# Physical quantity

A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as the combination of a magnitude and a unit. For example, the physical quantity mass can be quantified as n kg where n is magnitude and kg is the unit.

## Symbols, nomenclature

International recommendations for the use of symbols for quantities are set out in ISO/IEC 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 electric 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.

• Name reference: The quantity has a subscripted or superscripted single letter, group of letters, or complete word, to specify what concept or entity they refer to. These subscripts or superscripts tend to be written in upright roman typeface rather than italic while the basic symbol representing the quantity is in italic. For instance Ek or Ekinetic is usually used to denote kinetic energy and Ep or Epotential is usually used to denote potential energy.
• Quantity reference: The quantity has a subscripted or superscripted single letter, group of letters, or complete word, to specify what measurement/s they refer to. These subscrtipts or superscripts tend to be written in italic rather than upright roman typeface; the basic symbol is also in italic. For example cp or cisobaric is heat capacity at constant pressure.

The type of subscript is expressed by its typeface: 'k' and 'p' are abbreviations of the words kinetic and potential, whereas p (italic) is the symbol for the physical quantity pressure rather than an abbreviation of the word.

• Indices: The use of indices is for mathematical formalism using Index notation.

### Scalars

A scalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of the Latin or Greek alphabet, and are printed in italic type.

### Vectors

Vectors are physical quantities that possess both magnitude and direction. Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, if u is the speed of a particle, then the straightforward notations for its velocity are u, u, or ${\displaystyle {\vec {u}}\,\!}$.

### Numbers and elementary functions

Numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in 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.

Examples:

• Real numbers, such as 1 or 2,
• e, the base of natural logarithms,
• i, the imaginary unit,
• π for the ratio of a circle's circumference to its diameter, 3.14159265358979323846264338327950288...
• δx, Δy, dz, representing differences (finite or otherwise) in the quantities x, y and z
• sin α, sinh γ, log x

## Units and dimensions

### Units

There is often a choice of unit, though SI units (including submultiples and multiples of the basic unit) are usually used in scientific contexts due to their ease of use, international familiarity and prescription.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).

### Dimensions

The notion of dimension of a physical quantity was introduced by Joseph Fourier in 1822.[1] By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension.

## Base quantities

Base quantities are those quantities which are distinct in nature and in some cases have historically not been defined in terms of other quantities. Base quantities are those quantities on the basis of which other quantities can be expressed. 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 base units (e.g. the CGS and MKS systems of units).

International System of Quantities base quantities
Quantity SI unit Dimension
symbol
Name(s) (Common) symbol(s) Name Symbol
Length, width, height, depth, distance a, b, c, d, h, l, r, s, w, x, y, z metre m L
Time t, τ second s T
Mass m kilogram kg M
Absolute temperature T, θ kelvin K Θ
Amount of substance n mole mol N
Electric currenti, I ampere A I
Luminous intensityIv candela cd J
Solid angle ω, Ω steradian sr None

The last two angular units, plane angle and solid angle, are subsidiary units used in the SI, but are treated as dimensionless. The subsidiary units are used for convenience to differentiate between a truly dimensionless quantity (pure number) and an angle, which are different measurements.

## General derived quantities

Derived quantities are those whose definitions are based on other physical quantities (base quantities).

### Space

Important applied base units for space and time are below. Area and volume are thus of course derived from length, but included for completeness as they occur frequently in many derived quantities, in particular densities.

Quantity SI unit Dimensions
Description Symbols
(Spatial) position (vector) r, R, a, d m L
Angular position, angle of rotation (can be treated as vector or scalar) θ, θ rad None
Area, cross-section A, S, Ω m2 L2
Vector area (Magnitude of surface area, directed normal to tangential plane of surface) ${\displaystyle \mathbf {A} \equiv A\mathbf {\hat {n}} ,\quad \mathbf {S} \equiv S\mathbf {\hat {n}} \,\!}$ m2 L2
Volume τ, V m3 L3

### Densities, flows, gradients, and moments

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, ${\displaystyle \mathbf {\hat {t}} }$ 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.

