: 13192: 13182: 13111: 13015: 12954: 12949: 12909: 12899: 12889: 12884: 12788: 12690: 12685: 12604: 12599: 12561: 12043:

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

10471: 6149: 2339: 12648: 10762: 2754:

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

13389: 13141: 13116: 13055: 13005: 12990: 12980: 12944: 12835: 12798: 12767: 12638: 8311:

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

6823: 4508:

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

4108: 4034:

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

2738:

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

11420: 6667: 5183:

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

2557:

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

2373: 13177: 13025: 12959: 12857: 12803: 12793: 12335: 6915: 3190:

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

2586: 8607: 8485: 7528: 7493: 6960:

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

5081:, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: 4661:{\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.} 1462:{\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.} 13586: 13070: 12609: 12440: 11577: 10446: 6575: 6467: 3832:. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. 11524: 11207: 4135:) together the measured quantities to produce something with the same unit as some derived quantity 3038:), important in high speed flows where the velocity approaches or exceeds the local speed of sound: 12476: 12065: 10200: 6316: 5895: 5404: 5192: 5180:

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

4451: 112: 9731:{\displaystyle \sin(\theta \,1_{\text{z}}+\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})} 9125: 9097: 9069: 9009: 8953: 8925: 8865: 8837: 8781: 8721: 8693: 8665: 8178: 8151: 7935: 7172: 7141: 7106: 7077: 6734: 6610: 5570: 5023: 13698: 12633: 12584: 12435: 11785: 6255: 5828: 5225: 4435: 3456: 2188: 2063: 1863: 11282: 10342: 10184: 7897: 4996: 3944: 3110: 2841: 166: 13722: 12060: 11202: 10531: 10508: 9566: 7805:. The increase in deductive power gained by the use of directed length dimensions is apparent. 6198: 5424: 5346: 4443: 4337: 4132: 4024: 3160: 3137: 2964: 2169: 2078: 2032: 1867: 482: 11921: 11242:"A typechecker plugin for units of measure: domain-specific constraint solving in GHC Haskell" 10748: 10646: 9592: 7447: 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

13703: 12525: 11356:

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

10216: 10147:

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

7407: 6484: 6375: 4048: 4023:, and all other vectors are called derived units. As in any module, one may choose different 4013: 3927: 2821:

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

2478:

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

2192: 2164:

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

201: 12436:

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

11889: 10523: 8286: 5984: 2650:

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

13756: 12331: 12225:

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

12204: 12052: 12003: 11822: 11751: 10400: 9745: 9153: 8981: 8809: 8637: 6947: 6865: 5365: 4455: 4439: 4345: 4319:

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

3983: 3837: 3356: 3311:. From these we can form only one dimensionless product of powers of our chosen variables, 2149: 1886: 1871: 558: 278: 236: 205: 194: 69: 12265:

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

11173: 7018:

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

4028: 3386:

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

3136:

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

2207:

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: 13613: 13298: 12818: 12749: 12136: 12129: 8320: 8312: 7875: 5952: 5883: 5286: 5207: 5078: 5008: 4408: 4392: 3565: 3352: 3239: 3151: 2181: 1941: 572: 282: 224: 101: 73: 12208: 12056: 12007: 11826: 11755: 11354:

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

10412: 10404: 7071: 7070:

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

6547: 5523: 2651: 247:, being based on only universal constants, may be thought of as being "less arbitrary". 13638: 12813: 12589: 12321: 12170: 11701: 11653: 11474: 11377: 11154: 11119: 11001: 10979: 10978:

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

10888: 10501: 10416: 10390: 7029: 7008: 6803: 6022: 2411: 1890: 286: 11852: 7808:

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

2761:

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

12464: 12407: 12397: 12372: 12354: 12258: 12238: 12174: 12140: 12030: 12022: 12015: 11973: 11953: 11929: 11904: 11793: 11735: 11709: 11478: 11464: 11367: 11220: 11179: 11088: 11056: 11032: 11008: 10958: 10652: 10626: 10598: 10568: 10535: 10524: 10479: 10358: 10314: 10243: 10238: 5806: 5211: 4396: 3976: 3972: 3532: 3118: 2764: 2758: 2606: 2599: 2425:

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

2398: 2074: 216: 209: 45: 11742:

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

11455:. Advances in Mathematics for Applied Sciences. World Scientific. pp. 331–345. 11381: 11158: 11123: 10892: 10420: 3860:

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

3392:

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

13551: 12845: 12343: 12313: 12254: 12234: 12212: 12162: 12090: 12070: 12011: 11947: 11867: 11838: 11830: 11806: 11759: 11731: 11722:

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

11663: 11456: 11416: 11407: 11359: 11290:

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

11256: 11212: 11146: 11111: 11084: 10880: 10696: 10594: 10564: 10408: 10350: 10305: 10280: 10233: 9183: 6219: 5816: 5796: 5612: 4431: 4047:. However, it is not possible to take arbitrary fractional powers of units, due to 3954: 3181: 3177: 2743: 2417: 2391: 2108: 486: 274: 190: 97: 85: 65: 8336: 6992: 4031:

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

2618:. Since any quantity can be multiplied by 1 without changing it, the expression " 13010: 12452: 12182: 11981: 11811:"On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis" 11201:. Lecture Notes in Computer Science. Vol. 6299. Springer. pp. 268–305. 10729: 8419:

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

6012: 5972: 5628: 5416: 4416: 4384: 4333: 4097: 3841: 2849: 2837: 2422:

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

2165: 2142: 1396: 1246: 815: 490: 220: 12485: 11216: 10701: 10354: 13360: 12934: 12551: 11917: 11667: 11460: 10865: 10265:

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

9776: 6837: 6721: 6535: 6108: 6098: 5408: 4388: 4093: 3523: 3148:

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

3133: 3114: 2739: 2173: 147: 12458:

Units, quantities, and fundamental constants project dimensional analysis maps

12304:

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

11843: 11641: 10951:

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

10884: 10555:

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

5289:

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

5011:

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

3847: 3198:

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

13745: 12808: 12680: 12620: 12446:

Quantity System calculator for units conversion based on dimensional approach

12411: 11977: 11441: 11183: 10949: 10194: 9254:, which is not unreasonable. Analogous reasoning forces the conclusion that 8138:{\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}} 7814: 6988: 6961: 4101: 4020: 3950: 3207: 2908: 2775: 2445: 2208: 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: 11940: 11363: 11260: 7205:

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

5351:

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

4480:, where the logarithm is taken in any base, holds for dimensionless numbers 13412: 13365: 12653: 12643: 12628: 11985: 11967: 11949:

Multidimensional Analysis: Algebras and Systems for Science and Engineering

11115: 10920: 6906: 6898: 6858: 6119: 2554: 2294: 2177: 693: 124: 12431:

Unicalc Live web calculator doing units conversion by dimensional analysis

11872: 11834: 11453:

Advanced Mathematical and Computational Tools in Metrology and Testing XII

11137:

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

8475:, the following multiplication table for the orientation symbols results: 6836:

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

3852: 119:

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

13628: 12964: 12695: 12594: 12494: 12472: 12279: 11326: 10984: 10921:"Square bracket notation for dimensions and units: usage and conventions" 8324: 6807: 5377: 4341: 3031: 1859: 1640: 128: 33: 11409:

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

10395: 10112:

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

9782:

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

12325: 11311: 10210: 8269:{\displaystyle C={\frac {p_{\mathrm {x} }\rho r^{4}}{\eta {\dot {m}}}}} 2742:. More generally, dimensional analysis is used in interpreting various 530: 297: 294: 12094: 12074: 10284: 4100:

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

3480:

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

13569: 13381: 12445: 12430: 12217: 10585:

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

9198:

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

4512: 4447: 4328: 3514: 3499: 2751: 2145: 1855: 12457: 12317: 12166: 11150: 11102:

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

10780:"Dimensional Analysis and Numerical Experiments for a Rotating Disc" 4229:

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

4019:

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

3233: 13541: 11764: 11658: 10611:

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

10313:(in English and French) (v. 1.08, 9th ed.). pp. 136–137. 9239:

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

8387:-axis. Conventional analysis will yield the dimensionless variable 7273:{\displaystyle R\propto v_{\text{x}}^{a}\,v_{\text{y}}^{b}\,g^{c}.} 6922:

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

6864:

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

5722: 5712: 5561: 5169:{\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.} 3728:{\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,} 3099: 2665: 2258: 946: 509: 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

11926:

Proceedings of the Fifth SIAM Conference on Applied Linear Algebra

3961:, and the inverse of L is 1/L or L. L raised to any integer power 3856:

Dimensional analysis and numerical experiments for a rotating disc

3810:

to experiments to discover the form for the unknown function

2261:) has dimension TL—length from position, time due to the gradient; 13658: 13433: 13370: 12369:

Dimensional Analysis in the Identification of Mathematical Models

12191: 6946:

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

6827: 6815: 6515: 6026: 4678: 3926:

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

1851: 1473: 250:

There are many possible choices of base physical dimensions. The

37: 5364:

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

5332:

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

4407:) is also defined as a base dimension, N. In the interaction of 13603: 13576: 13492: 13482: 13040: 11642:"Angles in the SI: a detailed proposal for solving the problem" 11500:"NondimensionalizationTransform—Wolfram Language Documentation" 10171: 6525: 6140: 3505: 2561: 2201: 2197: 2184:), powers (like m for square metres), or combinations thereof. 1101: 266: 132: 53: 4705: = 0.01 minutes. Then the first term would be 4415:, connected with the symmetry properties of the collisionless 13516: 11340: 8373:-axis, with the force of gravity directed along the negative 8341: 6819: 5551: 5234: 3103: 3022:{\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.} 2590:

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

2405: 2312: 2212: 592: 169:

describes how every physically meaningful equation involving

12116:

Transactions of the American Institute of Chemical Engineers

12083:

Transactions of the American Society of Mechanical Engineers

11050: 8437:

to denote vector directions, and an orientationless symbol 1

8330: 2840:, dimensional analysis is performed to obtain dimensionless 13556: 13526: 13467: 11603:"QuantityVariableDimensions—Wolfram Language Documentation" 11075:

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

10476:

Essential of Fluid Mechanics: Fundamentals and Applications

10301: 6419:{\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}} 5962: 5938: 5702: 4422: 3848:

A third example: demand versus capacity for a rotating disc

2815:

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

2610:. For example, kPa and bar are both units of pressure, and 2141:

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

270: 262: 159: 61: 57: 12367:

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

11029:

Physics for Scientists and Engineers – with Modern Physics

10264: 7812:

It does not deal well with vector equations involving the

6459:{\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}} 3155:

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

175:

variables can be equivalently rewritten as an equation of

11199:

Central European Functional Programming School. CEFP 2009

7202:

the downward acceleration of gravity, with dimension TL.

6968:−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F, 6930:

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

5781:{\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I} 5206:, is an equation that remains valid independently of the 4671:

This is the height to which an object rises in time

4365:

Velocity, being expressible in terms of length and time (

2222: 563:. A quantity that has all exponents null is said to have 11550:"DimensionalCombinations—Wolfram Language Documentation" 11440: 11353: 10099:{\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1} 4377: 3251:

attached to an ideal linear spring with spring constant

2955:{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.} 2899:{\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.} 2574:, they are fundamentally different physical quantities. 300:

typeface. Mathematically, the dimension of the quantity

12081:

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

10905: 10863: 8148:

If we distinguish between inertial mass with dimension

6964:

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

4990: 4215:{\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.} 2856:), generally important in all types of fluid problems: 2459:

quantities (quantities with different dimensions), and

2062:

Express each of the quantities in the equation in some

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

is the analysis of the relationships between different

12284:"A mathematical formalisation of dimensional analysis" 11706:

Scaling, Self-Similarity, and Intermediate Asymptotics

10917:

For a review of the different conventions in use see:

10864:

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

8404:, but offers no insight into the relationship between 6882:

subtracting two positions should yield a displacement,

6847: 4908: 4815: 4721: 4559: 4309:{\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.} 3423: 3293:. The four quantities have the following dimensions: 2971:), used in problems in which pressure is of interest: 2343: 2200:

and time are normally chosen as base units. Units for

2111:

these equations to obtain the values of the exponents

1902:

is a variable that depends upon independent variables

12426:

List of dimensions for variety of physical quantities

12366: 11341:"CamFort: Specify, verify, and refactor Fortran code" 10014: 9808: 9748: 9631: 9595: 9569: 9381: 9314: 9156: 9128: 9100: 9072: 9042: 9012: 8984: 8956: 8928: 8898: 8868: 8840: 8812: 8784: 8754: 8724: 8696: 8668: 8640: 8610: 8578: 8548: 8518: 8488: 8289: 8211: 8181: 8154: 8071: 8018: 7938: 7900: 7589: 7531: 7496: 7450: 7410: 7292: 7218: 7175: 7144: 7109: 7080: 7032: 6998: 6737: 6670: 6613: 6550: 6487: 6433: 6390: 6319: 6258: 6152: 6039: 5987: 5898: 5831: 5734: 5640: 5573: 5526: 5131: 5118:{\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} } 5090: 5026: 4714: 4557: 4238: 4150: 3793:{\displaystyle E=Asf\left({\frac {\ell }{A}}\right),} 3753: 3674: 3577: 3459: 3411: 3169: 3044: 2977: 2921: 2862: 2733: 2553:

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

2342: 1657: 1490: 1413: 1263: 1118: 963: 832: 710: 609: 315: 231:

of a physical quantity is more fundamental than some

11343:. University of Cambridge; University of Kent. 2018. 7863:), and mass as a measure of the quantity of matter. 7490:

However, if we use directed length dimensions, then

6083:{\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu } 4327:

The dimension of physical quantities of interest in

3844:, which may be interpreted by dimensional analysis. 2328:

the integral of force with respect to the distance (

591:

As examples, the dimension of the physical quantity

11051:Martin, B.R.; Shaw, G.; Manchester Physics (2008), 10866:"Two alternative derivations of Bridgman's theorem" 10307:

SI Brochure: The International System of Units (SI)

8057:{\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}} 7829:

