definition of Wikipedia
Formally, the International Vocabulary of Metrology, 3rd edition (VIM3) defines quantity as:
Property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference.
Quantity calculus describes how to perform mathematical manipulations of quantities.
In practice, note that different observers may get different values of a quantity depending on the frame of reference; in turn the coordinate system and metric. Physical properties such as length, mass or time, by themselves, are not physically invariant. However, the laws of physics which include these properties are invariant.
Extensive quantity: its magnitude is additive for subsystems (volume, mass, etc.)
Intensive quantity: the magnitude is independent of the extent of the system (temperature, pressure, etc.)
General: Symbols for quantities should be chosen according to the international recommendations from ISO 80000, the IUPAP red book and the IUPAC green book. For example, the recommended symbol for the physical quantity 'mass' is m, and the recommended symbol for the quantity 'charge' is Q.
Subscripts and indices
Subscripts are used for two reasons, to simply attach a name to the quantity or associate it with another quantity, or represent a specific vector, matrix, or tensor component.
Vectors: Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. If, e.g., u is the speed of a particle, then the straightforward notation for its velocity is u, u, or .
Numbers and elementary functions
Numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes can be italic. Symbols for elementary functions (circular trigonometric, hyperbolic, logarithmic etc.), changes in a quantity like Δ in Δy or operators like d in dx, are also recommended to be printed in roman type.
Most physical quantities include a unit, but not all - some are dimensionless. Neither the name of a physical quantity, nor the symbol used to denote it, implies a particular choice of unit, though SI units are usually preferred and assumed today due to their ease of use and all-round applicability. For example, a quantity of mass might be represented by the symbol m, and could be expressed in the units kilograms (kg), pounds (lb), or Daltons (Da).
The notion of physical dimension of a physical quantity was introduced by Joseph Fourier in 1822. By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension.
The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units and dimensions are listed in the following table. Other conventions may have a different number of fundamental units (e.g. the CGS and MKS systems of units).
|Quantity name/s||(Common) Quantity symbol/s||SI unit name||SI unit symbol||Dimension symbol|
|Length, width, height, depth||a, b, c, d, h, l, r, s, w, x, y, z||metre||m||[L]|
|Amount of substance, number of moles||n||mole||mol||[N]|
|Electric current||i, I||ampere||A||[I]|
|Plane angle||α, β, γ, θ, φ, χ||radian||rad||dimensionless|
|Solid angle||ω, Ω||steradian||sr||dimensionless|
The last two angular units; plane angle and solid angle are subsidiary units used in the SI, but treated dimensionless. The subsidiary units are used for convenience to differentiate between a truly dimensionless quantity (pure number) and an angle, which are different measurements.
Physical quantities and units follow the same hierarchy; chosen base quantities have defined base units, from these any other quantities may be derived and have corresponding derived units.
Colour mixing analogy
Defining quantities is analogous to mixing colours, and could be classified a similar way, although this is not standard. Primary colours are to base quantities; as secondary (or tertiary etc.) colours are to derived quantities. Mixing colours is analogous to combining quantities using mathematical operations. But colours could be for light or paint, and analogously the system of units could be one of many forms: such as SI (now most common), CGS, Gaussian, old imperial units, a specific form of natural units or even arbitrarily defined units characteristic to the physical system in consideration (characteristic units).
The choice of a base system of quantities and units is arbitrary; but once chosen it must be adhered to throughout all analysis which follows for consistency. It makes no sense to mix up different systems of units. Choosing a system of units, one system out of the SI, CGS etc., is like choosing whether use paint or light colours.
In light of this analogy, primary definitions are base quantities with no defining equation, but defined standardized condition, "secondary" definitions are quantities defined purely in terms of base quantities, "tertiary" for quantities in terms of both base and "secondary" quantities, "quaternary" for quantities in terms of base, "secondary", and "tertiary" quantities, and so on.
Much of physics requires definitions to be made for the equations to make sense.
