Editing Physical quantity (section)

==== Densities, flows, gradients, and moments ====
Important and convenient derived quantities such as densities, [[flux]]es, [[Fluid dynamics|flows]], [[Electric current|current]]s 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 use ''q'' to stand for ''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<sub>m</sub>'', ''q<sub>n</sub>'', and '''F''' respectively. No symbol is necessarily required for the gradient of a scalar field, since only the [[Del|nabla/del operator]] ∇ or [[Gradient|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, <math> \mathbf{\hat{t}}</math> 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 [[Multivariable calculus|''n''-variable]] [[Function (mathematics)|function]] <math> X \equiv X \left ( x_1, x_2 \cdots x_n \right ) </math>, then

'''''Differential''''' The differential [[n-dimensional space|''n''-space]] [[volume element]] is <math> \mathrm{d}^n x \equiv \mathrm{d} V_n \equiv \mathrm{d} x_1 \mathrm{d} x_2 \cdots \mathrm{d} x_n  </math>,

:'''''Integral''''': The [[Multiple integral|''multiple'' integral]] of ''X'' over the ''n''-space volume is <math> \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  </math>.

{| class="wikitable"
! scope="col" width="150" | Quantity
! scope="col" width="150" | Typical symbols
! scope="col" width="250" | Definition
! scope="col" width="200" | Meaning, usage
! scope="col" width="100" | Dimensions
|-
| Quantity 
| ''q'' 
| ''q'' 
| Amount of a property
| [q]
|-
| Rate of change of quantity, [[time derivative]]
| <math> \dot{q} </math> 
| <math> \dot{q} \equiv \frac{\mathrm{d} q}{\mathrm{d} t} </math>
| Rate of change of property with respect to time
| [q]T<sup>−1</sup>
|-
| 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 ''ρ<sub>n</sub>'' is used. 
| <math> q = \int \rho_n  \mathrm{d} V_n </math> 
| Amount of property per unit n-space <br />
(length, area, volume or higher dimensions)
| [q]L<sup>−''n''</sup>
|-
| Specific quantity
| ''q<sub>m</sub>'' 
| <math> q_m = \frac{\mathrm{d} q}{\mathrm{d} m} </math> 
| Amount of property per unit mass
| [q]M<sup>−1</sup>
|-
| Molar quantity
| ''q<sub>n</sub>''
| <math> q_n = \frac{\mathrm{d} q}{\mathrm{d} n} </math> 
| Amount of property per mole of substance
| [q]N<sup>−1</sup>
|-
| Quantity gradient (if ''q'' is a [[scalar field]]).
|
| <math> \nabla q </math> 
| Rate of change of property with respect to position 
|| [q]L<sup>−1</sup>
|-
| Spectral quantity (for EM waves)
| ''q<sub>v</sub>, q<sub>ν</sub>, q<sub>λ</sub>''
| Two definitions are used, for frequency and wavelength:<br />
<math> q=\int q_\lambda \mathrm{d} \lambda </math><br />
<math> q=\int q_\nu \mathrm{d} \nu </math> 
| Amount of property per unit wavelength or frequency.
| [q]L<sup>−1</sup> (''q<sub>λ</sub>'')<br />
[q]T (''q<sub>ν</sub>'')
|-
| Flux, flow (synonymous) 
| ''Φ<sub>F</sub>'', ''F'' 
| Two definitions are used:<br />
[[Transport phenomena (engineering & physics)|Transport mechanics]], [[nuclear physics]]/[[particle physics]]: <br />
<math> q = \iiint F \mathrm{d} A \mathrm{d} t </math>

[[Vector field]]: <br />
<math> \Phi_F = \iint_S \mathbf{F} \cdot \mathrm{d} \mathbf{A}</math> 
| Flow of a property though a cross-section/surface boundary.
| [q]T<sup>−1</sup>L<sup>−2</sup>, [F]L<sup>2</sup>
|-
| Flux density 
| '''F''' 
| <math> \mathbf{F} \cdot \mathbf{\hat{n}} = \frac{\mathrm{d} \Phi_F}{\mathrm{d} A} </math> 
| Flow of a property though a cross-section/surface boundary per unit cross-section/surface area
| [F]
|-
| Current 
| ''i'', ''I''
| <math> I = \frac{\mathrm{d} q}{\mathrm{d} t} </math> 
| Rate of flow of property through a cross-section/surface boundary
| [q]T<sup>−1</sup>
|-
| Current density (sometimes called flux density in transport mechanics)
| '''j''', '''J'''
| <math> I = \iint \mathbf{J} \cdot \mathrm{d}\mathbf{S}</math>
| Rate of flow of property per unit cross-section/surface area
| [q]T<sup>−1</sup>L<sup>−2</sup>
|-
| [[Moment (physics)|Moment]] of quantity
| '''m''', '''M''' 
|
''k''-vector ''q'': <math> \mathbf{m} = \mathbf{r} \wedge q </math> 
* scalar ''q'': <math> \mathbf{m} = \mathbf{r} q </math> {{br}}
* 3D vector '''q''', equivalently{{efn|via [[Hodge duality]]}} <math> \mathbf{m} = \mathbf{r} \times \mathbf{q} </math>
| Quantity at position '''r''' has a moment about a point or axes, often relates to tendency of rotation or [[potential energy]].
| [q]L
|}