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

254:selects the following dimensions and corresponding 13596: 12128: 11000: 10648:The Mathematics of Measurement: A Critical History 10500: 10266: 10098: 9981: 9761: 9730: 9615: 9581: 9552: 9365:{\displaystyle \sin(\theta +\pi /2)=\cos(\theta )} 9364: 9169: 9141: 9113: 9085: 9057: 9025: 8997: 8969: 8941: 8913: 8881: 8853: 8825: 8797: 8769: 8737: 8709: 8681: 8653: 8625: 8593: 8563: 8533: 8503: 8303: 8283:is an undetermined constant (found to be equal to 8268: 8194: 8167: 8137: 8056: 7951: 7915: 7773: 7546: 7511: 7477: 7434: 7393: 7272: 7188: 7157: 7122: 7093: 7038: 6758: 6693: 6636: 6559: 6499: 6458: 6418: 6347: 6276: 6182: 6082: 5996: 5922: 5849: 5780: 5686: 5596: 5535: 5168: 5117: 5065: 4979: 4660: 4308: 4214: 3792: 3727: 3654: 3500:A more complex example: energy of a vibrating wire 3469: 3437: 3071: 3021: 2954: 2898: 2362: 2172:, e.g. 60 km/h. Other relations can involve 2159: 1830: 1628: 1461: 1384: 1234: 1089: 934: 803: 681: 428: 12264: 11446:"Type systems for programs respecting dimensions" 10764:A Treatise on Electricity and Magnetism, volume 1 10380: 9992:Dimensional homogeneity will now correctly yield 9788:, being in the xy-plane will thus have dimension 6919:an affine space), yielding a new affine quantity. 4027:, which yields different systems of units (e.g., 3234:A simple example: period of a harmonic oscillator 2805:is the continuously compounded interest rate and 2474: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 13743: 11772:Boucher; Alves (1960), "Dimensionless Numbers", 11578:"UnityDimensions—Wolfram Language Documentation" 11312:"Units of Measure in Rust with Refinement Types" 10167:are not, and are (correctly) deemed unphysical. 6893:This illustrates the subtle distinction between 3438:{\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}} 3269:of some dimensionless equation in the variables 2521:is meaningful, but the heterogeneous expression 11525:"UnitDimensions—Wolfram Language Documentation" 11178:(Phd). Vol. 391. University of Cambridge. 10580: 10578: 10296: 10294: 10268:"Dynamic similarity, the dimensionless science" 10222: 8453:with L specifying the dimension of length, and 3986:over the integers, with the dimensional symbol 12103:Bulletin of the Virginia Polytechnic Institute 12101:Murphy, N. F. (1949), "Dimensional Analysis", 11031:(6th ed.), San Francisco: W. H. Freeman, 10695:, Clarendon Press series, Oxford, p. 45, 10124:which is a series consisting of odd powers of 10006:, and orientational homogeneity requires that 7074:when fired with a vertical velocity component 6938:length, while displacements have dimension of 3744:is some unknown function, or, equivalently as 507:(with all other exponents zero) is known as a 135:in the absence of a more rigorous derivation. 13397: 12510: 12042: 11708:, Cambridge, UK: Cambridge University Press, 10469: 6983:1 K = 1 °C ≠ 1 °F = 1 °R. 6840:to extend type systems for units of measure. 5687:{\displaystyle mv^{2}\equiv pv\equiv p^{2}/m} 5403:It has been argued by some physicists, e.g., 5001:The value of a dimensional physical quantity 3072:{\displaystyle \mathrm {Ma} ={\frac {u}{c}},} 2664:dimensions), or the area of its surface, the 2247:derivative of position with respect to time ( 11853:"On the foundations of dimensional analysis" 11771: 11741: 11721: 11421:11245.1/fd7be191-700f-4468-a329-4c8ecd9007ba 11136: 10728:Rayleigh, Baron John William Strutt (1877), 10575: 10498: 10291: 9182:The orientational symbols form a group (the 7136:, the distance travelled, with dimension L, 3132:In 1822, the important Napoleonic scientist 2070: 11992: 11850: 11026: 6934:Properly then, positions have dimension of 6832:Griffioen's 2019 thesis extended Kennedy's 5477: 3477:from the original dimensionless equation). 2746:, economics ratios, and accounting ratios. 2406:Dimensional homogeneity (commensurability) 13404: 13390: 12524: 12517: 12503: 12483: 12290: 11805: 11700: 11280: 11003:The Cambridge Handbook of Physics Formulas 10879:: 255–261, See §5 General Results p. 259. 10499:de Jong, Frits J.; Quade, Wilhelm (1967). 10304:(2019). "2.3.3 Dimensions of quantities". 6954:unit only requires a unit of measurement. 4411:with strong laser pulses, a dimensionless 3938: 2827:is in the "denominator" of the derivative. 2679:-dimensional figure, the volume scales as 12216: 12064: 12027:Dimensional Analysis and Theory of Models 11871: 11842: 11763: 11657: 11405: 11206: 10983: 10911: 10700: 10521: 10394: 10260: 10258: 10206:Rayleigh's method of dimensional analysis 9978: 9957: 9890: 9859: 9836: 9825: 9714: 9675: 9644: 9531: 9505: 9472: 9446: 9413: 9396: 9209:and the side opposite has an orientation 8383:, at which point the mass returns to the 8331:Siano's extension: orientational analysis 7867:is defined by Huntley as a quantity only 7256: 7240: 5335: 5153: 5144: 5110: 5095: 4302: 4208: 3971:). Physically, 1/L can be interpreted as 3504:Consider the case of a vibrating wire of 2942: 2880: 1881:The method involves the following steps: 12391: 12244: 12021: 11898: 11784: 11685: 11044: 11027:Mosca, Gene; Tipler, Paul Allen (2007), 10947: 10851: 10754: 10727: 10684: 10669: 10625:, New York: Collier Books, p. 169, 10343:"Principles of the Theory of Dimensions" 10229:Covariance and contravariance of vectors 10213:– an application of dimensional analysis 10190:Dimensionless numbers in fluid mechanics 9308:are real scalars. An expression such as 7874:For example, consider the derivation of 6926:quantities which are vectors, and these 6797: 5357:in the Poiseuille's Law problem and the 4423:Polynomials and transcendental functions 4395:(the number of molecules divided by the 3851: 2390:In economics, one distinguishes between 489:of the SI basis with a dimension (Q) of 12487:An introduction to dimensional analysis 12348: 12303: 12224: 12195:(1915), "The Principle of Similitude", 11965: 11887: 11681: 11639: 11627: 11196: 11171: 10943: 10941: 10918: 10760: 10744: 10693:A Treatise on Electricity and Magnetism 10690: 10678:A Treatise on Electricity and Magnetism 10675: 10638: 10584: 10554: 9220:to indicate orientational equivalence) 9058:{\displaystyle \mathbf {1_{\text{z}}} } 8914:{\displaystyle \mathbf {1_{\text{y}}} } 8770:{\displaystyle \mathbf {1_{\text{x}}} } 8594:{\displaystyle \mathbf {1_{\text{z}}} } 8564:{\displaystyle \mathbf {1_{\text{y}}} } 8534:{\displaystyle \mathbf {1_{\text{x}}} } 8377:-axis. It is desired to find the range 7021: 3540:(LM/T), and we want to know the energy 2799:is the value of a bond (or portfolio), 1639:The dimension of the physical quantity 1472:The dimension of the physical quantity 1395:The dimension of the physical quantity 1245:The dimension of the physical quantity 1100:The dimension of the physical quantity 945:The dimension of the physical quantity 814:The dimension of the physical quantity 692:The dimension of the physical quantity 14: 13744: 13411: 12462: 12180: 12100: 11430:from the original on 21 February 2020. 11324: 11309: 11299:from the original on 10 November 2020. 11239: 11101: 11074: 10376: 10374: 10255: 9964: 9941: 9936: 9905: 9885: 9854: 8175:and quantity of matter with dimension 7746: 7727: 7692: 7673: 7638: 7619: 7593: 7375: 7359: 7328: 7312: 7295: 7065: 3878:(M/L), rotates at an angular velocity 2915:), modeling flow with a free surface: 2582:to convert 35 yards to 32.004 m. 2223:Percentages, derivatives and integrals 1818: 1814: 1804: 1801: 1797: 1787: 1784: 1780: 1770: 1766: 1743: 1735: 1722: 1705: 1695: 1688: 1609: 1601: 1588: 1578: 1575: 1571: 1559: 1552: 1539: 1522: 1451: 1444: 1374: 1361: 1344: 1332: 1325: 1312: 1295: 1224: 1211: 1194: 1183: 1173: 1166: 1150: 1079: 1063: 1046: 1027: 1018: 1011: 995: 924: 917: 901: 890: 874: 863: 793: 777: 765: 758: 742: 671: 655: 643: 638: 415: 401: 387: 373: 359: 345: 331: 13385: 12498: 12394:Physics: Principles with Applications 12152: 12126: 12113: 12080: 11572: 11570: 11444:; Nordvall-Forsberg, Fredrik (2022). 10715: 10644: 10620: 10470:Cimbala, John; Çengel, Yunus (2006). 10340: 10131:It is seen that the Taylor series of 7854: 6183:{\displaystyle pE\equiv mB\equiv IAB} 5224:For example, a quantity equation for 5186: 4378:Other fields of physics and chemistry 4226: 4141:can be expressed in the general form 3157:Newton's law of universal gravitation 3117:also treated the same problem of the 2363:{\displaystyle \textstyle \int F\ ds} 1988:Write the above equation in the form 11945: 11922:"The theory of dimensioned matrices" 11916: 11883:from the original on 16 January 2004 11394: 11270:from the original on 10 August 2017. 11175:Programming languages and dimensions 10998: 10977: 10938: 10432: 10300: 10197:– used to teach dimensional analysis 7580:. The dimensional equation becomes: 7101:and a horizontal velocity component 6852: 6214:= area (bounded by a current loop), 4991:Combining units and numerical values 4685:and the initial upward speed is 500 4531:makes sense (as an area), while for 3982:An abelian group is equivalent to a 2740:distinction between stocks and flows 2593: 1841: 84:physical quantities are of the same 12278: 11415:(Thesis). University of Amsterdam. 10906:Berberan-Santos & Pogliani 1999 10854:, 2. Dimensional Formulas pp. 17–27 10827: 10815: 10803: 10503:Dimensional analysis for economists 10371: 9049: 8905: 8761: 8585: 8555: 8525: 7821:nor does it handle well the use of 6848:Geometry: position vs. displacement 6802:Dimensional correctness as part of 4997:Physical quantity § Components 3913:thickness/radius or aspect ratio = 24: 13762:Conversion of units of measurement 13694:International System of Quantities 12547:International System of Units (SI) 12385: 11567: 10948:Thompson, Ambler (November 2009). 10777: 9866: 8227: 8094: 7753: 7699: 7645: 7600: 7538: 7503: 6999:Orientation and frame of reference 6694:{\displaystyle T\delta S/\delta r} 5157: 5154: 5145: 5111: 5099: 5096: 4966: 4892: 4874: 4866: 4863: 4860: 4789: 4786: 4783: 4757: 4748: 4642: 4634: 4595: 4586: 4511:Similarly, while one can evaluate 3975:, and 1/T as reciprocal time (see 3568:of the variables chosen, given by 3447:, for some dimensionless constant 3263:? That period is the solution for 3049: 3046: 2992: 2982: 2979: 2926: 2923: 2867: 2864: 2831: 2734:Finance, economics, and accounting 2325:(mass multiplied by acceleration); 2152:the variables with like exponents. 2066:in which the solution is required. 535:. A quantity that has only all of 25: 13788: 12419: 12247:Journal of the Franklin Institute 12227:Journal of the Franklin Institute 12193:J. W. Strutt (3rd Baron Rayleigh) 12155:Software: Practice and Experience 11488:from the original on 17 May 2022. 11172:Kennedy, Andrew J. (April 1996). 11104:Software: Practice and Experience 10873:Journal of Mathematical Chemistry 10587:Journal of the Franklin Institute 10557:Journal of the Franklin Institute 10530:. New York: McGraw-Hill. p. 9194:. For angles, consider an angle 8337:Angle § Dimensional analysis 6942:length. To assign a number to an 6826:, Python, and a code checker for 5400:is the dimension of the lattice. 4413:relativistic similarity parameter 3564:be two dimensionless products of 3257:suspended in gravity of strength 3217:The original meaning of the word 2624:5 bar × 100 kPa / 1 bar = 500 kPa 1889:that are likely to influence the 514:. A quantity that has only both 13718: 13717: 13672: 11894:(in French), Paris: Firmin Didot 11891:Theorie analytique de la chaleur 10919:Pisanty, E (17 September 2013). 10522:Waite, Lee; Fine, Jerry (2007). 9795:and the range of the projectile 9045: 8901: 8757: 8626:{\displaystyle \mathbf {1_{0}} } 8617: 8613: 8581: 8551: 8521: 8504:{\displaystyle \mathbf {1_{0}} } 8495: 8491: 7961:pressure gradient along the pipe 7547:{\displaystyle v_{\mathrm {y} }} 7512:{\displaystyle v_{\mathrm {x} }} 6950:, while to assign a number to a 6814:. There are implementations for 4369:), is redundant (the set is not 3344:for some dimensionless constant 3194:, which, after substituting his 2705:-ball in terms of the radius is 2685:, while the surface area, being 2444:However, the dimensions form an 185:dimensionless parameters, where 11674: 11633: 11620: 11595: 11542: 11517: 11492: 11434: 11399: 11388: 11347: 11333: 11318: 11303: 11274: 11233: 11190: 11165: 11130: 11095: 11068: 11020: 10992: 10971: 10899: 10857: 10845: 10821: 10771: 10721: 10708: 10614: 10605: 7784:and we may solve completely as 7015: 6897:quantities (ones modeled by an 5285:may be expressed in any units, 5007:is written as the product of a 3884:(T) and this leads to a stress 2637: 2564:and energy share the dimension 2244:) has the dimension L (length); 2160:Concrete numbers and base units 586: 575:that relate them. For example, 12275:, (5): 147, (6): 101, (7): 129 11310:Teller, David (January 2020). 11240:Gundry, Adam (December 2015). 11007:, Cambridge University Press, 10621:Mason, Stephen Finney (1962), 10548: 10515: 10492: 10472:"§7-2 Dimensional homogeneity" 10463: 10445:(3rd ed.), archived from 10426: 10383:Journal of High Energy Physics 10334: 10063: 10030: 9725: 9708: 9686: 9672: 9658: 9638: 9525: 9516: 9466: 9457: 9359: 9353: 9341: 9321: 7404:from which we may deduce that 7198:, both dimensioned as TL, and 7049:He introduced two approaches: 6905:quantities (ones modeled by a 5317:when expressed in seconds and 5242:multiplied by time difference 5060: 5054: 5045: 5039: 4879: 4856: 4794: 4773: 4767: 4735: 4646: 4624: 4605: 4573: 4293: 4242: 4192: 4178: 2645: 2630:, and bar/bar cancels out, so 153: 13: 1: 12557:US customary units (USCS/USC) 12306:American Mathematical Monthly 11774:Chemical Engineering Progress 11694: 11281:Garrigue, J.; Ly, D. (2017). 10761:Maxwell, James Clerk (1873), 10691:Maxwell, James Clerk (1873), 10676:Maxwell, James Clerk (1873), 10478:. McGraw-Hill. p. 203–. 