General derived quantities

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

Spaceedit

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)	${\displaystyle \mathbf{A}\equiv <!--\; \equiv \; -->A\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{n}},\mathbf{S}\equiv <!--\; \equiv \; -->S\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{n}}}$	m2	L2
Volume	τ, V	m3	L3

Densities, flows, gradients, and momentsedit

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 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, ${\displaystyle \stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{t}}}$ 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 ${\displaystyle X\equiv <!--\; \equiv \; -->X\left(x_1,x_2\cdots <!--\; \cdots \; -->x_n\right)}$, then:

Differential The differential n-space volume element is ${\displaystyle {\mathrm{d}}^{n}x\equiv <!--\; \equiv \; -->\mathrm{d}{V}_n\equiv <!--\; \equiv \; -->\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots <!--\; \cdots \; -->\mathrm{d}{x}_n}$,

Integral: The multiple integral of X over the n-space volume is ${\displaystyle \int <!--\; \int \; -->X{\mathrm{d}}^{n}x\equiv <!--\; \equiv \; -->\int <!--\; \int \; -->X\mathrm{d}{V}_n\equiv <!--\; \equiv \; -->\int <!--\; \int \; -->\cdots <!--\; \cdots \; -->\int <!--\; \int \; -->\int <!--\; \int \; -->X\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots <!--\; \cdots \; -->\mathrm{d}{x}_n}$.

Quantity	Typical symbols	Definition	Meaning, usage	Dimension
Quantity	q	q	Amount of a property	q
Rate of change of quantity, Time derivative	${\displaystyle \stackrel{\dot{}<!--\; \dot{}\; -->}{q}}$	${\displaystyle \stackrel{\dot{}<!--\; \dot{}\; -->}{q}\equiv <!--\; \equiv \; -->\frac{\mathrm{d}q}{\mathrm{d}t}}$	Rate of change of property with respect to time	qT−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.	${\displaystyle q=\int <!--\; \int \; -->{\mathrm{\rho <!--\; \rho \; -->}}_n\mathrm{d}{V}_n}$	Amount of property per unit n-space (length, area, volume or higher dimensions)	qL−n
Specific quantity	qm	${\displaystyle q_m=\frac{\mathrm{d}q}{\mathrm{d}m}}$	Amount of property per unit mass	qM−1
Molar quantity	qn	${\displaystyle q_n=\frac{\mathrm{d}q}{\mathrm{d}n}}$	Amount of property per mole of substance	qN−1
Quantity gradient (if q is a scalar field).		${\displaystyle \mathrm{\nabla <!--\; \nabla \; -->}q}$	Rate of change of property with respect to position	qL−1
Spectral quantity (for EM waves)	qv, qν, qλ	Two definitions are used, for frequency and wavelength: ${\displaystyle q=\int <!--\; \int \; -->q_{\mathrm{\lambda <!--\; \lambda \; -->}}\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}}$ ${\displaystyle q=\int <!--\; \int \; -->q_{\mathrm{\nu <!--\; \nu \; -->}}\mathrm{d}\mathrm{\nu <!--\; \nu \; -->}}$	Amount of property per unit wavelength or frequency.	qL−1 (qλ) qT (qν)
Flux, flow (synonymous)	ΦF, F	Two definitions are used; Transport mechanics, nuclear physics/particle physics: ${\displaystyle q=\iiint <!--\; \iiint \; -->F\mathrm{d}A\mathrm{d}t}$ Vector field: ${\displaystyle {\mathrm{\Phi <!--\; \Phi \; -->}}_F={\iint <!--\; \iint \; -->}_S\mathbf{F}\cdot <!--\; \cdot \; -->\mathrm{d}\mathbf{A}}$	Flow of a property though a cross-section/surface boundary.	qT−1L−2, FL2
Flux density	F	${\displaystyle \mathbf{F}\cdot <!--\; \cdot \; -->\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{n}}=\frac{\mathrm{d}{\mathrm{\Phi <!--\; \Phi \; -->}}_F}{\mathrm{d}A}}$	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	qT−1
Current density (sometimes called flux density in transport mechanics)	j, J	${\displaystyle I=\iint <!--\; \iint \; -->\mathbf{J}\cdot <!--\; \cdot \; -->\mathrm{d}\mathbf{S}}$	Rate of flow of property per unit cross-section/surface area	qT−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: ${\displaystyle \mathbf{m}=\mathbf{r}\times <!--\; \times \; -->\mathbf{q}}$	Quantity at position r has a moment about a point or axes, often relates to tendency of rotation or potential energy.	qL

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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.