Theoretical implications: Definitions are important since they can lead into new insights of a branch of physics. Two such examples occurred in classical physics. When entropy S was defined – the range of thermodynamics was greatly extended by associating chaos and disorder with a numerical quantity that could relate to energy and temperature, leading to the understanding of the second thermodynamic law and statistical mechanics. Also the action functional (also written S) (together with generalized coordinates and momenta and the lagrangian function), initially an alternative formulation of classical mechanics to Newton's laws, now extends the range of modern physics in general – notably quantum mechanics and particle physics.
Analytical convenience: They allow other equations to be written more compsier mathematical manipulation;
Ease of comparison: They allow comparisons of measurements to be made when they might appear ambiguous and unclear otherwise.
Defining equations are normally formulated in terms of elementary algebra and calculus, vector algebra and calculus, or for the most general applications tensor algebra and calculus, depending on the level of study and presentation, complexity of topic and scope of applicability. Functions may be incorporated into a definition, in for calculus this is necessary. Quantities may also be complex-valued for theoretical advantage, but for a physical measurement the real part is relevant, the imaginary part can be discarded. For more advanced treatments the equation may have to be written in an equivalent but alternative form using other defining equations for the definition to be useful. Often definitions can start from elementary algebra, then modify to vectors, then in the limiting cases calculus may be used. The various levels of maths used typically follows this pattern.
Typically definitions are explicit, meaning the defining quantity is the subject of the equation, but sometimes the equation is not written explicitly – although the defining quantity can be solved for to make the equation explicit. For vector equations, sometimes the defining quantity is in a cross or dot product and cannot be solved for explicitly as a vector, but the components can.
|Angular momentum||Electric current density|
|Angular momentum||Electric current density|
|Product form||Starting from
since L = 0 when p and r are parallel or antiparallel, and is a maximum when perpendicular, so that the only component of p which contributes to L is the tangential |p| sin θ, the magnitude of angular momentum L should be re-written as
Since r, p and L form a right-hand triad, this leads to the vector form
Alternatively for integral form
|Angular momentum||Electric current density|
|Product/integral form||Starting from
the components are Li, rj, pi, where i, j, k are each dummy indices each taking values 1, 2, 3, using the identity from tensor analysis
where εijk is the permutation/Levi-Cita tensor, leads to
|Using the Einstein summation convention,
Sometimes there is still freedom within the chosen units system, to define one or more quantities in more than one way. The situation splits into two cases:
There are two possibilities for each case:
Contradictions can be avoided by defining quantities successively; the order in which quantities are defined must be accounted for. Examples spanning these instances occur in electromagnetism, and are given below.
Definitions vs. functions: Defining quantities can vary as a function of parameters other than those in the definition. A defining equation only defines how calculate the defined quantity, it cannot describe how the quantity varies as a function of other parameters since the function would vary from one application to another.
Definitions vs. theorems: There is a very important difference between defining equations and general or derived results, theorems or laws. Defining equations do not find out any information about a physical system, they simply re-state one measurement in terms of others. Results, theorems, and laws, on the other hand do provide meaningful information, if only a little, since they represent a calculation for a quantity given other properties of the system, and describe how the system behaves as variables are changed.
Some equations, typically results from a derivation, contain quantities which may already be base quantities or have a definition, but be labelled in a different way with respect to the context of the result. These are not defining equations since they are results which apply to a physical situation – they are not quantity constructions, but can be used in the same way for calculations of the specific quantity within its scope of application.
Notice that these are all derived results from their respective theories – not proper definitions.
Important applied base units for space and time are below. Area and volume are of course derived from length, but included for completeness as they occur frequently in many derived quantities, in particular densities.
|(Common) Quantity name/s||(Common) Quantity symbol||SI unit||Dimension|
|(Spatial) position (vector)||r, R, a, d||m||[L]|
|Angular position, angle of rotation (can be treated as vector or scalar)||θ, θ||rad||dimensionless|
|Area, cross-section||A, S, Ω||m2||[L]2|
|Vector area (Magnitude of surface area, directed normal to tangential plane of surface)||m2||[L]2|
Important and convenient derived quantities such as densities, fluxes, flows, currents are associated with many quantities. Sometimes different terms such as current density and flux density, rate, frequency and current, are used interchangeably in the same context, sometimes they are used uniqueley.