10413:10.1088/1126-6708/2002/03/023 7072:distance a cannonball travels 6979:For temperature differences, 6348:{\displaystyle S/r\equiv L/r} 5923:{\displaystyle AIt\equiv ASt} 5492:International System of Units 5371: 4225:Consequently, every possible 4111:describes some number (e.g., 3933: 3872:(L). The disc has a density 1850:is a conceptual tool used in 1676:electric potential difference 13689:History of the metric system 12293:Journal of Applied Mechanics 12259:10.1016/0016-0032(85)90032-8 12239:10.1016/0016-0032(85)90031-6 12045:Journal of Applied Mechanics 12016:10.1016/0009-2509(55)80004-8 11996:Chemical Engineering Science 11736:10.1016/0004-3702(90)90038-2 11089:10.1016/0096-0551(77)90010-8 10599:10.1016/0016-0032(81)90475-0 10569:10.1016/0016-0032(71)90160-8 10223:Related areas of mathematics 9142:{\displaystyle 1_{\text{x}}} 9114:{\displaystyle 1_{\text{y}}} 9086:{\displaystyle 1_{\text{z}}} 9026:{\displaystyle 1_{\text{x}}} 8970:{\displaystyle 1_{\text{z}}} 8942:{\displaystyle 1_{\text{y}}} 8882:{\displaystyle 1_{\text{y}}} 8854:{\displaystyle 1_{\text{z}}} 8798:{\displaystyle 1_{\text{x}}} 8738:{\displaystyle 1_{\text{z}}} 8710:{\displaystyle 1_{\text{y}}} 8682:{\displaystyle 1_{\text{x}}} 8319:substance, the SI dimension 8195:{\displaystyle M_{\text{m}}} 8168:{\displaystyle M_{\text{i}}} 7952:{\displaystyle p_{\text{x}}} 7189:{\displaystyle v_{\text{y}}} 7158:{\displaystyle v_{\text{x}}} 7123:{\displaystyle v_{\text{x}}} 7094:{\displaystyle v_{\text{y}}} 6759:{\displaystyle Eq\equiv Bqv} 6637:{\displaystyle ma\equiv p/t} 5597:{\displaystyle S/t\equiv Pt} 5340: 5066:{\displaystyle Z=n\times =n} 4683:metres per second per second 4322: 4066:, one has the vector spaces 3085:is the local speed of sound. 2334:) the object has travelled ( 7: 11924:, in Lewis, John G. (ed.), 11292:(in French). hal-01503084. 11283:"Des unités dans le typeur" 11217:10.1007/978-3-642-17685-2_8 10830:"Similarly, one can define 10355:10.1007/978-1-349-00245-0_1 10178: 7046:of the dimensional matrix. 6277:{\displaystyle mv\equiv Ft} 5850:{\displaystyle pV\equiv NT} 5485: 4689:. It is not necessary for 3996:corresponding to the tuple 3470:{\displaystyle {\sqrt {C}}} 3228: 3102:, in a 1799 article at the 2580:1 yard = 0.9144 m 10: 13793: 12484:Dureisseix, David (2019). 12181:Porter, Alfred W. (1933), 11928:, SIAM, pp. 186–190, 11461:10.1142/9789811242380_0020 11325:Grecco, Hernan E. (2022). 10526:Applied Biofluid Mechanics 10347:Theory of Hydraulic Models 8334: 7916:{\displaystyle {\dot {m}}} 6856: 6834:Hindley–Milner type system 5489: 5344: 5329:when expressed in metres. 5202:, also sometimes called a 5190: 4994: 3942: 3894:) non-dimensional groups: 3089: 2597: 2448:under multiplication, so: 2415: 2409: 157: 107:Any physically meaningful 13777:Environmental engineering 13712: 13681: 13670: 13587:thermodynamic temperature 13432: 13427: 13419: 13353: 13327: 13271: 13211: 13165: 13084: 12973: 12776: 12742: 12735: 12704: 12673: 12666: 12619: 12610:English Engineering Units 12577: 12539: 12532: 12396:(7th ed.). Pearson. 11792:, Yale University Press, 11744:The Astrophysical Journal 10784:Ramsay Maunder Associates 10651:, Springer, p. 203, 10623:A history of the sciences 10507:. North Holland. p. 9582:{\displaystyle a=\theta } 9036: 8892: 8748: 8604: 7519:will be dimensioned as TL 6972:where the symbol ≘ means 6909:, such as displacement). 6599: 6596: 6585: 6576:magnetic vector potential 6468:root mean square velocity 6249: 6244: 6241: 6230: 5978: 5517: 5512: 5509: 5498: 5210:used when expressing the 4452:inhomogeneous polynomials 3546:(LM/T) in the wire. Let 3355:. They are often called 2699:. Thus the volume of the 2537:is meaningless. However, 1846:In dimensional analysis, 500:A quantity that has only 12477:University of Nottingham 12451:24 December 2017 at the 12184:The Method of Dimensions 11946:Hart, George W. (1995), 11888:Fourier, Joseph (1822), 11851:Drobot, S. (1953–1954), 11668:10.1088/1681-7575/ac023f 11327:"Pint: makes units easy" 10341:Yalin, M. Selim (1971). 10249: 10201:Numerical-value equation 9616:{\displaystyle b=\pi /2} 7478:{\displaystyle a+b+2c=0} 6901:, such as position) and 5478:Dimensional equivalences 5323:is the numeric value of 5311:is the numeric value of 5219:numerical-value equation 5193:Quantity theory of money 5191:Not to be confused with 4699:. For example, suppose 4456:dimensionless quantities 4436:transcendental functions 4049:representation-theoretic 3176:. By assuming a form of 2693:-dimensional, scales as 2081:involving the exponents 2059:are arbitrary exponents. 12649:Quantum chromodynamical 12441:Buckingham's pi-theorem 12312:(2): 115–138, 227–256, 12187:(3rd ed.), Methuen 12002:(3): 130–140, 167–177, 11966:Huntley, H. E. (1967), 11899:Gibbings, J.C. (2011), 11724:Artificial Intelligence 11364:10.1145/3276604.3276613 11261:10.1145/2887747.2804305 11055:(2nd ed.), Wiley, 10885:10.1023/A:1019102415633 10108:. In other words, that 9849: which means 7987:dynamic fluid viscosity 7435:{\displaystyle a+b+c=1} 6987:(Here °R refers to the 6500:{\displaystyle \rho Vv} 6474:= mass (of a molecule) 4677:if the acceleration of 4543:(3 m) + 3 m = 9 m + 3 m 3939:Mathematical properties 2774:In financial analysis, 2587:Newton's laws of motion 2264:the second derivative ( 2071:dimensional homogeneity 1864:functional relationship 121:dimensional homogeneity 13772:Mechanical engineering 13704:Systems of measurement 12762:centimetre–gram–second 12526:Systems of measurement 12465:"Dimensional Analysis" 12463:Bowley, Roger (2009). 12336:"Theory of Dimensions" 12135:, MIT Press, pp. 11640:Quincey, Paul (2021). 11406:Griffioen, P. (2019). 11116:10.1002/spe.4380150604 10925:Physics Stack Exchange 10645:Roche, John J (1998), 10100: 9983: 9763: 9732: 9617: 9583: 9554: 9366: 9171: 9143: 9115: 9087: 9059: 9027: 8999: 8971: 8943: 8915: 8883: 8855: 8827: 8799: 8771: 8739: 8711: 8683: 8655: 8627: 8595: 8565: 8535: 8505: 8305: 8304:{\displaystyle \pi /8} 8270: 8196: 8169: 8139: 8058: 7953: 7917: 7825:as physical variables. 7775: 7548: 7513: 7479: 7436: 7395: 7274: 7190: 7159: 7124: 7095: 7040: 6760: 6695: 6638: 6561: 6501: 6460: 6420: 6349: 6278: 6199:electric dipole moment 6184: 6084: 5998: 5997:{\displaystyle q\phi } 5924: 5851: 5782: 5688: 5598: 5537: 5347:Dimensionless quantity 5336:Dimensionless concepts 5170: 5119: 5067: 4981: 4662: 4310: 4216: 4177: 3857: 3794: 3729: 3656: 3513:(L) vibrating with an 3471: 3439: 3238:What is the period of 3161:gravitational constant 3125:instead of the Daviet 3073: 3023: 2956: 2900: 2364: 2079:simultaneous equations 2033:dimensionless constant 1832: 1630: 1463: 1386: 1236: 1091: 936: 805: 683: 430: 13284:Biblical and Talmudic 12750:metre–kilogram–second 12490:(lecture). INSA Lyon. 12127:Pesic, Peter (2005), 11873:10.4064/sm-14-1-84-99 11835:10.1103/PhysRev.4.345 11607:reference.wolfram.com 11582:reference.wolfram.com 11554:reference.wolfram.com 11529:reference.wolfram.com 11504:reference.wolfram.com 10836:as the dual space to 10217:System of measurement 10101: 9984: 9799:will be of the form: 9764: 9762:{\displaystyle 1_{0}} 9733: 9618: 9584: 9555: 9367: 9172: 9170:{\displaystyle 1_{0}} 9144: 9116: 9088: 9060: 9028: 9000: 8998:{\displaystyle 1_{0}} 8972: 8944: 8916: 8884: 8856: 8828: 8826:{\displaystyle 1_{0}} 8800: 8772: 8740: 8712: 8684: 8656: 8654:{\displaystyle 1_{0}} 8628: 8596: 8566: 8536: 8506: 8421:orientational symbols 8306: 8271: 8197: 8170: 8140: 8059: 7954: 7918: 7776: 7549: 7514: 7480: 7437: 7396: 7275: 7191: 7160: 7125: 7096: 7041: 6857:Further information: 6798:Programming languages 6761: 6696: 6639: 6562: 6502: 6461: 6421: 6350: 6279: 6185: 6085: 6025:(for changes this is 5999: 5925: 5852: 5783: 5689: 5599: 5538: 5171: 5120: 5068: 4982: 4663: 4545:does not make sense. 4311: 4217: 4157: 4014:scalar multiplication 3943:Further information: 3928:finite element method 3855: 3795: 3730: 3657: 3522:(L). The wire has a 3472: 3440: 3357:dimensionless numbers 3223:Theorie de la Chaleur 3074: 3024: 2957: 2901: 2416:Further information: 2365: 2193:system of measurement 1887:independent variables 1874:. It was named after 1833: 1631: 1464: 1387: 1237: 1092: 937: 806: 684: 431: 202:nondimensionalization 48:by identifying their 18:Unit commensurability 13767:Chemical engineering 13752:Dimensional analysis 12371:, World Scientific, 11969:Dimensional Analysis 11901:Dimensional Analysis 11790:Dimensional Analysis 11358:. pp. 121–132. 10957:. DIANE Publishing. 10702:2027/uc1.l0065867749 10452:on 23 September 2015 10185:Buckingham π theorem 10012: 9806: 9746: 9629: 9593: 9567: 9379: 9312: 9230: + ... ~ 1 9154: 9126: 9098: 9070: 9040: 9010: 8982: 8954: 8926: 8896: 8866: 8838: 8810: 8782: 8752: 8722: 8694: 8666: 8638: 8608: 8576: 8546: 8516: 8486: 8287: 8209: 8179: 8152: 8069: 8016: 7936: 7898: 7587: 7529: 7494: 7448: 7408: 7290: 7216: 7173: 7142: 7107: 7078: 7030: 7022:Huntley's extensions 6928:relative differences 6735: 6720:= displacement (see 6668: 6611: 6548: 6485: 6431: 6388: 6368:= angular momentum, 6317: 6256: 6150: 6037: 5985: 5896: 5829: 5732: 5638: 5571: 5524: 5366:back of the envelope 5273:= 5 m/s, where 5129: 5088: 5024: 4712: 4555: 4371:linearly independent 4346:linearly independent 4236: 4148: 3945:Buckingham π theorem 3838:dimensionless number 3822:, and so infer that 3751: 3672: 3575: 3457: 3409: 3111:Buckingham π theorem 3106:Academy of Science. 3042: 2975: 2919: 2860: 2719:, for some constant 2340: 2176:(often shown with a 1872:exponential equation 1655: 1488: 1411: 1261: 1116: 961: 830: 708: 607: 313: 279:absolute temperature 206:characteristic units 167:Buckingham π theorem 70:units of measurement 42:dimensional analysis 13614:amount of substance 12353:, Kluwer Academic, 12209:1915Natur..95...66R 12057:2005JAM....72..648M 12008:1955ChEnS...4..130K 11952:, Springer-Verlag, 11827:1914PhRv....4..345B 11756:1991ApJ...372..592B 10731:The Theory of Sound 10405:2002JHEP...03..023D 10089: 10062: 10047: 9974: 8441:. Thus, Huntley's L 8321:amount of substance 7255: 7239: 7066:Directed dimensions 6207:= magnetic moment, 5884:amount of substance 5212:physical quantities 5208:unit of measurement 4409:relativistic plasma 4393:amount of substance 3531:(M/L) and is under 3212:The Theory of Sound 3172:, thereby defining 3152:James Clerk Maxwell 2660:(the solid ball in 2616:100 kPa / 1 bar = 1 1942:functional equation 283:amount of substance 215:The dimension of a 193:of the dimensional 74:conversion of units 46:physical quantities 13639:luminous intensity 13413:SI base quantities 12756:metre–tonne–second 12552:UK imperial system 12023:Langhaar, Henry L. 11860:Studia Mathematica 11844:10338.dmlcz/101743 10096: 10069: 10048: 10033: 9979: 9958: 9759: 9728: 9613: 9579: 9550: 9362: 9167: 9139: 9111: 9083: 9055: 9023: 8995: 8967: 8939: 8911: 8879: 8851: 8823: 8795: 8767: 8735: 8707: 8679: 8651: 8623: 8591: 8561: 8531: 8501: 8301: 8266: 8192: 8165: 8135: 8054: 8000:radius of the pipe 7949: 7913: 7865:Quantity of matter 7855:Quantity of matter 7771: 7544: 7509: 7475: 7432: 7391: 7270: 7241: 7225: 7186: 7155: 7120: 7091: 7036: 7014:This leads to the 7009:frame of reference 6948:point of reference 6889:add two positions. 6779:= magnetic field, 6773:= electric field, 6756: 6691: 6634: 6560:{\displaystyle qA} 6557: 6497: 6456: 6416: 6345: 6274: 6180: 6080: 6023:electric potential 5994: 5920: 5847: 5778: 5684: 5594: 5536:{\displaystyle Fd} 5533: 5217:In contrast, in a 5187:Quantity equations 5166: 5115: 5063: 4977: 4975: 4917: 4824: 4730: 4658: 4568: 4344:the space, and be 4306: 4212: 3898:demand/capacity = 3858: 3790: 3725: 3652: 3650: 3467: 3435: 3432: 3069: 3019: 2952: 2896: 2778:can be defined as 2412:Apples and oranges 2399:debt-to-GDP ratios 2360: 2359: 2315:has the dimension 1944:can be written as 1891:dependent variable 1870:in the form of an 1828: 1626: 1459: 1382: 1232: 1087: 932: 801: 679: 573:conversion factors 426: 287:luminous intensity 210:physical constants 144:quantity dimension 140:physical dimension 13739: 13738: 13667: 13666: 13379: 13378: 13267: 13266: 12731: 12730: 12722:Foot–pound–second 12662: 12661: 12639:Heaviside–Lorentz 12403:978-0-321-62592-2 12378:978-981-02-0304-7 12360:978-0-7923-2031-9 12267:Petroleum Refiner 12146:978-0-262-16234-0 12095:10.1115/1.4018140 12075:10.1115/1.1943434 12036:978-0-88275-682-0 11959:978-0-387-94417-3 11935:978-0-89871-336-7 11910:978-1-84996-316-9 11807:Buckingham, Edgar 11799:978-0-548-91029-0 11715:978-0-521-43522-2 11702:Barenblatt, G. I. 11373:978-1-4503-6029-6 11226:978-3-642-17684-5 11186:. UCAM-CL-TR-391. 11062:978-0-470-03294-7 11038:978-0-7167-8964-2 11014:978-0-521-57507-2 10999:Woan, G. (2010), 10658:978-0-387-91581-4 10632:978-0-02-093400-4 10541:978-0-07-147217-3 10364:978-1-349-00247-4 10349:. pp. 1–34. 10320:978-92-822-2272-0 10285:10.1063/PT.3.1258 10244:Quantity calculus 10239:Geometric algebra 9945: 9916: 9898: 9850: 9722: 9699: 9683: 9652: 9539: 9513: 9480: 9454: 9421: 9404: 9277:has orientation 1 9180: 9179: 9136: 9108: 9080: 9051: 9020: 8964: 8936: 8907: 8876: 8848: 8792: 8763: 8732: 8704: 8676: 8587: 8557: 8527: 8264: 8260: 8189: 8162: 8133: 8124: 8052: 8042: 8007: 8006: 7946: 7910: 7283:Or dimensionally 7248: 7232: 7183: 7152: 7117: 7088: 7039:{\displaystyle m} 6853:Affine quantities 6795: 6794: 6581: 6580: 6454: 6414: 6226: 6225: 5807:moment of inertia 5204:complete equation 5200:quantity equation 5161: 5079:conversion factor 4916: 4823: 4782: 4747: 4729: 4687:metres per second 4633: 4585: 4567: 4541:, the expression 4527:, the expression 4450:functions, or to 4397:Avogadro constant 4078:, and can define 3977:reciprocal second 3973:reciprocal length 3957:is written as 1; 3781: 3709: 3696: 3643: 3612: 3465: 3433: 3431: 3119:parallelogram law 3064: 3014: 2947: 2946: 2891: 2765:Velocity of money 2759:debt-to-GDP ratio 2750:For example, the 2628:5 × 100 / 1 = 500 2607:conversion factor 2600:Conversion factor 2594:Conversion factor 2352: 2211:(N) is a unit of 1862:. It expresses a 1848:Rayleigh's method 1842:Rayleigh's method 1758: 1678: 1677: 1674: 1563: 1511: 1510: 1507: 1437: 1429: 1336: 1284: 1283: 1280: 1142: 1134: 1038: 984: 983: 980: 856: 848: 769: 731: 730: 727: 647: 630: 629: 626: 256:dimension symbols 217:physical quantity 96:are of different 16:(Redirected from 13784: 13721: 13720: 13676: 13675: 13650: 13620: 13598: 13593: 13571: 13566: 13562: 13560: 13552:electric current 13533: 13507: 13503: 13499: 13474: 13425: 13424: 13406: 13399: 13392: 13383: 13382: 12846:Mesures usuelles 12740: 12739: 12671: 12670: 12537: 12536: 12519: 12512: 12505: 12496: 12495: 12491: 12480: 12415: 12381: 12363: 12344:Internet Archive 12338:, chapter XI of 12332:Wilson, Edwin B. 12328: 12300: 12287: 12274: 12261: 12241: 12221: 12220: 12218:10.1038/095066c0 12188: 12177: 12149: 12134: 12123: 12110: 12097: 12089:(671): 671–678, 12077: 12068: 12039: 12018: 11989: 11962: 11938: 11913: 11895: 11884: 11882: 11875: 11857: 11847: 11846: 11802: 11781: 11768: 11767: 11738: 11718: 11689: 11678: 11672: 11671: 11661: 11637: 11631: 11624: 11618: 11617: 11615: 11613: 11599: 11593: 11592: 11590: 11588: 11574: 11565: 11564: 11562: 11560: 