If X is a n-variable function ${\displaystyle X\equiv X\left(x_{1},x_{2}\cdots x_{n}\right)}$, then:

Differential The differential n-space volume element is ${\displaystyle \mathrm {d} ^{n}x\equiv \mathrm {d} V_{n}\equiv \mathrm {d} x_{1}\mathrm {d} x_{2}\cdots \mathrm {d} x_{n}}$,
Integral: The multiple integral of X over the n-space volume is ${\displaystyle \int X\mathrm {d} ^{n}x\equiv \int X\mathrm {d} V_{n}\equiv \int \cdots \int \int X\mathrm {d} x_{1}\mathrm {d} x_{2}\cdots \mathrm {d} x_{n}\,\!}$.
Quantity Typical symbols Definition Meaning, usage Dimension
Quantity q q Amount of a property [q]
Rate of change of quantity, Time derivative ${\displaystyle {\dot {q}}\,\!}$ ${\displaystyle {\dot {q}}\equiv {\frac {\mathrm {d} q}{\mathrm {d} t}}}$ 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.

${\displaystyle q=\int \rho _{n}\mathrm {d} V_{n}}$ Amount of property per unit n-space

(length, area, volume or higher dimensions)

[q]Ln
Specific quantity qm ${\displaystyle q_{m}={\frac {\mathrm {d} q}{\mathrm {d} m}}\,\!}$ Amount of property per unit mass [q]M−1
Molar quantity qn ${\displaystyle q_{n}={\frac {\mathrm {d} q}{\mathrm {d} n}}\,\!}$ Amount of property per mole of substance [q]N−1
Quantity gradient (if q is a scalar field). ${\displaystyle \nabla q}$ 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:

${\displaystyle q=\int q_{\lambda }\mathrm {d} \lambda }$
${\displaystyle q=\int q_{\nu }\mathrm {d} \nu }$

Amount of property per unit wavelength or frequency. [q]L−1 (qλ)

[q]T (qν)

Flux, flow (synonymous) ΦF, F Two definitions are used;

Transport mechanics, nuclear physics/particle physics:
${\displaystyle q=\iiint F\mathrm {d} A\mathrm {d} t}$

Vector field:
${\displaystyle \Phi _{F}=\iint _{S}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }$

Flow of a property though a cross-section/surface boundary. [q]T−1L−2, [F]L2
Flux density F ${\displaystyle \mathbf {F} \cdot \mathbf {\hat {n}} ={\frac {\mathrm {d} \Phi _{F}}{\mathrm {d} A}}\,\!}$ Flow of a property though a cross-section/surface boundary per unit cross-section/surface area [F]
Current i, I ${\displaystyle I={\frac {\mathrm {d} q}{\mathrm {d} t}}}$ Rate of flow of property through a cross

section / surface boundary

[q]T−1
Current density (sometimes called flux density in transport mechanics) j, J ${\displaystyle I=\iint \mathbf {J} \cdot \mathrm {d} \mathbf {S} }$ Rate of flow of property per unit cross-section/surface area [q]T−1L−2
Moment of quantity m, M Two definitions can be used;

q is a scalar: ${\displaystyle \mathbf {m} =\mathbf {r} q}$
q is a vector: ${\displaystyle \mathbf {m} =\mathbf {r} \times \mathbf {q} }$

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, particularly 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 dependent on our measurement scheme, coordinate system and metric system.

## References

1. Fourier, Joseph. Théorie analytique de la chaleur, Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of physical dimensions for the physical quantities.)

## Sources

• Cook, Alan H. The observational foundations of physics, Cambridge, 1994. ISBN 0-521-45597-9
• Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
• Encyclopaedia of Physics, R.G. Lerner, G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005, pp 12–13
• Physics for Scientists and Engineers: With Modern Physics (6th Edition), P.A. Tipler, G. Mosca, W.H. Freeman and Co, 2008, 9-781429-202657