Physical quantity

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 are physical quantities that possess both magnitude and direction and whose operations obey the axioms of a vector space. 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 ${\displaystyle {\vec {u}}\,\!}$.

Scalars and vectors are the simplest tensors, which can be used to describe more general physical quantities. For example, the Cauchy stress tensor possesses magnitude, direction, and orientation qualities.

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).

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.

Important applied base units for space and time are below. Area and volume are thus, of course, derived from the 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

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 uniquely.

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] denotes the dimension of q.

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 use q_m, q_n, 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]L−n
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

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