11546: 11540: 11539: 11537: 11535: 11521: 11515: 11514: 11512: 11510: 11496: 11490: 11489: 11487: 11450: 11438: 11432: 11431: 11429: 11414: 11403: 11397: 11392: 11386: 11385: 11351: 11345: 11344: 11337: 11331: 11330: 11322: 11316: 11315: 11307: 11301: 11300: 11298: 11287: 11278: 11272: 11271: 11269: 11246: 11237: 11231: 11230: 11210: 11194: 11188: 11187: 11169: 11163: 11162: 11134: 11128: 11127: 11099: 11093: 11092: 11072: 11066: 11065: 11053:Particle Physics 11048: 11042: 11041: 11024: 11018: 11017: 11006: 10996: 10990: 10989: 10987: 10985:hep-th/0208093v3 10975: 10969: 10968: 10956: 10945: 10936: 10935: 10933: 10931: 10915: 10909: 10903: 10897: 10896: 10870: 10861: 10855: 10849: 10843: 10841: 10835: 10825: 10819: 10813: 10807: 10801: 10795: 10794: 10792: 10790: 10775: 10769: 10768: 10758: 10752: 10742: 10736: 10735: 10725: 10719: 10712: 10706: 10705: 10704: 10688: 10682: 10681: 10673: 10667: 10666: 10642: 10636: 10635: 10618: 10612: 10609: 10603: 10602: 10582: 10573: 10572: 10552: 10546: 10545: 10529: 10519: 10513: 10512: 10506: 10496: 10490: 10489: 10467: 10461: 10460: 10459: 10457: 10451: 10444: 10430: 10424: 10423: 10398: 10378: 10369: 10368: 10338: 10332: 10331: 10329: 10327: 10312: 10298: 10289: 10288: 10270: 10262: 10234:Exterior algebra 10166: 10158: 10146: 10138: 10127: 10123: 10111: 10107: 10105: 10103: 10102: 10097: 10088: 10077: 10061: 10056: 10046: 10041: 10029: 10024: 10023: 10005: 9998: 9988: 9986: 9985: 9980: 9973: 9968: 9967: 9956: 9955: 9950: 9946: 9944: 9939: 9934: 9927: 9926: 9921: 9917: 9915: 9914: 9909: 9908: 9901: 9900: 9899: 9896: 9889: 9888: 9881: 9871: 9870: 9869: 9858: 9857: 9851: 9848: 9846: 9845: 9835: 9834: 9824: 9823: 9798: 9794: 9787: 9770: 9768: 9766: 9765: 9760: 9758: 9757: 9739: 9737: 9735: 9734: 9729: 9724: 9723: 9720: 9701: 9700: 9697: 9685: 9684: 9681: 9668: 9654: 9653: 9650: 9622: 9620: 9619: 9614: 9609: 9588: 9586: 9585: 9580: 9559: 9557: 9556: 9551: 9546: 9542: 9541: 9540: 9537: 9515: 9514: 9511: 9487: 9483: 9482: 9481: 9478: 9456: 9455: 9452: 9428: 9424: 9423: 9422: 9419: 9406: 9405: 9402: 9371: 9369: 9368: 9363: 9337: 9307: 9303: 9299: 9276: 9268: 9262:has orientation 9261: 9253: 9238: 9219: 9216:. Since (using 9215: 9208: 9201: 9197: 9193: 9184:Klein four-group 9176: 9174: 9173: 9168: 9166: 9165: 9148: 9146: 9145: 9140: 9138: 9137: 9134: 9120: 9118: 9117: 9112: 9110: 9109: 9106: 9092: 9090: 9089: 9084: 9082: 9081: 9078: 9064: 9062: 9061: 9056: 9054: 9053: 9052: 9032: 9030: 9029: 9024: 9022: 9021: 9018: 9004: 9002: 9001: 8996: 8994: 8993: 8976: 8974: 8973: 8968: 8966: 8965: 8962: 8948: 8946: 8945: 8940: 8938: 8937: 8934: 8920: 8918: 8917: 8912: 8910: 8909: 8908: 8888: 8886: 8885: 8880: 8878: 8877: 8874: 8860: 8858: 8857: 8852: 8850: 8849: 8846: 8832: 8830: 8829: 8824: 8822: 8821: 8804: 8802: 8801: 8796: 8794: 8793: 8790: 8776: 8774: 8773: 8768: 8766: 8765: 8764: 8744: 8742: 8741: 8736: 8734: 8733: 8730: 8716: 8714: 8713: 8708: 8706: 8705: 8702: 8688: 8686: 8685: 8680: 8678: 8677: 8674: 8660: 8658: 8657: 8652: 8650: 8649: 8632: 8630: 8629: 8624: 8622: 8621: 8620: 8600: 8598: 8597: 8592: 8590: 8589: 8588: 8570: 8568: 8567: 8562: 8560: 8559: 8558: 8540: 8538: 8537: 8532: 8530: 8529: 8528: 8510: 8508: 8507: 8502: 8500: 8499: 8498: 8478: 8477: 8474: 8459: 8451: 8445: 8436: 8415: 8409: 8403: 8382: 8368: 8362: 8356: 8313:Poiseuille's law 8310: 8308: 8307: 8302: 8297: 8282: 8275: 8273: 8272: 8267: 8265: 8263: 8262: 8261: 8253: 8246: 8245: 8244: 8232: 8231: 8230: 8219: 8201: 8199: 8198: 8193: 8191: 8190: 8187: 8174: 8172: 8171: 8166: 8164: 8163: 8160: 8144: 8142: 8141: 8136: 8134: 8132: 8131: 8126: 8125: 8117: 8113: 8112: 8111: 8099: 8098: 8097: 8086: 8081: 8080: 8063: 8061: 8060: 8055: 8053: 8051: 8043: 8035: 8033: 8028: 8027: 7997: 7984: 7971: 7958: 7956: 7955: 7950: 7948: 7947: 7944: 7922: 7920: 7919: 7914: 7912: 7911: 7903: 7881: 7880: 7876:Poiseuille's Law 7845: 7839: 7833: 7804: 7797: 7790: 7780: 7778: 7777: 7772: 7770: 7769: 7764: 7760: 7759: 7758: 7757: 7756: 7750: 7749: 7741: 7740: 7739: 7731: 7730: 7716: 7715: 7710: 7706: 7705: 7704: 7703: 7702: 7696: 7695: 7687: 7686: 7685: 7677: 7676: 7662: 7661: 7656: 7652: 7651: 7650: 7649: 7648: 7642: 7641: 7633: 7632: 7631: 7623: 7622: 7605: 7604: 7603: 7597: 7596: 7578: 7573: 7568: 7563: 7558: 7553: 7551: 7550: 7545: 7543: 7542: 7541: 7523: 7518: 7516: 7515: 7510: 7508: 7507: 7506: 7486: 7484: 7482: 7481: 7476: 7441: 7439: 7438: 7433: 7400: 7398: 7397: 7392: 7390: 7389: 7384: 7380: 7379: 7378: 7372: 7371: 7363: 7362: 7349: 7348: 7337: 7333: 7332: 7331: 7325: 7324: 7316: 7315: 7299: 7298: 7279: 7277: 7276: 7271: 7266: 7265: 7254: 7249: 7246: 7238: 7233: 7230: 7209:may be written: 7208: 7201: 7197: 7195: 7193: 7192: 7187: 7185: 7184: 7181: 7166: 7164: 7162: 7161: 7156: 7154: 7153: 7150: 7135: 7131: 7129: 7127: 7126: 7121: 7119: 7118: 7115: 7100: 7098: 7097: 7092: 7090: 7089: 7086: 7045: 7043: 7042: 7037: 6790: 6784: 6778: 6772: 6765: 6763: 6762: 6757: 6729:Electromagnetic 6719: 6713: 6707: 6700: 6698: 6697: 6692: 6684: 6656: 6650: 6643: 6641: 6640: 6635: 6630: 6591: 6583: 6582: 6573: 6566: 6564: 6563: 6558: 6542:Electromagnetic 6533: 6523: 6513: 6506: 6504: 6503: 6498: 6465: 6463: 6462: 6457: 6455: 6453: 6449: 6448: 6435: 6425: 6423: 6422: 6417: 6415: 6413: 6409: 6408: 6395: 6373: 6367: 6361: 6354: 6352: 6351: 6346: 6341: 6327: 6308: 6302: 6296: 6290: 6283: 6281: 6280: 6275: 6236: 6228: 6227: 6220:electric current 6213: 6206: 6196: 6189: 6187: 6186: 6181: 6138: 6127: 6117: 6106: 6096: 6089: 6087: 6086: 6081: 6076: 6068: 6067: 6052: 6051: 6020: 6010: 6003: 6001: 6000: 5995: 5979:Electromagnetic 5970: 5960: 5950: 5936: 5929: 5927: 5926: 5921: 5881: 5875: 5869: 5863: 5856: 5854: 5853: 5848: 5817:angular velocity 5814: 5804: 5797:angular momentum 5794: 5787: 5785: 5784: 5779: 5774: 5769: 5768: 5747: 5746: 5720: 5710: 5700: 5693: 5691: 5690: 5685: 5680: 5675: 5674: 5653: 5652: 5626: 5620: 5610: 5603: 5601: 5600: 5595: 5581: 5559: 5549: 5542: 5540: 5539: 5534: 5504: 5496: 5495: 5473: 5466: 5459: 5452: 5446: 5440: 5429: 5421: 5413: 5399: 5393: 5386: 5362: 5356: 5328: 5322: 5316: 5310: 5301: 5284: 5278: 5272: 5263: 5247: 5241: 5232: 5175: 5173: 5172: 5167: 5162: 5160: 5148: 5139: 5125:is identical to 5124: 5122: 5121: 5116: 5114: 5102: 5072: 5070: 5069: 5064: 5016: 5006: 4986: 4984: 4983: 4978: 4976: 4969: 4961: 4960: 4948: 4944: 4943: 4918: 4909: 4904: 4895: 4887: 4886: 4877: 4873: 4855: 4851: 4850: 4825: 4816: 4811: 4802: 4801: 4792: 4780: 4766: 4765: 4764: 4755: 4745: 4731: 4722: 4718: 4704: 4694: 4688: 4684: 4676: 4667: 4665: 4664: 4659: 4645: 4641: 4631: 4620: 4619: 4604: 4603: 4602: 4593: 4583: 4569: 4560: 4544: 4540: 4530: 4526: 4520: 4507: 4501: 4491: 4485: 4479: 4406: 4404: 4368: 4315: 4313: 4312: 4307: 4292: 4291: 4267: 4266: 4254: 4253: 4221: 4219: 4218: 4213: 4207: 4206: 4205: 4204: 4190: 4189: 4176: 4171: 4140: 4130: 4116: 4091: 4077: 4071: 4065: 4059: 4046: 4037: 4011: 3995: 3994: 3991: 3970: 3966: 3960: 3922: 3910: 3893: 3889: 3883: 3877: 3871: 3865: 3831: 3821: 3815: 3808: 3799: 3797: 3796: 3791: 3786: 3782: 3774: 3743: 3734: 3732: 3731: 3726: 3715: 3711: 3710: 3702: 3697: 3695: 3684: 3661: 3659: 3658: 3653: 3651: 3644: 3636: 3627: 3626: 3613: 3611: 3600: 3591: 3590: 3563: 3554: 3545: 3539: 3530: 3521: 3512: 3495: 3485: 3476: 3474: 3473: 3468: 3466: 3461: 3452: 3446: 3444: 3442: 3441: 3436: 3434: 3424: 3422: 3402: 3391: 3385: 3381: 3377: 3373: 3369: 3365: 3349: 3343: 3330: 3310: 3304: 3300: 3296: 3292: 3286: 3280: 3274: 3268: 3262: 3256: 3250: 3246: 3201: 3197: 3193: 3182:Coulomb constant 3175: 3167: 3146: 3084: 3078: 3076: 3075: 3070: 3065: 3057: 3052: 3037: 3028: 3026: 3025: 3020: 3015: 3013: 3012: 3011: 2998: 2990: 2985: 2970: 2961: 2959: 2958: 2953: 2948: 2938: 2934: 2929: 2914: 2905: 2903: 2902: 2897: 2892: 2887: 2875: 2870: 2855: 2826: 2820: 2814: 2804: 2798: 2792: 2744:financial ratios 2729: 2718: 2704: 2698: 2692: 2684: 2678: 2670: 2657: 2633: 2629: 2625: 2621: 2617: 2613: 2581: 2573: 2570: 2567: 2552: 2536: 2520: 2504: 2495: 2486: 2418:Kind of quantity 2392:stocks and flows 2385: 2382: 2379: 2376:) has dimension 2371: 2369: 2367: 2366: 2361: 2350: 2333: 2324: 2321: 2318: 2303: 2300: 2297:) has dimension 2292: 2256: 2243: 2230: 2218: 2134: 2128: 2122: 2116: 2104: 2098: 2092: 2086: 2058: 2052: 2046: 2040: 2030: 2024: 1984: 1939: 1928: 1919: 1910: 1901: 1837: 1835: 1834: 1829: 1824: 1823: 1822: 1821: 1810: 1809: 1808: 1807: 1793: 1792: 1791: 1790: 1776: 1775: 1774: 1773: 1759: 1757: 1756: 1755: 1747: 1746: 1739: 1738: 1732: 1731: 1726: 1725: 1718: 1717: 1709: 1708: 1700: 1699: 1698: 1692: 1691: 1684: 1679: 1675: 1672: 1671: 1647: 1635: 1633: 1632: 1627: 1622: 1621: 1613: 1612: 1605: 1604: 1598: 1597: 1592: 1591: 1584: 1583: 1582: 1581: 1564: 1562: 1557: 1556: 1555: 1549: 1548: 1543: 1542: 1535: 1534: 1526: 1525: 1517: 1512: 1508: 1505: 1504: 1480: 1468: 1466: 1465: 1460: 1455: 1454: 1448: 1447: 1438: 1435: 1430: 1427: 1403: 1391: 1389: 1388: 1383: 1378: 1377: 1371: 1370: 1365: 1364: 1357: 1356: 1348: 1347: 1337: 1335: 1330: 1329: 1328: 1322: 1321: 1316: 1315: 1308: 1307: 1299: 1298: 1290: 1285: 1281: 1278: 1277: 1253: 1241: 1239: 1238: 1233: 1228: 1227: 1221: 1220: 1215: 1214: 1207: 1206: 1198: 1197: 1187: 1186: 1177: 1176: 1170: 1169: 1163: 1162: 1154: 1153: 1143: 1140: 1135: 1132: 1108: 1096: 1094: 1093: 1088: 1083: 1082: 1076: 1075: 1067: 1066: 1059: 1058: 1050: 1049: 1039: 1037: 1036: 1031: 1030: 1023: 1022: 1021: 1015: 1014: 1008: 1007: 999: 998: 990: 985: 981: 978: 977: 953: 941: 939: 938: 933: 928: 927: 921: 920: 914: 913: 905: 904: 894: 893: 887: 886: 878: 877: 867: 866: 857: 854: 849: 846: 822: 810: 808: 807: 802: 797: 796: 790: 789: 781: 780: 770: 768: 763: 762: 761: 755: 754: 746: 745: 737: 732: 728: 725: 724: 700: 688: 686: 685: 680: 675: 674: 668: 667: 659: 658: 648: 646: 641: 636: 631: 627: 624: 623: 599: 578: 555: 548: 541: 527: 520: 506: 496: 487: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: 13792: 13791: 13787: 13786: 13785: 13783: 13782: 13781: 13742: 13741: 13740: 13735: 13708: 13677: 13673: 13668: 13649: 13643: 13618: 13591: 13570: I 13564: 13558: 13531: 13505: 13501: 13497: 13472: 13461: 13456: 13448: 13420:Base quantities 13415: 13410: 13380: 13375: 13349: 13323: 13263: 13207: 13161: 13080: 12969: 12772: 12727: 12700: 12658: 12615: 12573: 12528: 12523: 12453:Wayback Machine 12422: 12404: 12388: 12386:Further reading 12379: 12361: 12318:10.2307/2315883 12167:10.1002/spe.401 12161:(11): 1067–76, 12147: 12131:Sky in a Bottle 12037: 11960: 11936: 11918:Hart, George W. 11911: 11880: 11855: 11815:Physical Review 11800: 11786:Bridgman, P. W. 11730:(1–2): 73–111, 11716: 11697: 11692: 11679: 11675: 11638: 11634: 11625: 11621: 11611: 11609: 11601: 11600: 11596: 11586: 11584: 11576: 11575: 11568: 11558: 11556: 11548: 11547: 11543: 11533: 11531: 11523: 11522: 11518: 11508: 11506: 11498: 11497: 11493: 11485: 11471: 11448: 11439: 11435: 11427: 11412: 11404: 11400: 11393: 11389: 11374: 11352: 11348: 11339: 11338: 11334: 11323: 11319: 11308: 11304: 11296: 11285: 11279: 11275: 11267: 11249:SIGPLAN Notices 11244: 11238: 11234: 11227: 11208:10.1.1.174.6901 11195: 11191: 11170: 11166: 11151:10.1109/52.2021 11135: 11131: 11100: 11096: 11073: 11069: 11063: 11049: 11045: 11039: 11025: 11021: 11015: 10997: 10993: 10976: 10972: 10965: 10954: 10946: 10939: 10929: 10927: 10916: 10912: 10904: 10900: 10868: 10862: 10858: 10850: 10846: 10837: 10831: 10826: 10822: 10814: 10810: 10802: 10798: 10788: 10786: 10778:Ramsay, Angus. 10776: 10772: 10759: 10755: 10743: 10739: 10726: 10722: 10713: 10709: 10689: 10685: 10674: 10670: 10659: 10643: 10639: 10633: 10619: 10615: 10610: 10606: 10583: 10576: 10553: 10549: 10542: 10520: 10516: 10497: 10493: 10486: 10468: 10464: 10455: 10453: 10449: 10442: 10431: 10427: 10396:physics/0110060 10379: 10372: 10365: 10339: 10335: 10325: 10323: 10321: 10310: 10299: 10292: 10263: 10256: 10252: 10225: 10181: 10160: 10148: 10140: 10132: 10125: 10113: 10109: 10078: 10073: 10057: 10052: 10042: 10037: 10025: 10019: 10015: 10013: 10010: 10009: 10007: 10000: 9993: 9969: 9963: 9962: 9951: 9940: 9935: 9933: 9929: 9928: 9922: 9910: 9904: 9903: 9902: 9895: 9891: 9884: 9883: 9882: 9880: 9876: 9875: 9865: 9864: 9860: 9853: 9852: 9847: 9841: 9837: 9830: 9826: 9819: 9815: 9807: 9804: 9803: 9796: 9793: 9789: 9783: 9753: 9749: 9747: 9744: 9743: 9741: 9719: 9715: 9696: 9692: 9680: 9676: 9664: 9649: 9645: 9630: 9627: 9626: 9624: 9605: 9594: 9591: 9590: 9568: 9565: 9564: 9536: 9532: 9510: 9506: 9501: 9497: 9477: 9473: 9451: 9447: 9442: 9438: 9418: 9414: 9401: 9397: 9392: 9388: 9380: 9377: 9376: 9333: 9313: 9310: 9309: 9305: 9301: 9282: 9280: 9270: 9267: 9263: 9255: 9252: 9248: 9244: 9240: 9237: 9233: 9221: 9217: 9214: 9210: 9207: 9203: 9199: 9195: 9192: 9188: 9161: 9157: 9155: 9152: 9151: 9133: 9129: 9127: 9124: 9123: 9105: 9101: 9099: 9096: 9095: 9077: 9073: 9071: 9068: 9067: 9048: 9044: 9043: 9041: 9038: 9037: 9017: 9013: 9011: 9008: 9007: 8989: 8985: 8983: 8980: 8979: 8961: 8957: 8955: 8952: 8951: 8933: 8929: 8927: 8924: 8923: 8904: 8900: 8899: 8897: 8894: 8893: 8873: 8869: 8867: 8864: 8863: 8845: 8841: 8839: 8836: 8835: 8817: 8813: 8811: 8808: 8807: 8789: 8785: 8783: 8780: 8779: 8760: 8756: 8755: 8753: 8750: 8749: 8729: 8725: 8723: 8720: 8719: 8701: 8697: 8695: 8692: 8691: 8673: 8669: 8667: 8664: 8663: 8645: 8641: 8639: 8636: 8635: 8616: 8612: 8611: 8609: 8606: 8605: 8584: 8580: 8579: 8577: 8574: 8573: 8554: 8550: 8549: 8547: 8544: 8543: 8524: 8520: 8519: 8517: 8514: 8513: 8494: 8490: 8489: 8487: 8484: 8483: 8473: 8467: 8461: 8458: 8454: 8452: 8449: 8446: 8443: 8440: 8435: 8431: 8427: 8423: 8411: 8405: 8388: 8378: 8364: 8358: 8346: 8339: 8333: 8293: 8288: 8285: 8284: 8280: 8279:where now only 8252: 8251: 8247: 8240: 8236: 8226: 8225: 8221: 8220: 8218: 8210: 8207: 8206: 8186: 8182: 8180: 8177: 8176: 8159: 8155: 8153: 8150: 8149: 8127: 8116: 8115: 8114: 8107: 8103: 8093: 8092: 8088: 8087: 8085: 8076: 8072: 8070: 8067: 8066: 8044: 8034: 8032: 8023: 8019: 8017: 8014: 8013: 7995: 7982: 7969: 7943: 7939: 7937: 7934: 7933: 7902: 7901: 7899: 7896: 7895: 7857: 7846: 7843: 7840: 7837: 7834: 7831: 7799: 7792: 7785: 7765: 7752: 7751: 7745: 7744: 7743: 7742: 7732: 7726: 7725: 7724: 7723: 7722: 7718: 7717: 7711: 7698: 7697: 7691: 7690: 7689: 7688: 7678: 7672: 7671: 7670: 7669: 7668: 7664: 7663: 7657: 7644: 7643: 7637: 7636: 7635: 7634: 7624: 7618: 7617: 7616: 7615: 7614: 7610: 7609: 7599: 7598: 7592: 7591: 7590: 7588: 7585: 7584: 7579: 7576: 7571: 7569: 