Editing Flux (section)

=== General mathematical definition (transport) ===
[[File:General flux diagram.svg|thumb|upright=1.5|The [[field line]]s of a [[vector field]] {{math|'''F'''}} through surfaces with [[unit vector|unit]] normal {{math|'''n'''}}, the angle from {{math|'''n'''}} to {{math|'''F'''}} is {{mvar|θ}}. Flux is a measure of how much of the field passes through a given surface. {{math|'''F'''}} is decomposed into components perpendicular (⊥) and parallel {{nowrap|( ‖ )}} to {{math|'''n'''}}. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. <br />'''Top:''' Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. <br />'''Bottom:''' Field line through a [[curved surface]], showing the setup of the unit normal and surface element to calculate flux.]]
[[Image:Surface integral - definition.svg|thumb|upright=1.5|To calculate the flux of a vector field {{math|'''F'''}} ''(red arrows)'' through a surface {{mvar|S}} the surface is divided into small patches {{mvar|dS}}.  The flux through each patch is equal to the normal (perpendicular) component of the field, the [[dot product]] of {{math|'''F'''('''x''')}} with the unit normal vector {{math|'''n'''('''x''')}} ''(blue arrows)'' at the point {{math|'''x'''}} multiplied by the area {{mvar|dS}}.  The sum of {{math|'''F''' · '''n''', ''dS''}} for each patch on the surface is the flux through the surface.]]
Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol ''j'', (or ''J'') is used for flux, ''q'' for the [[physical quantity]] that flows, ''t'' for time, and ''A'' for area. These identifiers will be written in bold when and only when they are vectors.

First, flux as a (single) scalar:
<math display="block">j = \frac{I}{A},</math>
where
<math display="block">I = \lim_{\Delta t \to 0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}.</math>
In this case the surface in which flux is being measured is fixed and has area ''A''. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface.

Second, flux as a [[scalar field]] defined along a surface, i.e. a function of points on the surface:
<math display="block">j(\mathbf{p}) = \frac{\partial I}{\partial A}(\mathbf{p}),</math>
<math display="block">I(A,\mathbf{p}) = \frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}).</math>
As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. ''q'' is now a function of '''p''', a point on the surface, and ''A'', an area. Rather than measure the total flow through the surface, ''q'' measures the flow through the disk with area ''A'' centered at ''p'' along the surface.

Finally, flux as a [[vector field]]:
<math display="block">\mathbf{j}(\mathbf{p}) = \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}),</math>
<math display="block">\mathbf{I}(A,\mathbf{p}) = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}}\; \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}}).</math>
In this case, there is no fixed surface we are measuring over. ''q'' is a function of a point, an area, and a direction (given by a unit vector <math>\mathbf{\hat{n}}</math>), and measures the flow through the disk of area A perpendicular to that unit vector. ''I'' is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an [[abuse of notation]] because the "arg{{nnbsp}}max" cannot directly compare vectors; we take the vector with the biggest norm instead.)

==== Properties ====

These direct definitions, especially the last, are rather unwieldy {{Citation needed|date=May 2025}}. For example, the arg{{nnbsp}}max construction is artificial from the perspective of empirical measurements, when with a [[weathervane]] or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway.

If the flux '''j''' passes through the area at an angle θ to the area normal <math>\mathbf{\hat{n}}</math>, then the [[dot product]]
<math display="block">\mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta.</math>
That is, the component of flux passing through the surface (i.e. normal to it) is ''j''{{nnbsp}}cos{{nnbsp}}''θ'', while the component of flux passing tangential to the area is ''j''{{nnbsp}}sin{{nnbsp}}''θ'', but there is ''no'' flux actually passing ''through'' the area in the tangential direction. The ''only'' component of flux passing normal to the area is the cosine component.

For vector flux, the [[surface integral]] of '''j''' over a [[Surface (mathematics)|surface]] ''S'', gives the proper flowing per unit of time through the surface:
<math display="block">\frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot \mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A},</math>
where '''A''' (and its infinitesimal) is the [[vector area]]{{snd}} combination <math>\mathbf{A} = A \mathbf{\hat{n}}</math> of the magnitude of the area ''A'' through which the property passes and a [[unit vector]] <math>\mathbf{\hat{n}}</math> normal to the area.
Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration ''t''<sub>1</sub> to ''t''<sub>2</sub>, getting the total amount of the property flowing through the surface in that time (''t''<sub>2</sub>&nbsp;−&nbsp;''t''<sub>1</sub>):
<math display="block">q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\, dt.</math>