To clarify these effective template derived quantities, we let q be any quantity within some scope of context (not necessarily base quantities) and present in the table below some of the most commonly used symbols where applicable, their definitions, usage, SI units and SI dimensions - where [q] is the dimension of q.
For time derivatives, specific, molar, and flux densities of quantities there is no one symbol, nomenclature depends on subject, though time derivatives can be generally written using overdot notation. For generality we use qm, qn, and F respectively. No symbol is necessarily required for the gradient of a scalar field, since only the nabla/del operator ∇ or grad needs to be written. For spatial density, current, current density and flux, the notations are common from one context to another, differing only by a change in subscripts.
For current density, is a unit vector in the direction of flow, i.e. tangent to a flowline. Notice the dot product with the unit normal for a surface, since the amount of current passing through the surface is reduced when the current is not normal to the area. Only the current passing perpendicular to the surface contributes to the current passing through the surface, no current passes in the (tangential) plane of the surface.
The calculus notations below can be used synonymously.
|Quantity||Typical symbols||Definition||Meaning, usage||Dimension|
|Quantity||q||q||Amount of a property||[q]|
|Rate of change of quantity, Time derivative||Rate of change of property with respect to time||[q] [T]−1|
|Quantity spatial density||ρ = volume density (n = 3), σ = surface density (n = 2), λ = linear density (n = 1)
No common symbol for n-space density, here ρn is used.
|Amount of property per unit n-space
(length, area, volume or higher dimensions)
|Specific quantity||qm||Amount of property per unit mass||[q][L]-n|
|Molar quantity||qn||Amount of property per mole of substance||[q][L]-n|
|Quantity gradient (if q is a scalar field.||Rate of change of property with respect to position||[q] [L]−1|
|Spectral quantity (for EM waves)||qv, qν, qλ||Two definitions are used, for frequency and wavelength:
||Amount of property per unit wavelength or frequency.||[q][L]−1 (qλ)
|Flux, flow (synonymous)||ΦF, F||Two definitions are used;
||Flow of a property though a cross-section/surface boundary.||[q] [T]−1 [L]−2, [F] [L]2|
|Flux density||F||Flow of a property though a cross-section/surface boundary per unit cross-section/surface area||[F]|
|Current||i, I||Rate of flow of property through a cross
section/ surface boundary
|Current density (sometimes called flux density in transport mechanics)||j, J||Rate of flow of property per unit cross-section/surface area||[q] [T]−1 [L]−2|
|Moment of quantity||m, M||Two definitions can be used;
q is a scalar:
|Quantity at position r has a moment about a point or axes, often relates to tendency of rotation or potential energy.||[q] [L]|
The meaning of the term physical quantity is generally well understood (everyone understands what is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). The term physical quantity does not imply a physically invariant quantity. Length for example is a physical quantity, yet it is variant under coordinate change in special and general relativity. The notion of physical quantities is so basic and intuitive in the realm of science, that it does not need to be explicitly spelled out or even mentioned. It is universally understood that scientists will (more often than not) deal with quantitative data, as opposed to qualitative data. Explicit mention and discussion of physical quantities is not part of any standard science program, and is more suited for a philosophy of science or philosophy program.
The notion of physical quantities is seldom used in physics, nor is it part of the standard physics vernacular. The idea is often misleading, as its name implies "a quantity that can be physically measured", yet is often incorrectly used to mean a physical invariant. Due to the rich complexity of physics, many different fields possess different physical invariants. There is no known physical invariant sacred in all possible fields of physics. Energy, space, momentum, torque, position, and length (just to name a few) are all found to be experimentally variant in some particular scale and system. Additionally, the notion that it is possible to measure "physical quantities" comes into question, particular in quantum field theory and normalization techniques. As infinities are produced by the theory, the actual “measurements” made are not really those of the physical universe (as we cannot measure infinities), they are those of the renormalization scheme which is expressly depended on our measurement scheme, coordinate system and metric system.
It is not always possible to define the distance between two points of any quality space, and this distance is —inside a given theoretical context— not uniquely defined. The notion of a distance, even in the context of quality space, relies on a concept of a metric space. Without a metric space, any notion of distance, physical or otherwise is undefined.
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