7566: 7561: 7559: 7556: 7537: 7536: 7532: 7530: 7527: 7526: 7524: 7521: 7502: 7501: 7497: 7495: 7492: 7491: 7449: 7446: 7445: 7443: 7409: 7406: 7405: 7385: 7374: 7373: 7364: 7358: 7357: 7356: 7355: 7351: 7350: 7338: 7327: 7326: 7317: 7311: 7310: 7309: 7308: 7304: 7303: 7294: 7293: 7291: 7288: 7287: 7261: 7257: 7250: 7245: 7234: 7229: 7217: 7214: 7213: 7206: 7199: 7180: 7176: 7174: 7171: 7170: 7168: 7149: 7145: 7143: 7140: 7139: 7137: 7133: 7114: 7110: 7108: 7105: 7104: 7102: 7085: 7081: 7079: 7076: 7075: 7068: 7056: 7031: 7028: 7027: 7024: 7001: 6861: 6855: 6850: 6838:dependent types 6831: 6810:, and later in 6800: 6786: 6780: 6774: 6768: 6736: 6733: 6732: 6715: 6714:= temperature, 6709: 6703: 6680: 6669: 6666: 6665: 6657:= acceleration 6652: 6646: 6626: 6612: 6609: 6608: 6587: 6569: 6549: 6546: 6545: 6529: 6519: 6509: 6486: 6483: 6482: 6444: 6440: 6436: 6434: 6432: 6429: 6428: 6404: 6400: 6396: 6394: 6389: 6386: 6385: 6369: 6363: 6357: 6337: 6323: 6318: 6315: 6314: 6304: 6298: 6292: 6286: 6257: 6254: 6253: 6232: 6209: 6208: 6202: 6192: 6151: 6148: 6147: 6134: 6133: 6123: 6113: 6112: 6102: 6092: 6072: 6063: 6059: 6047: 6043: 6038: 6035: 6034: 6016: 6013:electric charge 6006: 5986: 5983: 5982: 5973:Poynting vector 5966: 5956: 5946: 5932: 5897: 5894: 5893: 5877: 5876:= temperature, 5871: 5865: 5859: 5830: 5827: 5826: 5810: 5800: 5790: 5770: 5764: 5760: 5742: 5738: 5733: 5730: 5729: 5716: 5706: 5696: 5676: 5670: 5666: 5648: 5644: 5639: 5636: 5635: 5622: 5616: 5606: 5577: 5572: 5569: 5568: 5555: 5545: 5525: 5522: 5521: 5500: 5494: 5488: 5480: 5468: 5461: 5454: 5448: 5442: 5436: 5425: 5417: 5409: 5405:Michael J. Duff 5395: 5388: 5382: 5374: 5358: 5352: 5349: 5343: 5338: 5324: 5318: 5312: 5306: 5293: 5280: 5274: 5268: 5252: 5243: 5237: 5228: 5196: 5189: 5149: 5140: 5138: 5130: 5127: 5126: 5106: 5091: 5089: 5086: 5085: 5025: 5022: 5021: 5012: 5002: 4999: 4993: 4974: 4973: 4965: 4956: 4952: 4939: 4935: 4931: 4907: 4905: 4903: 4897: 4896: 4891: 4882: 4878: 4869: 4859: 4846: 4842: 4838: 4814: 4812: 4810: 4804: 4803: 4797: 4793: 4776: 4760: 4756: 4751: 4738: 4720: 4715: 4713: 4710: 4709: 4700: 4690: 4686: 4682: 4672: 4637: 4627: 4615: 4611: 4598: 4594: 4589: 4576: 4558: 4556: 4553: 4552: 4542: 4532: 4528: 4522: 4516: 4503: 4497: 4487: 4481: 4462: 4425: 4417:Vlasov equation 4402: 4400: 4385:electric charge 4380: 4366: 4334:change of basis 4325: 4287: 4283: 4262: 4258: 4249: 4245: 4237: 4234: 4233: 4200: 4196: 4195: 4191: 4185: 4181: 4172: 4161: 4149: 4146: 4145: 4136: 4128: 4122: 4118: 4112: 4098:natural pairing 4079: 4073: 4067: 4061: 4055: 4042: 4035: 4021:base quantities 4016:in the module. 3997: 3992: 3989: 3987: 3968: 3962: 3958: 3947: 3941: 3936: 3914: 3899: 3891: 3885: 3879: 3873: 3867: 3866:(L) and radius 3861: 3850: 3842:Reynolds number 3823: 3817: 3811: 3804: 3773: 3769: 3752: 3749: 3748: 3739: 3701: 3688: 3683: 3682: 3678: 3673: 3670: 3669: 3649: 3648: 3635: 3628: 3622: 3618: 3615: 3614: 3604: 3599: 3592: 3586: 3582: 3578: 3576: 3573: 3572: 3562: 3556: 3553: 3547: 3541: 3535: 3526: 3517: 3508: 3502: 3491: 3481: 3460: 3458: 3455: 3454: 3448: 3421: 3410: 3407: 3406: 3404: 3398: 3387: 3383: 3379: 3375: 3371: 3367: 3363: 3345: 3338: 3332: 3318: 3312: 3306: 3302: 3298: 3294: 3288: 3282: 3276: 3270: 3264: 3258: 3252: 3248: 3242: 3236: 3231: 3221:, in Fourier's 3199: 3195: 3191: 3189: 3173: 3163: 3138: 3098:, a student of 3096:François Daviet 3092: 3080: 3056: 3045: 3043: 3040: 3039: 3035: 3007: 3003: 2999: 2991: 2989: 2978: 2976: 2973: 2972: 2968: 2933: 2922: 2920: 2917: 2916: 2912: 2876: 2874: 2863: 2861: 2858: 2857: 2853: 2850:Reynolds number 2838:fluid mechanics 2834: 2832:Fluid mechanics 2822: 2816: 2806: 2800: 2794: 2779: 2736: 2728: 2720: 2714: 2706: 2700: 2694: 2686: 2680: 2674: 2666: 2653: 2648: 2640: 2632:5 bar = 500 kPa 2631: 2627: 2623: 2620:100 kPa / 1 bar 2619: 2615: 2612:100 kPa = 1 bar 2611: 2602: 2596: 2579: 2571: 2568: 2565: 2551: 2544: 2538: 2535: 2528: 2522: 2519: 2512: 2506: 2503: 2497: 2494: 2488: 2485: 2479: 2468: 2457:incommensurable 2442: 2420: 2414: 2408: 2383: 2380: 2377: 2341: 2338: 2337: 2335: 2329: 2322: 2319: 2316: 2301: 2298: 2265: 2248: 2239: 2228: 2225: 2216: 2166:concrete number 2162: 2143:non-dimensional 2130: 2124: 2118: 2112: 2100: 2094: 2088: 2082: 2054: 2048: 2042: 2036: 2026: 2023: 2014: 2008: 2002: 1989: 1982: 1973: 1966: 1959: 1945: 1938: 1930: 1927: 1921: 1918: 1912: 1909: 1903: 1897: 1885:Gather all the 1844: 1817: 1813: 1812: 1811: 1800: 1796: 1795: 1794: 1783: 1779: 1778: 1777: 1769: 1765: 1764: 1763: 1748: 1742: 1741: 1740: 1734: 1733: 1727: 1721: 1720: 1719: 1710: 1704: 1703: 1702: 1701: 1694: 1693: 1687: 1686: 1685: 1683: 1673:electric charge 1670: 1656: 1653: 1652: 1643: 1614: 1608: 1607: 1606: 1600: 1599: 1593: 1587: 1586: 1585: 1574: 1570: 1569: 1568: 1558: 1551: 1550: 1544: 1538: 1537: 1536: 1527: 1521: 1520: 1519: 1518: 1516: 1503: 1489: 1486: 1485: 1476: 1450: 1449: 1443: 1442: 1434: 1426: 1412: 1409: 1408: 1399: 1397:electric charge 1373: 1372: 1366: 1360: 1359: 1358: 1349: 1343: 1342: 1341: 1331: 1324: 1323: 1317: 1311: 1310: 1309: 1300: 1294: 1293: 1292: 1291: 1289: 1276: 1262: 1259: 1258: 1249: 1223: 1222: 1216: 1210: 1209: 1208: 1199: 1193: 1192: 1191: 1182: 1181: 1172: 1171: 1165: 1164: 1155: 1149: 1148: 1147: 1139: 1131: 1117: 1114: 1113: 1104: 1078: 1077: 1068: 1062: 1061: 1060: 1051: 1045: 1044: 1043: 1032: 1026: 1025: 1024: 1017: 1016: 1010: 1009: 1000: 994: 993: 992: 991: 989: 976: 962: 959: 958: 949: 923: 922: 916: 915: 906: 900: 899: 898: 889: 888: 879: 873: 872: 871: 862: 861: 853: 845: 831: 828: 827: 818: 792: 791: 782: 776: 775: 774: 764: 757: 756: 747: 741: 740: 739: 738: 736: 723: 709: 706: 705: 696: 670: 669: 660: 654: 653: 652: 642: 637: 635: 622: 608: 605: 604: 595: 589: 576: 550: 543: 536: 522: 515: 501: 494: 491: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: 13790: 13780: 13779: 13774: 13769: 13764: 13759: 13754: 13737: 13736: 13734: 13733: 13726: 13713: 13710: 13709: 13707: 13706: 13701: 13696: 13691: 13685: 13683: 13679: 13678: 13671: 13669: 13665: 13664: 13661: 13656: 13654: 13651: 13647: 13641: 13635: 13634: 13631: 13626: 13624: 13621: 13616: 13610: 13609: 13606: 13601: 13599: 13594: 13589: 13583: 13582: 13579: 13574: 13572: 13567: 13554: 13548: 13547: 13544: 13539: 13537: 13534: 13529: 13523: 13522: 13519: 13514: 13512: 13509: 13495: 13489: 13488: 13485: 13480: 13478: 13475: 13470: 13468:time, duration 13464: 13463: 13458: 13453: 13451: 13444: 13441: 13437: 13436: 13431: 13429: 13423: 13421: 13417: 13416: 13409: 13408: 13401: 13394: 13386: 13377: 13376: 13374: 13373: 13368: 13363: 13361:Absolute scale 13357: 13355: 13351: 13350: 13348: 13347: 13342: 13337: 13331: 13329: 13325: 13324: 13322: 13321: 13316: 13311: 13306: 13301: 13296: 13291: 13286: 13281: 13275: 13273: 13269: 13268: 13265: 13264: 13262: 13261: 13256: 13251: 13246: 13241: 13236: 13231: 13226: 13221: 13215: 13213: 13209: 13208: 13206: 13205: 13200: 13195: 13190: 13185: 13180: 13175: 13169: 13167: 13163: 13162: 13160: 13159: 13154: 13149: 13144: 13139: 13134: 13129: 13124: 13119: 13114: 13109: 13104: 13099: 13094: 13088: 13086: 13082: 13081: 13079: 13078: 13073: 13068: 13063: 13058: 13053: 13048: 13043: 13038: 13033: 13028: 13023: 13018: 13013: 13008: 13003: 12998: 12993: 12988: 12983: 12977: 12975: 12971: 12970: 12968: 12967: 12962: 12957: 12952: 12947: 12942: 12937: 12932: 12927: 12922: 12917: 12912: 12907: 12902: 12897: 12892: 12887: 12882: 12877: 12872: 12867: 12866: 12865: 12855: 12850: 12849: 12848: 12843: 12833: 12828: 12823: 12822: 12821: 12816: 12806: 12801: 12796: 12791: 12786: 12780: 12778: 12774: 12773: 12771: 12770: 12765: 12759: 12753: 12746: 12744: 12737: 12733: 12732: 12729: 12728: 12726: 12725: 12719: 12714: 12708: 12706: 12702: 12701: 12699: 12698: 12693: 12688: 12683: 12677: 12675: 12668: 12664: 12663: 12660: 12659: 12657: 12656: 12651: 12646: 12641: 12636: 12631: 12625: 12623: 12617: 12616: 12614: 12613: 12607: 12602: 12597: 12592: 12587: 12581: 12579: 12575: 12574: 12572: 12571: 12570: 12569: 12559: 12554: 12549: 12543: 12541: 12534: 12530: 12529: 12522: 12521: 12514: 12507: 12499: 12493: 12492: 12481: 12460: 12455: 12443: 12438: 12433: 12428: 12421: 12420:External links 12418: 12417: 12416: 12402: 12387: 12384: 12383: 12382: 12377: 12364: 12359: 12346: 12329: 12301: 12288: 12276: 12262: 12253:(6): 285–302, 12242: 12233:(6): 267–283, 12222: 12203:(2368): 66–8, 12189: 12178: 12150: 12145: 12124: 12111: 12098: 12078: 12066:10.1.1.422.610 12051:(5): 648–657, 12040: 12035: 12019: 11990: 11988:, LOC 67-17978 11963: 11958: 11943: 11934: 11914: 11909: 11896: 11885: 11848: 11821:(4): 345–376, 11803: 11798: 11782: 11769: 11765:10.1086/170003 11739: 11719: 11714: 11696: 11693: 11691: 11690: 11673: 11632: 11619: 11594: 11566: 11541: 11516: 11491: 11469: 11442:McBride, Conor 11433: 11398: 11387: 11372: 11346: 11332: 11317: 11302: 11273: 11232: 11225: 11189: 11164: 11129: 11110:(6): 555–569. 11094: 11067: 11061: 11043: 11037: 11019: 11013: 10991: 10970: 10963: 10937: 10910: 10898: 10856: 10844: 10820: 10808: 10796: 10770: 10753: 10745:Fourier (1822) 10737: 10720: 10707: 10683: 10668: 10657: 10637: 10631: 10613: 10604: 10593:(5): 331–337. 10574: 10563:(6): 391–340. 10547: 10540: 10514: 10491: 10484: 10462: 10425: 10370: 10363: 10333: 10319: 10290: 10253: 10251: 10248: 10247: 10246: 10241: 10236: 10231: 10224: 10221: 10220: 10219: 10214: 10208: 10203: 10198: 10195:Fermi estimate 10192: 10187: 10180: 10177: 10095: 10092: 10087: 10084: 10081: 10076: 10072: 10068: 10065: 10060: 10055: 10051: 10045: 10040: 10036: 10032: 10028: 10022: 10018: 9990: 9989: 9977: 9972: 9966: 9961: 9954: 9949: 9943: 9938: 9932: 9925: 9920: 9913: 9907: 9894: 9887: 9879: 9874: 9868: 9863: 9856: 9844: 9840: 9833: 9829: 9822: 9818: 9814: 9811: 9791: 9756: 9752: 9727: 9718: 9713: 9710: 9707: 9704: 9695: 9691: 9688: 9679: 9674: 9671: 9667: 9663: 9660: 9657: 9648: 9643: 9640: 9637: 9634: 9612: 9608: 9604: 9601: 9598: 9578: 9575: 9572: 9561: 9560: 9549: 9545: 9535: 9530: 9527: 9524: 9521: 9518: 9509: 9504: 9500: 9496: 9493: 9490: 9486: 9476: 9471: 9468: 9465: 9462: 9459: 9450: 9445: 9441: 9437: 9434: 9431: 9427: 9417: 9412: 9409: 9400: 9395: 9391: 9387: 9384: 9361: 9358: 9355: 9352: 9349: 9346: 9343: 9340: 9336: 9332: 9329: 9326: 9323: 9320: 9317: 9278: 9265: 9250: 9246: 9242: 9235: 9231: 9212: 9205: 9190: 9178: 9177: 9164: 9160: 9149: 9132: 9121: 9104: 9093: 9076: 9065: 9047: 9034: 9033: 9016: 9005: 8992: 8988: 8977: 8960: 8949: 8932: 8921: 8903: 8890: 8889: 8872: 8861: 8844: 8833: 8820: 8816: 8805: 8788: 8777: 8759: 8746: 8745: 8728: 8717: 8700: 8689: 8672: 8661: 8648: 8644: 8633: 8619: 8615: 8602: 8601: 8583: 8571: 8553: 8541: 8523: 8511: 8497: 8493: 8481: 8469: 8463: 8456: 8448: 8442: 8438: 8433: 8429: 8425: 8332: 8329: 8300: 8296: 8292: 8277: 8276: 8259: 8256: 8250: 8243: 8239: 8235: 8229: 8224: 8217: 8214: 8185: 8158: 8146: 8145: 8130: 8123: 8120: 8110: 8106: 8102: 8096: 8091: 8084: 8079: 8075: 8064: 8050: 8047: 8041: 8038: 8031: 8026: 8022: 8005: 8004: 8001: 7998: 7992: 7991: 7988: 7985: 7979: 7978: 7975: 7972: 7966: 7965: 7962: 7959: 7942: 7930: 7929: 7926: 7925:mass flow rate 7923: 7909: 7906: 7892: 7891: 7888: 7885: 7870: 7856: 7853: 7842: 7836: 7830: 7827: 7826: 7819: 7782: 7781: 7768: 7763: 7755: 7748: 7738: 7735: 7729: 7721: 7714: 7709: 7701: 7694: 7684: 7681: 7675: 7667: 7660: 7655: 7647: 7640: 7630: 7627: 7621: 7613: 7608: 7602: 7595: 7575: 7565: 7555: 7540: 7535: 7520: 7505: 7500: 7474: 7471: 7468: 7465: 7462: 7459: 7456: 7453: 7431: 7428: 7425: 7422: 7419: 7416: 7413: 7402: 7401: 7388: 7383: 7377: 7370: 7367: 7361: 7354: 7347: 7344: 7341: 7336: 7330: 7323: 7320: 7314: 7307: 7302: 7297: 7281: 7280: 7269: 7264: 7260: 7253: 7244: 7237: 7228: 7224: 7221: 7179: 7148: 7113: 7084: 7067: 7064: 7063: 7062: 7059: 7054: 7035: 7023: 7020: 7000: 6997: 6985: 6984: 6974:corresponds to 6970: 6969: 6932: 6931: 6920: 6891: 6890: 6883: 6880: 6877: 6854: 6851: 6849: 6846: 6799: 6796: 6793: 6792: 6766: 6755: 6752: 6749: 6746: 6743: 6740: 6730: 6726: 6725: 6722:entropic force 6701: 6690: 6687: 6683: 6679: 6676: 6673: 6663: 6659: 6658: 6644: 6633: 6629: 6625: 6622: 6619: 6616: 6606: 6602: 6601: 6598: 6595: 6579: 6578: 6567: 6556: 6553: 6543: 6539: 6538: 6536:phase velocity 6507: 6496: 6493: 6490: 6480: 6476: 6475: 6452: 6447: 6443: 6439: 6426: 6412: 6407: 6403: 6399: 6393: 6383: 6379: 6378: 6355: 6344: 6340: 6336: 6333: 6330: 6326: 6322: 6311: 6310: 6284: 6273: 6270: 6267: 6264: 6261: 6251: 6247: 6246: 6243: 6240: 6224: 6223: 6190: 6179: 6176: 6173: 6170: 6167: 6164: 6161: 6158: 6155: 6144: 6143: 6109:magnetic field 6099:electric field 6090: 6079: 6075: 6071: 6066: 6062: 6058: 6055: 6050: 6046: 6042: 6031: 6030: 6004: 5993: 5990: 5980: 5976: 5975: 5930: 5919: 5916: 5913: 5910: 5907: 5904: 5901: 5891: 5887: 5886: 5857: 5846: 5843: 5840: 5837: 5834: 5824: 5820: 5819: 5788: 5777: 5773: 5767: 5763: 5759: 5756: 5753: 5750: 5745: 5741: 5737: 5726: 5725: 5694: 5683: 5679: 5673: 5669: 5665: 5662: 5659: 5656: 5651: 5647: 5643: 5632: 5631: 5604: 5593: 5590: 5587: 5584: 5580: 5576: 5565: 5564: 5543: 5532: 5529: 5519: 5515: 5514: 5511: 5508: 5490:Main article: 5487: 5484: 5479: 5476: 5373: 5370: 5345:Main article: 5342: 5339: 5337: 5334: 5303: 5302: 5265: 5264: 5188: 5185: 5177: 5176: 5165: 5159: 5156: 5152: 5147: 5143: 5137: 5134: 5113: 5109: 5105: 5101: 5098: 5094: 5074: 5073: 5062: 5059: 5056: 5053: 5050: 5047: 5044: 5041: 5038: 5035: 5032: 5029: 4995:Main article: 4992: 4989: 4988: 4987: 4972: 4968: 4964: 4959: 4955: 4951: 4947: 4942: 4938: 4934: 4930: 4927: 4924: 4921: 4915: 4912: 4906: 4902: 4899: 4898: 4894: 4890: 4885: 4881: 4876: 4872: 4868: 4865: 4862: 4858: 4854: 4849: 4845: 4841: 4837: 4834: 4831: 4828: 4822: 4819: 4813: 4809: 4806: 4805: 4800: 4796: 4791: 4788: 4785: 4779: 4775: 4772: 4769: 4763: 4759: 4754: 4750: 4744: 4741: 4737: 4734: 4728: 4725: 4719: 4717: 4669: 4668: 4657: 4654: 4651: 4648: 4644: 4640: 4636: 4630: 4626: 4623: 4618: 4614: 4610: 4607: 4601: 4597: 4592: 4588: 4582: 4579: 4575: 4572: 4566: 4563: 4492:, but it does 4471:) = log 4424: 4421: 4389:thermodynamics 4379: 4376: 4375: 4374: 4363: 4360:span the space 4324: 4321: 4317: 4316: 4305: 4301: 4298: 4295: 4290: 4286: 4282: 4279: 4276: 4273: 4270: 4265: 4261: 4257: 4252: 4248: 4244: 4241: 4223: 4222: 4211: 4203: 4199: 4194: 4188: 4184: 4180: 4175: 4170: 4167: 4164: 4160: 4156: 4153: 4133:exponentiating 4124: 4120: 4094:tensor product 4051:obstructions. 