## General derived quantities::Physical quantity

### ::concepts

Mathrm::physical    Quantity::mathbf    Width::scope    Surface::property    Symbol::density    Current::units

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 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 1
Area, cross-section A, S, Ω m2 L2
Vector area (Magnitude of surface area, directed normal to tangential plane of surface) $\mathbf{A} \equiv A\mathbf{\hat{n}}, \quad \mathbf{S}\equiv S\mathbf{\hat{n}} \,\!$ 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, $\mathbf{\hat{t}}$ 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 $X \equiv X \left ( x_1, x_2 \cdots x_n \right )$, then:

Differential The differential n-space volume element is $\mathrm{d}^n x \equiv \mathrm{d} V_n \equiv \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n$,
Integral: The multiple integral of X over the n-space volume is $\int X \mathrm{d}^n x \equiv \int X \mathrm{d} V_n \equiv \int \cdots \int \int X \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n \,\!$.
Quantity Typical symbols Definition Meaning, usage Dimension
Quantity q q Amount of a property [q]
Rate of change of quantity, Time derivative $\dot{q} \,\!$ $\dot{q} \equiv \frac{\mathrm{d} q}{\mathrm{d} t}$ 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.

$q = \int \rho_n \mathrm{d} V_n$ Amount of property per unit n-space

(length, area, volume or higher dimensions)

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

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

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:
$q = \iiint F \mathrm{d} A \mathrm{d} t$

Vector field:
$\Phi_F = \iint_S \mathbf{F} \cdot \mathrm{d} \mathbf{A}$

Flow of a property though a cross-section/surface boundary. [q]T−1L−2, [F]L2
Flux density F $\mathbf{F} \cdot \mathbf{\hat{n}} = \frac{\mathrm{d} \Phi_F}{\mathrm{d} A} \,\!$ Flow of a property though a cross-section/surface boundary per unit cross-section/surface area [F]
Current i, I $I = \frac{\mathrm{d} q}{\mathrm{d} t}$ 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 $I = \iint \mathbf{J} \cdot \mathrm{d}\mathbf{S}$ 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: $\mathbf{m} = \mathbf{r} q$
q is a vector: $\mathbf{m} = \mathbf{r} \times \mathbf{q}$

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, 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.

Physical quantity sections
Intro  Symbols, nomenclature  Units and dimensions  Base quantities  General derived quantities  See also  References  Sources