Editing Flux (section)

=== General mathematical definition (surface integral) ===
[[Image:Flux diagram.png|thumb|upright=1.2|The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.]]
As a mathematical concept, flux is represented by the [[surface integral#Surface integrals of vector fields|surface integral of a vector field]],<ref>{{cite book |author1=Murray R. Spiegel |url=https://archive.org/details/vectoranalysis0000unse_t6w7 |title=Vector Analysis |title-link= |author2=S. Lipcshutz |author3=D. Spellman |publisher=McGraw Hill |year=2009 |isbn=978-0-07-161545-7 |edition=2nd |series=Schaum's Outlines |page=100}}</ref>
<math display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathrm{d}\mathbf{A}</math>
<math display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathbf{n}\,\mathrm{d}A</math>
where '''F''' is a [[vector field]], and d'''A''' is the [[vector area]] of the surface ''A'', directed as the [[Normal (geometry)|surface normal]]. For the second, '''n''' is the outward pointed [[unit normal vector]] to the surface.

The surface has to be [[orientability|orientable]], i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is usually directed by the [[right-hand rule]].

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive [[divergence]] (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the [[curve]] encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the [[inner product]] of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the '''influx''' is counted positive; the opposite is the '''outflux'''.

The [[divergence theorem]] states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the [[divergence]]).

If the surface is not closed, it has an oriented curve as boundary. [[Stokes' theorem]] states that the flux of the [[Curl (mathematics)|curl]] of a vector field is the [[line integral]] of the vector field over this boundary. This path integral is also called [[Circulation (fluid dynamics)|circulation]], especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.