3940: 3937: 3935: 3932: 3924: 3923: 3911: 3849: 3846: 3801: 3800: 3789: 3785: 3780: 3777: 3772: 3768: 3765: 3762: 3759: 3756: 3736: 3735: 3724: 3721: 3718: 3714: 3708: 3705: 3700: 3694: 3691: 3687: 3681: 3677: 3663: 3662: 3647: 3642: 3639: 3634: 3631: 3629: 3625: 3621: 3617: 3616: 3610: 3607: 3603: 3598: 3595: 3593: 3589: 3585: 3581: 3580: 3560: 3551: 3524:linear density 3501: 3498: 3464: 3430: 3427: 3420: 3417: 3414: 3336: 3331:, and putting 3316: 3235: 3232: 3230: 3227: 3187: 3134:Joseph Fourier 3115:Simeon Poisson 3091: 3088: 3087: 3086: 3068: 3063: 3060: 3055: 3051: 3048: 3029: 3018: 3010: 3006: 3002: 2997: 2994: 2988: 2984: 2981: 2962: 2951: 2945: 2941: 2937: 2932: 2928: 2925: 2906: 2895: 2890: 2886: 2883: 2879: 2873: 2869: 2866: 2833: 2830: 2829: 2828: 2772: 2768: 2762: 2757:In economics, 2755: 2735: 2732: 2724: 2710: 2647: 2644: 2639: 2636: 2598:Main article: 2595: 2592: 2549: 2542: 2533: 2526: 2517: 2510: 2501: 2492: 2483: 2450: 2424: 2407: 2404: 2388: 2387: 2358: 2355: 2349: 2346: 2326: 2306: 2305: 2262: 2245: 2224: 2221: 2217:1 N = 1 kg⋅m⋅s 2174:multiplication 2161: 2158: 2154: 2153: 2136: 2106: 2067: 2060: 2019: 2012: 2006: 2000: 1986: 1978: 1971: 1964: 1957: 1934: 1925: 1916: 1907: 1894: 1843: 1840: 1839: 1838: 1827: 1820: 1816: 1806: 1803: 1799: 1789: 1786: 1782: 1772: 1768: 1762: 1754: 1751: 1745: 1737: 1730: 1724: 1716: 1713: 1707: 1697: 1690: 1682: 1669: 1666: 1663: 1660: 1637: 1636: 1625: 1620: 1617: 1611: 1603: 1596: 1590: 1580: 1577: 1573: 1567: 1561: 1554: 1547: 1541: 1533: 1530: 1524: 1515: 1502: 1499: 1496: 1493: 1470: 1469: 1458: 1453: 1446: 1441: 1433: 1425: 1422: 1419: 1416: 1393: 1392: 1381: 1376: 1369: 1363: 1355: 1352: 1346: 1340: 1334: 1327: 1320: 1314: 1306: 1303: 1297: 1288: 1275: 1272: 1269: 1266: 1243: 1242: 1231: 1226: 1219: 1213: 1205: 1202: 1196: 1190: 1185: 1180: 1175: 1168: 1161: 1158: 1152: 1146: 1138: 1130: 1127: 1124: 1121: 1098: 1097: 1086: 1081: 1074: 1071: 1065: 1057: 1054: 1048: 1042: 1035: 1029: 1020: 1013: 1006: 1003: 997: 988: 975: 972: 969: 966: 943: 942: 931: 926: 919: 912: 909: 903: 897: 892: 885: 882: 876: 870: 865: 860: 852: 844: 841: 838: 835: 812: 811: 800: 795: 788: 785: 779: 773: 767: 760: 753: 750: 744: 735: 722: 719: 716: 713: 690: 689: 678: 673: 666: 663: 657: 651: 645: 640: 634: 621: 618: 615: 612: 588: 585: 577:1 in = 2.54 cm 556:is known as a 528: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: 13789: 13778: 13775: 13773: 13770: 13768: 13765: 13763: 13760: 13758: 13755: 13753: 13750: 13749: 13747: 13732: 13731: 13727: 13725: 13724: 13715: 13714: 13711: 13705: 13702: 13700: 13699:2019 revision 13697: 13695: 13692: 13690: 13687: 13686: 13684: 13680: 13662: 13660: 13657: 13655: 13652: 13646: 13642: 13640: 13637: 13636: 13632: 13630: 13627: 13625: 13622: 13617: 13615: 13612: 13611: 13607: 13605: 13602: 13600: 13595: 13590: 13588: 13585: 13584: 13580: 13578: 13575: 13573: 13568: 13555: 13553: 13550: 13549: 13545: 13543: 13540: 13538: 13535: 13530: 13528: 13525: 13524: 13520: 13518: 13515: 13513: 13510: 13496: 13494: 13491: 13490: 13486: 13484: 13481: 13479: 13476: 13471: 13469: 13466: 13465: 13459: 13454: 13452: 13450: 13445: 13442: 13439: 13438: 13435: 13430: 13426: 13422: 13418: 13414: 13407: 13402: 13400: 13395: 13393: 13388: 13387: 13384: 13372: 13369: 13367: 13364: 13362: 13359: 13358: 13356: 13352: 13346: 13343: 13341: 13338: 13336: 13333: 13332: 13330: 13328:List articles 13326: 13320: 13317: 13315: 13312: 13310: 13307: 13305: 13302: 13300: 13297: 13295: 13292: 13290: 13287: 13285: 13282: 13280: 13277: 13276: 13274: 13270: 13260: 13257: 13255: 13252: 13250: 13247: 13245: 13242: 13240: 13237: 13235: 13232: 13230: 13227: 13225: 13222: 13220: 13217: 13216: 13214: 13212:South America 13210: 13204: 13201: 13199: 13196: 13194: 13191: 13189: 13186: 13184: 13181: 13179: 13176: 13174: 13171: 13170: 13168: 13166:North America 13164: 13158: 13155: 13153: 13150: 13148: 13147:South African 13145: 13143: 13140: 13138: 13135: 13133: 13130: 13128: 13125: 13123: 13120: 13118: 13115: 13113: 13110: 13108: 13105: 13103: 13100: 13098: 13095: 13093: 13090: 13089: 13087: 13083: 13077: 13074: 13072: 13069: 13067: 13064: 13062: 13059: 13057: 13054: 13052: 13049: 13047: 13044: 13042: 13039: 13037: 13034: 13032: 13029: 13027: 13024: 13022: 13019: 13017: 13014: 13012: 13009: 13007: 13004: 13002: 12999: 12997: 12994: 12992: 12989: 12987: 12984: 12982: 12979: 12978: 12976: 12972: 12966: 12963: 12961: 12958: 12956: 12953: 12951: 12948: 12946: 12943: 12941: 12938: 12936: 12933: 12931: 12928: 12926: 12923: 12921: 12918: 12916: 12913: 12911: 12908: 12906: 12903: 12901: 12898: 12896: 12895:Luxembourgian 12893: 12891: 12888: 12886: 12883: 12881: 12878: 12876: 12873: 12871: 12868: 12864: 12861: 12860: 12859: 12856: 12854: 12851: 12847: 12844: 12842: 12839: 12838: 12837: 12834: 12832: 12829: 12827: 12824: 12820: 12817: 12815: 12812: 12811: 12810: 12807: 12805: 12802: 12800: 12797: 12795: 12792: 12790: 12787: 12785: 12782: 12781: 12779: 12775: 12769: 12768:gravitational 12766: 12763: 12760: 12757: 12754: 12751: 12748: 12747: 12745: 12741: 12738: 12734: 12723: 12720: 12718: 12715: 12713: 12710: 12709: 12707: 12703: 12697: 12694: 12692: 12689: 12687: 12684: 12682: 12679: 12678: 12676: 12672: 12669: 12665: 12655: 12652: 12650: 12647: 12645: 12642: 12640: 12637: 12635: 12632: 12630: 12627: 12626: 12624: 12622: 12618: 12611: 12608: 12606: 12603: 12601: 12598: 12596: 12593: 12591: 12588: 12586: 12585:Apothecaries' 12583: 12582: 12580: 12576: 12568: 12565: 12564: 12563: 12560: 12558: 12555: 12553: 12550: 12548: 12545: 12544: 12542: 12538: 12535: 12531: 12527: 12520: 12515: 12513: 12508: 12506: 12501: 12500: 12497: 12489: 12488: 12482: 12478: 12474: 12470: 12469:Sixty Symbols 12466: 12461: 12459: 12456: 12454: 12450: 12447: 12444: 12442: 12439: 12437: 12434: 12432: 12429: 12427: 12424: 12423: 12413: 12409: 12405: 12399: 12395: 12390: 12389: 12380: 12374: 12370: 12365: 12362: 12356: 12352: 12347: 12345: 12341: 12337: 12333: 12330: 12327: 12323: 12319: 12315: 12311: 12307: 12302: 12298: 12294: 12289: 12285: 12281: 12277: 12272: 12268: 12263: 12260: 12256: 12252: 12248: 12243: 12240: 12236: 12232: 12228: 12223: 12219: 12214: 12210: 12206: 12202: 12198: 12194: 12190: 12186: 12185: 12179: 12176: 12172: 12168: 12164: 12160: 12156: 12151: 12148: 12142: 12138: 12133: 12132: 12125: 12121: 12117: 12112: 12108: 12104: 12099: 12096: 12092: 12088: 12084: 12079: 12076: 12072: 12067: 12062: 12058: 12054: 12050: 12046: 12041: 12038: 12032: 12028: 12024: 12020: 12017: 12013: 12009: 12005: 12001: 11997: 11991: 11987: 11983: 11979: 11975: 11971: 11970: 11964: 11961: 11955: 11951: 11950: 11944: 11942: 11937: 11931: 11927: 11923: 11919: 11915: 11912: 11906: 11902: 11897: 11893: 11892: 11886: 11879: 11874: 11869: 11865: 11861: 11854: 11849: 11845: 11840: 11836: 11832: 11828: 11824: 11820: 11816: 11812: 11808: 11804: 11801: 11795: 11791: 11787: 11783: 11779: 11775: 11770: 11766: 11761: 11757: 11753: 11749: 11745: 11740: 11737: 11733: 11729: 11725: 11720: 11717: 11711: 11707: 11703: 11699: 11698: 11687: 11683: 11677: 11669: 11665: 11660: 11655: 11652:(5): 053002. 11651: 11647: 11643: 11636: 11629: 11623: 11608: 11604: 11598: 11583: 11579: 11573: 11571: 11555: 11551: 11545: 11530: 11526: 11520: 11505: 11501: 11495: 11484: 11480: 11476: 11472: 11470:9789811242380 11466: 11462: 11458: 11454: 11447: 11443: 11437: 11426: 11422: 11418: 11411: 11410: 11402: 11396: 11391: 11383: 11379: 11375: 11369: 11365: 11361: 11357: 11350: 11342: 11336: 11328: 11321: 11313: 11306: 11295: 11291: 11284: 11277: 11266: 11262: 11258: 11255:(12): 11–22. 11254: 11250: 11243: 11236: 11228: 11222: 11218: 11214: 11209: 11204: 11200: 11193: 11185: 11181: 11177: 11176: 11168: 11160: 11156: 11152: 11148: 11144: 11140: 11139:IEEE Software 11133: 11125: 11121: 11117: 11113: 11109: 11105: 11098: 11090: 11086: 11083:(3): 93–111. 11082: 11078: 11071: 11064: 11058: 11054: 11047: 11040: 11034: 11030: 11023: 11016: 11010: 11005: 11004: 10995: 10986: 10981: 10974: 10966: 10964:9781437915594 10960: 10953: 10952: 10944: 10942: 10926: 10922: 10914: 10908:, p. 256 10907: 10902: 10894: 10890: 10886: 10882: 10878: 10874: 10867: 10860: 10853: 10852:Bridgman 1922 10848: 10840: 10834: 10829: 10824: 10817: 10812: 10805: 10800: 10785: 10781: 10774: 10766: 10765: 10757: 10750: 10746: 10741: 10733: 10732: 10724: 10717: 10711: 10703: 10698: 10694: 10687: 10679: 10672: 10665: 10660: 10654: 10650: 10649: 10641: 10634: 10628: 10624: 10617: 10608: 10600: 10596: 10592: 10588: 10581: 10579: 10570: 10566: 10562: 10558: 10551: 10543: 10537: 10533: 10528: 10527: 10518: 10510: 10505: 10504: 10495: 10487: 10485:9780073138350 10481: 10477: 10473: 10466: 10448: 10441: 10440: 10435: 10429: 10422: 10418: 10414: 10410: 10406: 10402: 10397: 10392: 10388: 10384: 10377: 10375: 10366: 10360: 10356: 10352: 10348: 10344: 10337: 10322: 10316: 10309: 10308: 10303: 10297: 10295: 10286: 10282: 10278: 10274: 10273:Physics Today 10269: 10261: 10259: 10254: 10245: 10242: 10240: 10237: 10235: 10232: 10230: 10227: 10226: 10218: 10215: 10212: 10209: 10207: 10204: 10202: 10199: 10196: 10193: 10191: 10188: 10186: 10183: 10182: 10176: 10173: 10168: 10164: 10156: 10152: 10144: 10136: 10129: 10121: 10117: 10093: 10090: 10085: 10082: 10079: 10074: 10070: 10066: 10058: 10053: 10049: 10043: 10038: 10034: 10026: 10020: 10016: 10003: 9996: 9975: 9970: 9959: 9952: 9947: 9930: 9923: 9918: 9911: 9892: 9877: 9872: 9861: 9842: 9838: 9831: 9827: 9820: 9816: 9812: 9809: 9802: 9801: 9800: 9786: 9780: 9778: 9772: 9754: 9750: 9716: 9711: 9705: 9702: 9693: 9689: 9677: 9669: 9665: 9661: 9655: 9646: 9641: 9635: 9632: 9610: 9606: 9602: 9599: 9596: 9576: 9573: 9570: 9547: 9543: 9533: 9528: 9522: 9519: 9507: 9502: 9498: 9494: 9491: 9488: 9484: 9474: 9469: 9463: 9460: 9448: 9443: 9439: 9435: 9432: 9429: 9425: 9415: 9410: 9407: 9398: 9393: 9389: 9385: 9382: 9375: 9374: 9373: 9356: 9350: 9347: 9344: 9338: 9334: 9330: 9327: 9324: 9318: 9315: 9297: 9293: 9289: 9285: 9274: 9259: 9229: 9225: 9185: 9162: 9158: 9150: 9130: 9122: 9102: 9094: 9074: 9066: 9035: 9014: 9006: 8990: 8986: 8978: 8958: 8950: 8930: 8922: 8891: 8870: 8862: 8842: 8834: 8818: 8814: 8806: 8786: 8778: 8747: 8726: 8718: 8698: 8690: 8670: 8662: 8646: 8642: 8634: 8603: 8572: 8542: 8512: 8482: 8480: 8479: 8476: 8472: 8466: 8422: 8417: 8414: 8408: 8402: 8398: 8395: 8391: 8386: 8381: 8376: 8372: 8367: 8361: 8354: 8350: 8343: 8338: 8328: 8326: 8322: 8316: 8314: 8298: 8294: 8290: 8257: 8254: 8248: 8241: 8237: 8233: 8222: 8215: 8212: 8205: 8204: 8203: 8183: 8156: 8128: 8121: 8118: 8108: 8104: 8100: 8089: 8082: 8077: 8073: 8065: 8048: 8045: 8039: 8036: 8029: 8024: 8020: 8012: 8011: 8010: 8002: 7999: 7994: 7993: 7989: 7986: 7981: 7980: 7976: 7973: 7968: 7967: 7963: 7960: 7940: 7932: 7931: 7927: 7924: 7907: 7904: 7894: 7893: 7889: 7886: 7883: 7882: 7879: 7877: 7872: 7868: 7866: 7862: 7861:inertial mass 7852: 7848: 7824: 7820: 7817: 7816: 7815:cross product 7811: 7810: 7809: 7806: 7802: 7795: 7788: 7766: 7761: 7736: 7733: 7719: 7712: 7707: 7682: 7679: 7665: 7658: 7653: 7628: 7625: 7611: 7606: 7583: 7582: 7581: 7533: 7498: 7488: 7472: 7469: 7466: 7463: 7460: 7457: 7454: 7451: 7429: 7426: 7423: 7420: 7417: 7414: 7411: 7386: 7381: 7368: 7365: 7352: 7345: 7342: 7339: 7334: 7321: 7318: 7305: 7300: 7286: 7285: 7284: 7267: 7262: 7258: 7251: 7242: 7235: 7226: 7222: 7219: 7212: 7211: 7210: 7203: 7177: 7146: 7111: 7082: 7073: 7060: 7052: 7051: 7050: 7047: 7033: 7019: 7017: 7012: 7010: 7006: 6996: 6994: 6993:Réaumur scale 6990: 6989:Rankine scale 6982: 6981: 6980: 6977: 6975: 6967: 6966: 6965: 6963: 6962:absolute zero 6958: 6955: 6953: 6949: 6945: 6941: 6937: 6929: 6925: 6921: 6918: 6917: 6912: 6911: 6910: 6908: 6904: 6900: 6896: 6888: 6884: 6881: 6878: 6875: 6874: 6873: 6869: 6867: 6860: 6845: 6841: 6839: 6835: 6829: 6825: 6821: 6817: 6813: 6809: 6805: 6804:type checking 6789: 6783: 6777: 6771: 6767: 6753: 6750: 6747: 6744: 6741: 6738: 6731: 6728: 6727: 6723: 6718: 6712: 6706: 6702: 6688: 6685: 6681: 6677: 6674: 6671: 6664: 6661: 6660: 6655: 6649: 6645: 6631: 6627: 6623: 6620: 6617: 6614: 6607: 6604: 6603: 6600:Nomenclature 6594: 6590: 6584: 6577: 6572: 6568: 6554: 6551: 6544: 6541: 6540: 6537: 6532: 6527: 6522: 6517: 6512: 6508: 6494: 6491: 6488: 6481: 6478: 6477: 6473: 6469: 6450: 6445: 6441: 6437: 6427: 6410: 6405: 6401: 6397: 6391: 6384: 6381: 6380: 6377: 6372: 6366: 6360: 6356: 6342: 6338: 6334: 6331: 6328: 6324: 6320: 6313: 6312: 6307: 6301: 6295: 6289: 6285: 6271: 6268: 6265: 6262: 6259: 6252: 6248: 6245:Nomenclature 6239: 6235: 6229: 6221: 6217: 6212: 6205: 6200: 6195: 6191: 6177: 6174: 6171: 6168: 6165: 6162: 6159: 6156: 6153: 6146: 6145: 6142: 6137: 6131: 6126: 6121: 6116: 6110: 6105: 6100: 6095: 6091: 6077: 6073: 6069: 6064: 6060: 6056: 6053: 6048: 6044: 6040: 6033: 6032: 6028: 6024: 6019: 6014: 6009: 6005: 5991: 5988: 5981: 5977: 5974: 5969: 5964: 5959: 5954: 5949: 5944: 5940: 5935: 5931: 5917: 5914: 5911: 5908: 5905: 5902: 5899: 5892: 5889: 5888: 5885: 5880: 5874: 5868: 5862: 5858: 5844: 5841: 5838: 5835: 5832: 5825: 5822: 5821: 5818: 5813: 5808: 5803: 5798: 5793: 5789: 5775: 5771: 5765: 5761: 5757: 5754: 5751: 5748: 5743: 5739: 5735: 5728: 5727: 5724: 5719: 5714: 5709: 5704: 5699: 5695: 5681: 5677: 5671: 5667: 5663: 5660: 5657: 5654: 5649: 5645: 5641: 5634: 5633: 5630: 5625: 5619: 5614: 5609: 5605: 5591: 5588: 5585: 5582: 5578: 5574: 5567: 5566: 5563: 5558: 5553: 5548: 5544: 5530: 5527: 5520: 5516: 5513:Nomenclature 5507: 5503: 5497: 5493: 5483: 5475: 5471: 5464: 5457: 5451: 5445: 5439: 5432: 5430: 5428: 5422: 5420: 5414: 5412: 5406: 5401: 5398: 5392: 5385: 5379: 5369: 5367: 5361: 5355: 5348: 5333: 5330: 5327: 5321: 5315: 5309: 5300: 5296: 5292: 5291: 5290: 5288: 5283: 5277: 5271: 5262: 5259: 5255: 5251: 5250: 5249: 5246: 5240: 5236: 5231: 5227: 5222: 5220: 5215: 5213: 5209: 5205: 5201: 5194: 5184: 5181: 5163: 5150: 5141: 5135: 5132: 5107: 5103: 5092: 5084: 5083: 5082: 5080: 5057: 5051: 5048: 5042: 5036: 5033: 5030: 5027: 5020: 5019: 5018: 5015: 5010: 5005: 4998: 4970: 4962: 4957: 4953: 4949: 4945: 4940: 4936: 4932: 4928: 4925: 4922: 4919: 4913: 4910: 4900: 4888: 4883: 4870: 4852: 4847: 4843: 4839: 4835: 4832: 4829: 4826: 4820: 4817: 4807: 4798: 4777: 4770: 4761: 4752: 4742: 4739: 4732: 4726: 4723: 4708: 4707: 4706: 4703: 4698: 4693: 4680: 4675: 4655: 4652: 4649: 4638: 4628: 4621: 4616: 4612: 4608: 4599: 4590: 4580: 4577: 4570: 4564: 4561: 4551: 4550: 4549: 4546: 4539: 4535: 4525: 4519: 4514: 4509: 4506: 4500: 4495: 4490: 4484: 