Aphysical quantity (or simplyquantity)[1][a] is a property of a material or system that can bequantified bymeasurement. A physical quantity can be expressed as avalue, which is a pair of anumerical value and aunit of measurement. For example, the physical quantitymass, symbolm, can be quantified asm=nkg, wheren is the numerical value and kg is the unit symbol (forkilogram).Vector quantities have, besides numerical value and unit, direction or orientation in space.
The notion ofdimension of a physical quantity was introduced byJoseph Fourier in 1822.[2] By convention, physical quantities are organized in a dimensional system built uponbase quantities, each of which is regarded as having its own dimension.The dimension of a quantityZ is denoteddimZ ordim(Z).[1]
Some physical quantities arecommensurable, meaning that they can be added, subtracted, and compared with one another. To be commensurable, quantities must have the same dimension, but this alone is not sufficient; the quantities must also be the samekind.[1] For example, bothkinematic viscosity andthermal diffusivity have dimension of square length per time (units ofm2/s), but they are not commensurable. Quantities of the same kind share extra commonalities beyond their dimension and units allowing their comparison (see also:dimensional equivalence).
There is often a choice of unit, thoughSIunits 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 symbolm, and could be expressed in the unitskilograms (kg),pounds (lb), ordaltons (Da).The unit of a quantityZ is denoted[Z].[1]
FollowingISO 80000-1,[1] any value ormagnitude of a physical quantity is expressed as a comparison to a unit of that quantity. Thevalue of a physical quantityZ is expressed as the product of anumerical value {Z} (a pure number) and a unit [Z]:
For example, let be "2 metres"; then, is the numerical value and is the unit.Conversely, the numerical value expressed in an arbitrary unit can be obtained as:
The multiplication sign is usually left out, just as it is left out between variables in the scientific notation of formulas. The convention used to express quantities is referred to asquantity calculus. In formulas, the unit [Z] can be treated as if it were a specific magnitude of a kind of physical dimension: seeDimensional analysis for more on this treatment.
International recommendations for the use of symbols for quantities are set out inISO/IEC 80000, theIUPAP red book and theIUPAC green book. For example, the recommended symbol for the physical quantity "mass" ism, and the recommended symbol for the quantity "electric charge" isQ.
Physical quantities are normally typeset in italics.Purely numerical quantities, even those denoted by letters, are usually printed in roman (upright) type, though sometimes in italics. 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.
Ascalar is a physical quantity that has magnitude but no direction. Symbols for physical quantities are usually chosen to be a single letter of theLatin orGreek alphabet, and are printed in italic type.
Vectors are physical quantities that possess both magnitude and direction and whose operations obey theaxioms of avector space. Symbols for physical quantities that are vectors are in bold type, underlined or with an arrow above. For example, ifu is the speed of a particle, then the straightforward notations for its velocity areu,u, or.
Scalar and vector quantities are the simplesttensor quantities, which aretensors that can be used to describe more general physical properties. For example, theCauchy stress tensor possesses magnitude, direction, and orientation qualities.
A system of quantities relates physical quantities, and due to this dependence, a limited number of quantities can serve as a basis in terms of which the dimensions of all the remaining quantities of the system can be defined. A set of mutually independent quantities may be chosen by convention to act as such a set, and are called base quantities. The seven base quantities of theInternational System of Quantities (ISQ) and their correspondingSI units and dimensions are listed in the following table.[3]: 136 Other conventions may have a different number ofbase units (e.g. theCGS andMKS systems of units).
The angular quantities,plane angle andsolid angle, are defined as derived dimensionless quantities in the SI. For some relations, their unitsradian andsteradian can be written explicitly to emphasize the fact that the quantity involves plane or solid angles.[3]: 137
Important applied base units for space and time are below.Area andvolume are thus, of course, derived from the length, but included for completeness as they occur frequently in many derived quantities, in particular densities.
Important and convenient derived quantities such as densities,fluxes,flows,currents are associated with many quantities. Sometimes different terms such ascurrent density andflux density,rate,frequency andcurrent, are used interchangeably in the same context; sometimes they are used uniquely.
To clarify these effective template-derived quantities, we useq to stand forany 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] denotes the dimension ofq.
For time derivatives, specific, molar, and flux densities of quantities, there is no one symbol; nomenclature depends on the subject, though time derivatives can be generally written using overdot notation. For generality we useqm,qn, andF respectively. No symbol is necessarily required for the gradient of a scalar field, since only thenabla/del operator ∇ orgrad 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 thedot 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 passingthrough the surface, no current passesin the (tangential) plane of the surface.
The calculus notations below can be used synonymously.
^"The concept 'quantity' may be generically divided into, e.g. 'physical quantity', 'chemical quantity', and 'biological quantity', or 'base quantity' and 'derived quantity'."[1]
^Fourier, Joseph.Théorie analytique de la chaleur, Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept ofphysical dimensions for the physical quantities.)
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