4478: 4474: 4470: 4466: 4459: 4457: 4453: 4449: 4445: 4444:trigonometric 4441: 4437: 4434:arguments to 4433: 4429: 4420: 4418: 4414: 4410: 4398: 4394: 4390: 4386: 4372: 4364: 4361: 4357: 4356: 4355: 4352: 4349: 4347: 4343: 4339: 4335: 4330: 4320: 4303: 4299: 4296: 4288: 4284: 4280: 4277: 4274: 4271: 4268: 4263: 4259: 4255: 4250: 4246: 4239: 4232: 4231: 4230: 4228: 4209: 4201: 4197: 4186: 4182: 4173: 4168: 4165: 4162: 4158: 4154: 4151: 4144: 4143: 4142: 4139: 4134: 4127: 4115: 4110: 4105: 4103: 4102:dimensionless 4099: 4095: 4090: 4086: 4082: 4076: 4070: 4064: 4058: 4052: 4050: 4045: 4039: 4032: 4030: 4026: 4022: 4017: 4015: 4009: 4005: 4001: 3985: 3980: 3978: 3974: 3965: 3956: 3952: 3951:abelian group 3946: 3931: 3929: 3921: 3917: 3912: 3909: 3905: 3902: 3897: 3896: 3895: 3888: 3882: 3876: 3870: 3864: 3854: 3845: 3843: 3839: 3833: 3830: 3826: 3820: 3814: 3807: 3787: 3783: 3778: 3775: 3770: 3766: 3763: 3760: 3757: 3754: 3747: 3746: 3745: 3742: 3722: 3719: 3716: 3712: 3706: 3703: 3698: 3692: 3689: 3685: 3679: 3675: 3668: 3667: 3666: 3645: 3640: 3637: 3632: 3630: 3623: 3619: 3608: 3605: 3601: 3596: 3594: 3587: 3583: 3571: 3570: 3569: 3567: 3559: 3550: 3544: 3538: 3534: 3529: 3525: 3520: 3516: 3511: 3507: 3497: 3494: 3487: 3484: 3478: 3462: 3451: 3428: 3425: 3418: 3415: 3412: 3401: 3395: 3390: 3362:The variable 3360: 3358: 3354: 3348: 3342: 3335: 3329: 3325: 3322: 3315: 3309: 3291: 3285: 3279: 3273: 3267: 3261: 3255: 3245: 3241: 3226: 3224: 3220: 3215: 3213: 3209: 3208:Lord Rayleigh 3203: 3186: 3183: 3180:in which the 3179: 3178:Coulomb's law 3171: 3166: 3162: 3159:in which the 3158: 3153: 3149: 3147: 3145: 3141: 3135: 3130: 3128: 3124: 3120: 3116: 3112: 3107: 3105: 3101: 3097: 3083: 3066: 3061: 3058: 3053: 3033: 3030: 3016: 3008: 3004: 3000: 2995: 2986: 2966: 2963: 2949: 2943: 2939: 2935: 2930: 2910: 2909:Froude number 2907: 2893: 2888: 2884: 2881: 2877: 2871: 2851: 2848: 2847: 2846: 2843: 2839: 2825: 2819: 2813: 2809: 2803: 2797: 2791: 2787: 2783: 2777: 2776:bond duration 2773: 2769: 2766: 2763: 2760: 2756: 2753: 2749: 2748: 2747: 2745: 2741: 2731: 2727: 2723: 2717: 2713: 2709: 2703: 2697: 2690: 2683: 2677: 2672: 2669: 2663: 2659: 2656: 2652:volume of an 2643: 2635: 2609: 2608: 2601: 2591: 2588: 2583: 2575: 2563: 2558: 2556: 2548: 2541: 2532: 2525: 2516: 2509: 2500: 2491: 2482: 2477: 2472: 2466: 2462: 2458: 2454: 2451:One may take 2449: 2447: 2446:abelian group 2440: 2436: 2432: 2428: 2423: 2419: 2413: 2403: 2400: 2395: 2393: 2375: 2356: 2353: 2347: 2344: 2332: 2327: 2314: 2311: 2310: 2309: 2296: 2291: 2287: 2283: 2279: 2275: 2271: 2268: 2263: 2260: 2255: 2251: 2246: 2242: 2237: 2236: 2235: 2232: 2220: 2214: 2210: 2205: 2203: 2199: 2194: 2190: 2185: 2183: 2182:juxtaposition 2179: 2175: 2171: 2167: 2157: 2151: 2147: 2144: 2140: 2137: 2133: 2127: 2121: 2115: 2110: 2107: 2103: 2097: 2091: 2085: 2080: 2076: 2072: 2068: 2065: 2061: 2057: 2051: 2045: 2039: 2034: 2029: 2022: 2018: 2011: 2005: 1999: 1996: 1992: 1987: 1981: 1977: 1970: 1963: 1956: 1952: 1948: 1943: 1937: 1933: 1924: 1915: 1906: 1900: 1895: 1892: 1888: 1884: 1883: 1882: 1879: 1877: 1876:Lord Rayleigh 1873: 1869: 1865: 1861: 1857: 1853: 1849: 1825: 1760: 1752: 1749: 1728: 1714: 1711: 1680: 1667: 1664: 1661: 1658: 1651: 1650: 1649: 1646: 1642: 1623: 1618: 1615: 1594: 1565: 1545: 1531: 1528: 1513: 1500: 1497: 1494: 1491: 1484: 1483: 1482: 1479: 1475: 1456: 1439: 1431: 1423: 1420: 1417: 1414: 1407: 1406: 1405: 1402: 1398: 1379: 1367: 1353: 1350: 1338: 1318: 1304: 1301: 1286: 1273: 1270: 1267: 1264: 1257: 1256: 1255: 1252: 1248: 1229: 1217: 1203: 1200: 1188: 1178: 1159: 1156: 1144: 1136: 1128: 1125: 1122: 1119: 1112: 1111: 1110: 1107: 1103: 1084: 1072: 1069: 1055: 1052: 1040: 1033: 1004: 1001: 986: 973: 970: 967: 964: 957: 956: 955: 952: 948: 929: 910: 907: 895: 883: 880: 868: 858: 850: 842: 839: 836: 833: 826: 825: 824: 821: 817: 798: 786: 783: 771: 751: 748: 733: 720: 717: 714: 711: 704: 703: 702: 699: 695: 676: 664: 661: 649: 632: 619: 616: 613: 610: 603: 602: 601: 598: 594: 584: 580: 574: 568: 566: 565:dimension one 562: 560: 553: 546: 539: 534: 532: 525: 518: 513: 511: 504: 498: 492: 488: 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: 13728: 13716: 13644: 13446: 13309:Mesopotamian 13203:Puerto Rican 12600:Astronomical 12486: 12468: 12393: 12368: 12350: 12339: 12309: 12305: 12296: 12292: 12280:Tao, Terence 12270: 12266: 12250: 12246: 12230: 12226: 12200: 12196: 12183: 12158: 12154: 12130: 12119: 12115: 12106: 12102: 12086: 12082: 12048: 12044: 12026: 11999: 11995: 11968: 11948: 11925: 11903:, Springer, 11900: 11890: 11863: 11859: 11818: 11814: 11789: 11777: 11773: 11747: 11743: 11727: 11723: 11705: 11680:Siano ( 11676: 11649: 11645: 11635: 11628:Huntley 1967 11622: 11610:. Retrieved 11606: 11597: 11585:. Retrieved 11581: 11557:. Retrieved 11553: 11544: 11532:. Retrieved 11528: 11519: 11507:. Retrieved 11503: 11494: 11452: 11436: 11408: 11401: 11390: 11355: 11349: 11335: 11320: 11305: 11289: 11276: 11252: 11248: 11235: 11198: 11192: 11174: 11167: 11145:(3): 21–27. 11142: 11138: 11132: 11107: 11103: 11097: 11080: 11077:Comput. Lang 11076: 11070: 11052: 11046: 11028: 11022: 11002: 10994: 10973: 10950: 10928:. Retrieved 10924: 10913: 10901: 10876: 10872: 10859: 10847: 10838: 10832: 10823: 10811: 10799: 10787:. Retrieved 10783: 10773: 10763: 10756: 10740: 10730: 10723: 10710: 10692: 10686: 10677: 10671: 10662: 10647: 10640: 10622: 10616: 10607: 10590: 10586: 10560: 10556: 10550: 10525: 10517: 10502: 10494: 10475: 10465: 10454:, retrieved 10447:the original 10438: 10428: 10386: 10382: 10346: 10336: 10324:. Retrieved 10306: 10279:(9): 42–47. 10276: 10272: 10169: 10162: 10154: 10150: 10142: 10134: 10130: 10119: 10115: 10001: 9994: 9991: 9784: 9781: 9773: 9562: 9295: 9291: 9287: 9283: 9272: 9257: 9227: 9223: 9181: 8470: 8464: 8420: 8418: 8412: 8406: 8400: 8396: 8393: 8389: 8384: 8379: 8374: 8370: 8365: 8359: 8352: 8348: 8340: 8323:, with unit 8317: 8278: 8147: 8008: 7873: 7869:proportional 7864: 7860: 7858: 7849: 7828: 7822: 7813: 7807: 7800: 7793: 7786: 7783: 7489: 7403: 7282: 7204: 7069: 7048: 7025: 7013: 7004: 7002: 6986: 6978: 6973: 6971: 6959: 6956: 6951: 6943: 6939: 6935: 6933: 6927: 6923: 6914: 6907:vector space 6902: 6899:affine space 6894: 6892: 6886: 6885:but one may 6870: 6862: 6859:Affine space 6842: 6801: 6787: 6785:= velocity, 6781: 6775: 6769: 6716: 6710: 6704: 6653: 6647: 6592: 6588: 6570: 6530: 6520: 6510: 6471: 6376:displacement 6370: 6364: 6358: 6305: 6299: 6297:= velocity, 6293: 6287: 6237: 6233: 6215: 6210: 6203: 6193: 6135: 6130:permeability 6124: 6120:permittivity 6114: 6103: 6093: 6017: 6007: 5967: 5957: 5947: 5933: 5878: 5872: 5866: 5864:= pressure, 5860: 5823:Ideal gases 5811: 5801: 5791: 5717: 5707: 5697: 5623: 5617: 5607: 5556: 5546: 5505: 5501: 5481: 5469: 5462: 5455: 5449: 5443: 5437: 5433: 5426: 5418: 5410: 5402: 5396: 5390: 5383: 5375: 5359: 5353: 5350: 5331: 5325: 5319: 5313: 5307: 5304: 5298: 5294: 5281: 5275: 5269: 5266: 5260: 5257: 5253: 5244: 5238: 5229: 5226:displacement 5223: 5218: 5216: 5203: 5199: 5197: 5182: 5178: 5075: 5013: 5003: 5000: 4701: 4696: 4691: 4673: 4670: 4547: 4537: 4533: 4523: 4517: 4510: 4504: 4498: 4493: 4488: 4482: 4476: 4475:− log 4472: 4468: 4464: 4460: 4430: 4426: 4381: 4370: 4359: 4353: 4350: 4340:: they must 4326: 4318: 4227:commensurate 4224: 4137: 4125: 4113: 4106: 4088: 4084: 4080: 4074: 4068: 4062: 4056: 4053: 4043: 4040: 4033: 4018: 4007: 4003: 3999: 3981: 3963: 3948: 3925: 3919: 3915: 3907: 3903: 3900: 3886: 3880: 3874: 3868: 3862: 3859: 3840:such as the 3834: 3828: 3824: 3818: 3812: 3805: 3802: 3740: 3737: 3664: 3557: 3548: 3542: 3536: 3527: 3518: 3509: 3503: 3492: 3488: 3482: 3479: 3449: 3399: 3393: 3388: 3361: 3346: 3340: 3333: 3327: 3323: 3320: 3313: 3307: 3289: 3283: 3277: 3271: 3265: 3259: 3253: 3243: 3237: 3222: 3218: 3216: 3211: 3204: 3184: 3168:is taken as 3164: 3150: 3143: 3139: 3131: 3126: 3122: 3108: 3093: 3081: 2965:Euler number 2835: 2823: 2817: 2811: 2807: 2801: 2795: 2789: 2785: 2781: 2737: 2725: 2721: 2715: 2711: 2707: 2701: 2695: 2688: 2681: 2675: 2667: 2661: 2654: 2649: 2641: 2638:Applications 2605: 2603: 2584: 2576: 2559: 2555:sanity check 2546: 2539: 2530: 2523: 2514: 2507: 2498: 2489: 2480: 2475: 2473: 2469: 2464: 2460: 2456: 2452: 2443: 2438: 2434: 2430: 2426: 2421: 2396: 2389: 2330: 2307: 2295:acceleration 2289: 2285: 2281: 2277: 2273: 2269: 2266: 2253: 2249: 2240: 2233: 2226: 2206: 2186: 2178:centered dot 2163: 2155: 2131: 2125: 2119: 2113: 2101: 2095: 2089: 2083: 2055: 2049: 2043: 2037: 2027: 2020: 2016: 2009: 2003: 1997: 1994: 1990: 1979: 1975: 1968: 1961: 1954: 1950: 1946: 1935: 1931: 1922: 1913: 1904: 1898: 1880: 1847: 1845: 1644: 1638: 1477: 1471: 1400: 1394: 1250: 1244: 1141:displacement 1105: 1099: 950: 944: 855:acceleration 819: 813: 697: 694:acceleration 691: 596: 590: 587:Simple cases 581: 569: 564: 557: 551: 544: 537: 529: 523: 516: 508: 502: 499: 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: 13757:Measurement 13173:Costa Rican 13137:Seychellois 13046:Singaporean 12841:Traditional 12696:Metrication 12634:Geometrised 12590:Avoirdupois 12473:Brady Haran 12340:Aeronautics 10767:, p. 5 10734:, Macmillan 10680:, p. 4 10326:1 September 9777:normal form 8357:at a speed 7058:consistent. 6808:Standard ML 6708:= entropy, 6605:Mechanical 6597:Expression 6250:Mechanical 6242:Expression 5518:Mechanical 5510:Expression 5378:Ising model 4529:(3 m) = 9 m 4448:logarithmic 4440:exponential 4405:10 mol 3394:irrelevance 3240:oscillation 3127:homogeneity 3032:Mach number 2673:: being an 2646:Mathematics 2073:, obtain a 1940:, then the 1860:engineering 1641:capacitance 252:SI standard 154:Formulation 34:engineering 13746:Categories 13259:Venezuelan 13244:Paraguayan 13198:Nicaraguan 13076:Vietnamese 13051:Sri Lankan 13036:Philippine 12996:Indonesian 12920:Portuguese 12814:Winchester 12717:Comparison 12667:Background 12605:Electrical 11941:postscript 11695:References 11659:2108.05704 11646:Metrologia 10747:, p. 10716:Pesic 2005 10389:(3): 023, 10211:Similitude 9563:which for 8447:becomes L1 8369:above the 8363:and angle 8355:) = (0, 0) 8335:See also: 7890:Dimension 7016:extensions 6991:, not the 6362:= action, 6231:Momentum, 5943:wave front 5870:= volume, 5372:Formalisms 5248:would be: 4454:, must be 3934:Properties 3453:(equal to 3382:, because 3247:of a mass 2476:expression 2439:subtracted 2410:See also: 2238:position ( 2229:1% = 1/100 2189:base units 2146:parameters 2139:Substitute 2064:base units 298:sans serif 113:inequality 94:quantities 13447:Dimension 13428:Quantity 13254:Uruguayan 13239:Colombian 13229:Brazilian 13219:Argentine 13157:Tanzanian 13127:Mauritian 13097:Ethiopian 13061:Taiwanese 13031:Pakistani 13011:Mongolian 12986:Cambodian 12905:Norwegian 12875:Icelandic 12870:Hungarian 12863:Byzantine 12819:Exchequer 12567:Hong Kong 12412:853154197 12175:206506776 12061:CiteSeerX 12029:, Wiley, 11978:682090763 11972:, Dover, 11866:: 84–99, 11750:: 592–6, 11479:243831207 11395:Hart 1995 11203:CiteSeerX 11184:1476-2986 9873:∼ 9839:θ 9712:θ 9706:⁡ 9662:π 9642:θ 9636:⁡ 9603:π 9577:θ 9523:⁡ 9495:⁡ 9464:⁡ 9436:⁡ 9386:⁡ 9357:θ 9351:⁡ 9331:π 9325:θ 9319:⁡ 8291:π 8258:˙ 8249:η 8234:ρ 8122:˙ 8101:ρ 8074:π 8046:η 8040:˙ 8021:π 7908:˙ 7734:− 7680:− 7626:− 7366:− 7319:− 7223:∝ 7005:direction 6791:= charge 6745:≡ 6686:δ 6675:δ 6621:≡ 6489:ρ 6332:≡ 6303:= force, 6266:≡ 6169:≡ 6160:≡ 6078:μ 6057:≡ 6041:ε 5992:ϕ 5953:intensity 5909:≡ 5839:≡ 5758:≡ 5755:ω 5749:≡ 5740:ω 5664:≡ 5655:≡ 5586:≡ 5341:Constants 5287:converted 5037:× 4963:⋅ 4950:⋅ 4929:⋅ 4923:− 4920:⋅ 4889:⋅ 4836:⋅ 4830:− 4827:⋅ 4771:⋅ 4740:− 4733:⋅ 4695:to be in 4650:⋅ 4609:⋅ 4578:− 4571:⋅ 4513:monomials 4329:mechanics 4323:Mechanics 4285:π 4260:π 4247:π 4183:π 4159:∏ 4083: := 4036:(0, 0, 0) 3969:L × L = L 3892:5 − 3 = 2 3776:ℓ 3704:ℓ 3638:ℓ 3620:π 3584:π 3515:amplitude 3419:κ 3359:as well. 3219:dimension 3123:dimension 3001:ρ 2993:Δ 2889:μ 2878:ρ 2752:P/E ratio 2345:∫ 2187:A set of 2069:By using 1868:variables 1856:chemistry 1802:− 1785:− 1750:− 1712:− 1662:⁡ 1616:− 1576:− 1529:− 1495:⁡ 1432:× 1418:⁡ 1351:− 1302:− 1268:⁡ 1201:− 1179:× 1157:− 1137:× 1123:⁡ 1070:− 1053:− 1002:− 968:⁡ 908:− 881:− 869:× 851:× 837:⁡ 784:− 749:− 715:⁡ 662:− 614:⁡ 531:kinematic 510:geometric 388:Θ 320:⁡ 229:dimension 150:in 1822. 92:physical 52:(such as 13723:Category 13682:See also 13542:kilogram 13340:Obsolete 13335:Humorous 13289:Egyptian 13249:Peruvian 13224:Bolivian 13188:Honduran 13152:Tunisian 13132:Moroccan 13122:Malagasy 13107:Eritrean 13102:Egyptian 13092:Algerian 13021:Nepalese 13001:Japanese 12935:Scottish 12925:Romanian 12826:Estonian 12736:Historic 12712:Overview 12681:Overview 12578:Specific 12475:for the 12449:Archived 12282:(2012). 12025:(1951), 11986:6128830M 11920:(1994), 11878:archived 11809:(1914), 11788:(1922), 11704:(1996), 11612:19 April 11587:19 April 11559:19 April 11534:19 April 11509:19 April 11483:Archived 11425:Archived 11382:53089559 11294:Archived 11265:Archived 11159:22450087 11124:40558757 10893:14833238 10828:Tao 2012 10816:Tao 2012 10804:Tao 2012 10789:15 April 10436:(2012), 10421:15806354 10179:See also 10153:) + sin( 9300:, where 7887:Variable 6924:relative 6662:Thermal 6651:= mass, 6451:⟩ 6438:⟨ 6411:⟩ 6398:⟨ 6382:Thermal 6291:= mass, 6222:in loop 5723:momentum 5713:velocity 5621:= time, 5562:distance 5499:Energy, 5486:SI units 5394:, where 4496:hold if 4438:such as 4123:, ..., π 4104:scalar. 4029:choosing 3955:identity 3229:Examples 3100:Lagrange 2842:pi terms 2793:, where 2626:because 2461:multiply 2427:compared 2259:velocity 2170:division 2150:grouping 2025:, where 1866:of some 947:pressure 561:quantity 533:quantity 512:quantity 493:, since 285:(N) and 221:rational 109:equation 13730:Outline 13659:candela 13561: 13557: 13508:, etc. 13462:symbol 13443:Symbol 13434:SI unit 13371:Modulor 13345:Unusual 13314:Persian 13272:Ancient 13234:Chilean 13193:Mexican 13183:Haitian 13112:Guinean 13016:Myanmar 12955:Swedish 12950:Spanish 12940:Serbian 12930:Russian 12910:Ottoman 12900:Maltese 12890:Latvian 12885:Italian 12831:Finnish 12809:English 12789:Cypriot 12784:Cornish 12691:History 12686:Outline 12621:Natural 12562:Chinese 12540:General 12533:Current 12334:(1920) 12326:2315883 12205:Bibcode 12053:Bibcode 12004:Bibcode 11823:Bibcode 11780:: 55–64 11752:Bibcode 11686:1985-II 10930:15 July 10401:Bibcode 10106:⁠ 10008:⁠ 9769:⁠ 9742:⁠ 9738:⁠ 9625:⁠ 9623:yields 8432: 1 8428: 1 7974:density 7485:⁠ 7444:⁠ 7196:⁠ 7169:⁠ 7165:⁠ 7138:⁠ 7130:⁠ 7103:⁠ 6916:acts on 6828:Fortran 6816:Haskell 6586:Force, 6516:density 6309:= time 6027:voltage 5951:= wave 4697:seconds 4681:is 9.8 4679:gravity 4367:V = L/T 4109:nullity 4092:as the 3533:tension 3445:⁠ 3405:⁠ 3305:; and 3192:Q = TLM 3090:History 2671:-sphere 2431:equated 2370:⁠ 2336:⁠ 2129:, ..., 2099:, ..., 2053:, ..., 1974:, ..., 1929:, ..., 1852:physics 1509:current 1474:voltage 1428:current 559:dynamic 189:is the 125:derived 38:science 13604:kelvin 13597:Θ 13577:ampere 13493:length 13483:second 13449:symbol 13366:N-body 13304:Indian 13279:Arabic 13142:Somali 13117:Libyan 13085:Africa 13056:Syrian 13006:Korean 12991:Indian 12981:Afghan 12945:Slovak 12915:Polish 12853:German 12836:French 12799:Danish 12777:Europe 12743:Metric 12674:Metric 12654:Stoney 12644:Planck 12629:Atomic 12410: 12400: 12375: 12357: 12342:, via 12324: 12299:(A–34) 12273:(4): 5 12197:Nature 12173: 12143: 12063: 12033: 11984: 11976: 11956: 11932: 11907: 11796: 11712: 11682:1985-I 11477: 11467: 11380: 11370: 11223: 11205: 11182: 11157: 11122: 11059: 11035: 11011: 10961: 10891: 10655: 10629: 10538: 10482: 10456:2 June 10419: 10361: 10317: 10172:radian 9269:while 8342:Angles 7884:Symbol 7823:angles 6952:vector 6944:affine 6940:vector 6936:affine 6903:vector 6895:affine 6866:origin 6822:, and 6526:volume 6479:Waves 6141:volume 5890:Waves 5613:action 5447:, and 5423:, and 5305:where 5142:0.3048 5108:0.3048 4781: 4746: 4632: 4584: 4432:Scalar 4387:. In 3984:module 3953:: The 3803:where 3738:where 3566:powers 3506:length 3378:, and 3287:, and 3200:Q = TL 3196:M = TL 3174:M = TL 3079:where 2562:torque 2465:divide 2453:ratios 2351: 2209:newton 2202:volume 2198:length 2191:for a 1858:, and 1279:energy 1102:energy 625:length 549:, and 495:Q = TI 439:where 267:length 227:. The 195:matrix 133:system 68:) and 64:, and 54:length 13517:metre 13457:name 13440:Name 13354:Other 13319:Roman 13299:Hindu 13294:Greek 13178:Cuban 13066:Tatar 13026:Omani 12965:Welsh 12960:Swiss 12880:Irish 12858:Greek 12804:Dutch 12794:Czech 12764:(CGS) 12758:(MTS) 12752:(MKS) 12724:(FPS) 12705:UK/US 12322:JSTOR 12171:S2CID 12137:227–8 12122:(251) 11881:(PDF) 11856:(PDF) 11654:arXiv 11486:(PDF) 11475:S2CID 11449:(PDF) 11428:(PDF) 11413:(PDF) 11378:S2CID 11297:(PDF) 11286:(PDF) 11268:(PDF) 11245:(PDF) 11155:S2CID 11120:S2CID 10980:arXiv 10955:(PDF) 10889:S2CID 10869:(PDF) 10664:well. 10450:(PDF) 10443:(PDF) 10417:S2CID 10391:arXiv 10311:(PDF) 10250:Notes 10118:)cos( 7574:as TL 7554:as TL 6820:OCaml 6139:= 3d 5629:power 5552:force 5235:speed 4338:basis 4025:bases 3959:L = 1 3370:with 3353:group 3170:unity 3104:Turin 2658:-ball 2467:them. 2437:, or 2435:added 2313:force 2213:force 2109:Solve 2031:is a 1506:power 1247:power 1133:force 979:force 816:force 726:speed 593:speed 483:basis 295:roman 281:(Θ), 277:(I), 273:(M), 269:(L), 265:(T), 233:scale 225:power 111:, or 102:units 98:kinds 13633:mol 13629:mole 13527:mass 13460:Unit 13455:Unit 13071:Thai 13041:Pegu 12974:Asia 12612:(US) 12595:Troy 12408:OCLC 12398:ISBN 12373:ISBN 12355:ISBN 12141:ISBN 12031:ISBN 11974:OCLC 11954:ISBN 11930:ISBN 11905:ISBN 11794:ISBN 11710:ISBN 11614:2023 11589:2023 11561:2023 11536:2023 11511:2023 11465:ISBN 11368:ISBN 11221:ISBN 11180:ISSN 11057:ISBN 11033:ISBN 11009:ISBN 10959:ISBN 10932:2014 10842:..." 10791:2017 10653:ISBN 10627:ISBN 10536:ISBN 10480:ISBN 10458:2015 10434:JCGM 10387:2002 10359:ISBN 10328:2021 10315:ISBN 10302:BIPM 10161:exp( 10159:and 10149:cos( 10141:cos( 10139:and 10133:sin( 10114:sin( 9999:and 9997:= −1 9589:and 9304:and 9294:sin( 9290:) + 9286:cos( 9271:cos( 9256:sin( 9226:) = 9222:tan( 8410:and 8325:mole 7990:TLM 7964:TLM 7803:= −1 7798:and 7570:and 7564:as L 7442:and 6824:Rust 6593:TLM 6238:TLM 5963:time 5939:area 5703:mass 5506:TLM 5467:and 5389:~ 1/ 5297:= 5 5279:and 5267:for 5009:unit 4937:0.01 4844:0.01 4778:0.01 4502:and 4486:and 4463:log( 4446:and 4401:6.02 4399:, ≈ 4342:span 4072:and 4060:and 3555:and 2691:− 1) 2496:and 2374:work 2288:) / 2035:and 2015:... 1436:time 1282:time 982:area 954:is 847:mass 729:time 628:time 521: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 13663:cd 13546:kg 12314:doi 12255:doi 12251:320 12235:doi 12231:320 12213:doi 12163:doi 12109:(6) 12091:doi 12071:doi 12012:doi 11939:As 11868:doi 11839:hdl 11831:doi 11760:doi 11748:372 11732:doi 11664:doi 11457:doi 11417:hdl 11360:doi 11257:doi 11213:doi 11147:doi 11112:doi 11085:doi 10881:doi 10749:156 10697:hdl 10595:doi 10591:311 10565:doi 10561:292 10532:260 10409:doi 10351:doi 10281:doi 10004:= 2 9703:cos 9633:sin 9520:cos 9492:sin 9461:cos 9433:sin 9383:sin 9348:cos 9316:sin 9249:= 1 8468:= 1 7977:LM 7928:TM 7841:, L 7835:, L 7796:= 1 7789:= 1 6887:not 5941:of 5472:→ 0 5465:→ 0 5458:→ ∞ 5233:as 4926:9.8 4833:9.8 4743:9.8 4629:500 4581:9.8 4494:not 3979:). 3297:; 2836:In 2550:man 2543:man 2534:man 2527:man 2518:rat 2511:man 2502:man 2493:rat 2484:man 2463:or 2455:of 2180:or 2148:by 2077:of 2075:set 1896:If 1659:dim 1648:is 1492:dim 1481:is 1415:dim 1404:is 1265:dim 1254:is 1120:dim 1109:is 965:dim 834:dim 823:is 712:dim 701:is 611:dim 600:is 554:≠ 0 547:≠ 0 540:≠ 0 526:≠ 0 519:≠ 0 505:≠ 0 317:dim 258:: 235:or 142:or 32:In 13748:: 13653:J 13623:N 13608:K 13581:A 13563:, 13536:M 13521:m 13511:L 13504:, 13500:, 13487:s 13477:T 12471:. 12467:. 12406:. 12320:, 12310:75 12308:, 12297:68 12295:, 12271:32 12269:, 12249:, 12229:, 12211:, 12201:95 12199:, 12169:, 12159:31 12157:, 12139:, 12120:40 12118:, 12107:42 12105:, 12087:66 12085:, 12069:, 12059:, 12049:72 12047:, 12010:, 11998:, 11982:OL 11980:, 11876:, 11864:14 11862:, 11858:, 11837:, 11829:, 11817:, 11813:, 11778:55 11776:, 11758:, 11746:, 11728:45 11726:, 11684:, 11662:. 11650:58 11648:. 11644:. 11605:. 11580:. 11569:^ 11552:. 11527:. 11502:. 11481:. 11473:. 11463:. 11451:. 11423:. 11376:. 11366:. 11288:. 11263:. 11253:50 11251:. 11247:. 11219:. 11211:. 11153:. 11141:. 11118:. 11108:15 11106:. 11079:. 10940:^ 10923:. 10887:. 10877:26 10875:. 10871:. 10782:. 10661:, 10589:. 10577:^ 10559:. 10534:. 10509:28 10474:. 10415:, 10407:, 10399:, 10385:, 10373:^ 10357:. 10345:. 10293:^ 10277:64 10275:. 10271:. 10257:^ 10128:. 9771:. 9245:/1 9234:/1 8416:. 8392:= 8351:, 8315:. 8003:L 7791:, 7560:, 7525:, 7167:, 7011:. 6818:, 6812:F# 6724:) 6574:= 6534:= 6528:, 6524:= 6518:, 6514:= 6470:, 6466:= 6374:= 6218:= 6201:, 6197:= 6132:, 6128:= 6122:, 6118:= 6111:, 6107:= 6101:, 6097:= 6029:) 6021:= 6015:, 6011:= 5971:= 5965:, 5961:= 5955:, 5945:, 5937:= 5882:= 5815:= 5809:, 5805:= 5799:, 5795:= 5721:= 5715:, 5711:= 5705:, 5701:= 5627:= 5615:, 5611:= 5560:= 5554:, 5550:= 5460:, 5441:, 5415:, 5256:= 5214:. 5198:A 5017:. 4954:60 4536:+ 4442:, 4373:). 4362:). 4348:. 4119:{π 4087:⊗ 4038:. 4006:, 4002:, 3901:ρR 3829:ℓs 3827:= 3496:. 3374:, 3339:= 3319:= 3301:; 3281:, 3275:, 3214:. 3202:. 3144:ma 3142:= 3129:. 3113:. 3036:Ma 2969:Eu 2913:Fr 2854:Re 2824:dr 2812:dr 2808:dV 2788:)/ 2786:dr 2782:dV 2634:. 2529:+ 2513:+ 2487:, 2433:, 2429:, 2372:, 2293:, 2290:dt 2286:dt 2282:dx 2276:= 2274:dt 2257:, 2254:dt 2250:dx 2231:. 2219:. 2123:, 2117:, 2093:, 2087:, 2047:, 2041:, 1993:= 1967:, 1960:, 1949:= 1920:, 1911:, 1878:. 1854:, 567:. 542:, 497:. 475:, 469:, 463:, 457:, 451:, 445:, 223:) 180:− 115:, 60:, 56:, 40:, 13648:v 13645:I 13619:n 13592:T 13565:i 13559:I 13532:m 13506:r 13502:x 13498:l 13473:t 13405:e 13398:t 13391:v 12518:e 12511:t 12504:v 12479:. 12414:. 12316:: 12286:. 12257:: 12237:: 12215:: 12207:: 12165:: 12093:: 12073:: 12055:: 12014:: 12006:: 12000:4 11870:: 11841:: 11833:: 11825:: 11819:4 11762:: 11754:: 11734:: 11688:) 11670:. 11666:: 11656:: 11630:) 11626:( 11616:. 11591:. 11563:. 11538:. 11513:. 11459:: 11419:: 11384:. 11362:: 11329:. 11314:. 11259:: 11229:. 11215:: 11161:. 11149:: 11143:5 11126:. 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9503:b 9499:( 9489:+ 9485:) 9479:z 9475:1 9470:b 9467:( 9458:) 9453:z 9449:1 9444:a 9440:( 9430:= 9426:) 9420:z 9416:1 9411:b 9408:+ 9403:z 9399:1 9394:a 9390:( 9360:) 9354:( 9345:= 9342:) 9339:2 9335:/ 9328:+ 9322:( 9306:b 9302:a 9298:) 9296:θ 9292:b 9288:θ 9284:a 9279:0 9275:) 9273:θ 9266:z 9264:1 9260:) 9258:θ 9251:z 9247:x 9243:y 9241:1 9236:x 9232:y 9228:θ 9224:θ 9218:~ 9213:y 9211:1 9206:x 9204:1 9200:θ 9196:θ 9191:z 9189:1 9163:0 9159:1 9135:x 9131:1 9107:y 9103:1 9079:z 9075:1 9050:z 9046:1 9019:x 9015:1 8991:0 8987:1 8963:z 8959:1 8935:y 8931:1 8906:y 8902:1 8875:y 8871:1 8847:z 8843:1 8819:0 8815:1 8791:x 8787:1 8762:x 8758:1 8731:z 8727:1 8703:y 8699:1 8675:x 8671:1 8647:0 8643:1 8618:0 8614:1 8586:z 8582:1 8556:y 8552:1 8526:x 8522:1 8496:0 8492:1 8471:i 8465:i 8462:1 8457:x 8455:1 8450:x 8444:x 8439:0 8434:z 8430:y 8426:x 8424:1 8413:θ 8407:R 8401:v 8399:/ 8397:g 8394:R 8390:π 8385:x 8380:R 8375:y 8371:x 8366:θ 8360:v 8353:y 8349:x 8347:( 8299:8 8295:/ 8281:C 8255:m 8242:4 8238:r 8228:x 8223:p 8216:= 8213:C 8188:m 8184:M 8161:i 8157:M 8129:2 8119:m 8109:5 8105:r 8095:x 8090:p 8083:= 8078:2 8049:r 8037:m 8030:= 8025:1 7996:r 7983:η 7970:ρ 7945:x 7941:p 7905:m 7844:z 7838:y 7832:x 7818:, 7801:c 7794:b 7787:a 7767:c 7762:) 7754:y 7747:L 7737:2 7728:T 7720:( 7713:b 7708:) 7700:y 7693:L 7683:1 7674:T 7666:( 7659:a 7654:) 7646:x 7639:L 7629:1 7620:T 7612:( 7607:= 7601:x 7594:L 7577:y 7572:g 7567:x 7562:R 7557:y 7539:y 7534:v 7522:x 7504:x 7499:v 7473:0 7470:= 7467:c 7464:2 7461:+ 7458:b 7455:+ 7452:a 7430:1 7427:= 7424:c 7421:+ 7418:b 7415:+ 7412:a 7387:c 7382:) 7376:L 7369:2 7360:T 7353:( 7346:b 7343:+ 7340:a 7335:) 7329:L 7322:1 7313:T 7306:( 7301:= 7296:L 7268:. 7263:c 7259:g 7252:b 7247:y 7243:v 7236:a 7231:x 7227:v 7220:R 7207:R 7200:g 7182:y 7178:v 7151:x 7147:v 7134:R 7116:x 7112:v 7087:y 7083:v 7055:x 7034:m 6830:. 6788:q 6782:v 6776:B 6770:E 6754:v 6751:q 6748:B 6742:q 6739:E 6717:r 6711:T 6705:S 6689:r 6682:/ 6678:S 6672:T 6654:a 6648:m 6632:t 6628:/ 6624:p 6618:a 6615:m 6589:F 6571:A 6555:A 6552:q 6531:v 6521:V 6511:ρ 6495:v 6492:V 6472:m 6446:2 6442:v 6406:2 6402:v 6392:m 6371:r 6365:L 6359:S 6343:r 6339:/ 6335:L 6329:r 6325:/ 6321:S 6306:t 6300:F 6294:v 6288:m 6272:t 6269:F 6263:v 6260:m 6234:p 6216:I 6211:A 6204:m 6194:p 6178:B 6175:A 6172:I 6166:B 6163:m 6157:E 6154:p 6136:V 6125:μ 6115:ε 6104:B 6094:E 6074:/ 6070:V 6065:2 6061:B 6054:V 6049:2 6045:E 6018:ϕ 6008:q 5989:q 5968:S 5958:t 5948:I 5934:A 5918:t 5915:S 5912:A 5906:t 5903:I 5900:A 5879:N 5873:T 5867:V 5861:p 5845:T 5842:N 5836:V 5833:p 5812:ω 5802:I 5792:L 5776:I 5772:/ 5766:2 5762:L 5752:L 5744:2 5736:I 5718:p 5708:v 5698:m 5682:m 5678:/ 5672:2 5668:p 5661:v 5658:p 5650:2 5646:v 5642:m 5624:P 5618:t 5608:S 5592:t 5589:P 5583:t 5579:/ 5575:S 5557:d 5547:F 5531:d 5528:F 5502:E 5470:G 5463:ħ 5456:c 5450:G 5444:c 5438:ħ 5427:G 5419:ħ 5411:c 5397:d 5391:χ 5384:χ 5360:κ 5354:C 5326:d 5320:D 5314:t 5308:T 5299:T 5295:D 5282:d 5276:t 5270:s 5261:t 5258:s 5254:d 5245:t 5239:s 5230:d 5195:. 5164:. 5158:t 5155:f 5151:1 5146:m 5136:= 5133:1 5112:m 5104:= 5100:t 5097:f 5093:1 5061:] 5058:Z 5055:[ 5052:n 5049:= 5046:] 5043:Z 5040:[ 5034:n 5031:= 5028:Z 5014:n 5004:Z 4971:. 4967:m 4958:2 4946:) 4941:2 4933:( 4914:2 4911:1 4901:= 4893:m 4884:2 4880:) 4875:s 4871:/ 4867:n 4864:i 4861:m 4857:( 4853:) 4848:2 4840:( 4821:2 4818:1 4808:= 4799:2 4795:) 4790:n 4787:i 4784:m 4774:( 4768:) 4762:2 4758:s 4753:/ 4749:m 4736:( 4727:2 4724:1 4702:t 4692:t 4674:t 4656:. 4653:t 4647:) 4643:s 4639:/ 4635:m 4625:( 4622:+ 4617:2 4613:t 4606:) 4600:2 4596:s 4591:/ 4587:m 4574:( 4565:2 4562:1 4538:x 4534:x 4524:x 4518:x 4515:( 4505:b 4499:a 4489:b 4483:a 4477:b 4473:a 4469:b 4467:/ 4465:a 4403:× 4304:. 4300:0 4297:= 4294:) 4289:m 4281:, 4278:. 4275:. 4272:. 4269:, 4264:2 4256:, 4251:1 4243:( 4240:f 4210:. 4202:i 4198:k 4193:) 4187:i 4179:( 4174:m 4169:1 4166:= 4163:i 4155:= 4152:X 4138:X 4129:} 4126:m 4121:1 4114:m 4089:V 4085:V 4081:V 4075:V 4069:V 4063:L 4057:M 4044:V 4010:) 4008:k 4004:j 4000:i 3998:( 3993:M 3990:L 3988:T 3964:p 3920:R 3918:/ 3916:t 3908:S 3906:/ 3904:ω 3887:S 3881:ω 3875:ρ 3869:R 3863:t 3825:E 3819:ℓ 3813:f 3806:f 3788:, 3784:) 3779:A 3771:( 3767:f 3764:s 3761:A 3758:= 3755:E 3741:F 3723:, 3720:0 3717:= 3713:) 3707:A 3699:, 3693:s 3690:A 3686:E 3680:( 3676:F 3646:. 3641:A 3633:= 3624:2 3609:s 3606:A 3602:E 3597:= 3588:1 3561:2 3558:π 3552:1 3549:π 3543:E 3537:s 3528:ρ 3519:A 3510:ℓ 3493:κ 3483:g 3463:C 3450:κ 3429:k 3426:m 3416:= 3413:T 3400:g 3389:g 3384:g 3380:T 3376:m 3372:k 3368:g 3364:g 3347:C 3341:C 3337:1 3334:G 3328:m 3326:/ 3324:k 3321:T 3317:1 3314:G 3308:g 3303:k 3299:m 3295:T 3290:g 3284:k 3278:m 3272:T 3266:T 3260:g 3254:k 3249:m 3244:T 3188:e 3185:k 3165:G 3140:F 3082:c 3067:, 3062:c 3059:u 3054:= 3050:a 3047:M 3034:( 3017:. 3009:2 3005:u 2996:p 2987:= 2983:u 2980:E 2967:( 2950:. 2944:L 2940:g 2936:u 2931:= 2927:r 2924:F 2911:( 2894:. 2885:d 2882:u 2872:= 2868:e 2865:R 2852:( 2818:r 2810:/ 2802:r 2796:V 2790:V 2784:/ 2780:( 2726:n 2722:C 2716:r 2712:n 2708:C 2702:n 2696:x 2689:n 2687:( 2682:x 2676:n 2668:n 2662:n 2655:n 2572:M 2569:L 2566:T 2547:L 2545:/ 2540:m 2531:L 2524:m 2515:m 2508:m 2499:L 2490:m 2481:m 2441:. 2386:. 2384:M 2381:L 2378:T 2357:s 2354:d 2348:F 2331:s 2323:M 2320:L 2317:T 2304:. 2302:L 2299:T 2284:/ 2280:( 2278:d 2272:/ 2270:x 2267:d 2252:/ 2241:x 2135:. 2132:m 2126:c 2120:b 2114:a 2105:. 2102:m 2096:c 2090:b 2084:a 2056:m 2050:c 2044:b 2038:a 2028:C 2021:n 2017:R 2013:3 2010:R 2007:2 2004:R 2001:1 1998:R 1995:C 1991:R 1985:. 1983:) 1980:n 1976:R 1972:3 1969:R 1965:2 1962:R 1958:1 1955:R 1953:( 1951:F 1947:R 1936:n 1932:R 1926:3 1923:R 1917:2 1914:R 1908:1 1905:R 1899:R 1893:. 1826:. 1819:2 1815:I 1805:1 1798:M 1788:2 1781:L 1771:4 1767:T 1761:= 1753:1 1744:I 1736:M 1729:2 1723:L 1715:3 1706:T 1696:I 1689:T 1681:= 1668:= 1665:C 1645:C 1624:. 1619:1 1610:I 1602:M 1595:2 1589:L 1579:3 1572:T 1566:= 1560:I 1553:M 1546:2 1540:L 1532:3 1523:T 1514:= 1501:= 1498:V 1478:V 1457:. 1452:I 1445:T 1440:= 1424:= 1421:Q 1401:Q 1380:. 1375:M 1368:2 1362:L 1354:3 1345:T 1339:= 1333:T 1326:M 1319:2 1313:L 1305:2 1296:T 1287:= 1274:= 1271:P 1251:P 1230:. 1225:M 1218:2 1212:L 1204:2 1195:T 1189:= 1184:L 1174:M 1167:L 1160:2 1151:T 1145:= 1129:= 1126:E 1106:E 1085:. 1080:M 1073:1 1064:L 1056:2 1047:T 1041:= 1034:2 1028:L 1019:M 1012:L 1005:2 996:T 987:= 974:= 971:P 951:P 930:. 925:M 918:L 911:2 902:T 896:= 891:L 884:2 875:T 864:M 859:= 843:= 840:F 820:F 799:. 794:L 787:2 778:T 772:= 766:T 759:L 752:1 743:T 734:= 721:= 718:a 698:a 677:. 672:L 665:1 656:T 650:= 644:T 639:L 633:= 620:= 617:v 597:v 552:c 545:b 538:a 524:b 517:a 503: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