Talk:Flux/Archive 1

headings
I added the headings to this article in hopes that people will be more willing to contribute to the idea of flux, and how it is used in their specific field without making the whole article too genre specific. If anyone can think of more headings to add please do. Also, I just made headings for each of the specific definitions already there. For instance I think fluid systems would fall under Physics, but I don't know enough about fluid systems to tell whether or not it warrants it's own heading.

Also there was some information in there that was commented out that sounded like it was a def. of flux in some specific field. I deleted it(editing around the comments was irksome), but maybe someone can go back and take a look at it to see if they know what it would fall under. Thanks! Starfoxy 03:03, 5 Feb 2005 (UTC)


 * Looks good. I'm going to edit the top paragraph/definition for clarity. I feel that the mathematics definition is in most basic terms; we can then use it to expound on other types of flux in other disciplines. If you don't like this, let me know, we can debate it and do some reverting ;) Tygar 06:12, Feb 7, 2005 (UTC)


 * I've created and added a diagram for the concept of flux. How is it?  Would you change anything at all? -RadRafe 05:22, 16 Mar 2005 (UTC)

Split the article?
I think that this article actually covers two distinct concepts. One concept is that of the quantity defined by the integral of a vector field across a surface. This is what we see in electromagnetism. The other concept is that of material flow across a surface. This is what we see in diffusion.

If matters were this clear, I would not hesitate to create Flux (of a vector field) and Flux (of substance), and set about trying to get the disambiguation page moved to this location. However, there are other kinds of flux that do not fall neatly into one category or the other. For instance, the fluid-mechanics definition of flux clearly deals with a material flow, but I imagine that one could (though it would be nontrivial) define a vector field which, upon integration over a surface, yield the amount of fluid passing through the surface.

So what do people think? Is this really a single article, or two articles lashed together? --Smack (talk) 04:57, 28 Mar 2005 (UTC)


 * I would not be as quick for splitting the article. It looks to me that all these concepts share the same idea, that of a flow through a surface, be that flow of energy or electrons (well, here I am not sure), or of liquid, or the abstract math definition with vector field. This article would indeed need more work, and some sections could (but don't have to) be made shorter and with the more in-depth stuff moved to new articles. But I would say that an article trying to show the big picture behind all those fluxes is necessary. Oleg Alexandrov 18:51, 28 Mar 2005 (UTC)

The two definitions are more related than may be apparent. The difference is not vector field vs. material flow. The distinction is between a vector field and the integral of that vector field over a surface. (Note that using the first definition the flux of material flow is in general a vector field) Some areas of science call the integral the "flux". The vector field can then be called "flux density". Other areas call the vector field the "flux". The integral could then be called the surface integral of the flux.

Worse, the usage doesn't break cleanly by subject area. In the study of electromagnetic radiation, it's common to use the first definition of flux (which makes sense, since one is really dealing with energy transport).--Srleffler 05:41, 11 November 2005 (UTC)

Entry from Pages Needing Attention

 * Flux - a haphazard jumble of vague special-case definitions
 * I split the special case definitions into a series of headings, and added a blanket explaination to the beginning. Flux is used so specifically in each branch that splitting it up seemed like the only way to go. I think some equations with explanations would be in order if someone has the tex skills to put them in. Starfoxy 03:07, 5 Feb 2005 (UTC)
 * I think the page is sufficiently improved now. The general definition of flux is better treated, with examples, analogies, and a diagram.  Specialized definitions are clearly separated into their respective subjects.  —RadRafe 07:40, 19 Mar 2005 (UTC)

I removed this from the physics pages needing attention. I think the article is no longer in desperate need of fixing and is really rather nice right now. Starfoxy 16:10, 30 Apr 2005 (UTC)

Definition of flux / lead para
I think this definition is starting to get muddied again by introducing a diametrically opposite statement to the first, right in the second sentence. Also I feel the term 'vector density' whilst being no doubt correct, is going to start to overload some peoples brains (like mine). Can we put vector density in braces after the words flux density? Anyway good to see youre still around Patrick.--Light current 22:28, 25 September 2005 (UTC)


 * Thanks. I rephrased the intro (and made various other additions and changes).--Patrick 10:15, 26 September 2005 (UTC)

I have changed the first paragraph to include very clear language stating that in Transport Phenomenon flux is implied to be per area per time, while in Electromagnetics flux is the integral of the vector field over a finite area. Neither of these definitions is "correct", and as the article now stands, the E+M definition is given the most prominent position. I would suggest reading Maxwell(1892 p.13 Treatise on Electricty and Magentism) for a very early statement that confirms the Transport definition. "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the Surface-integral of the flux. It represents the quantity which passes through the surface." This definition is clearly not what is being presented as the primary defintion in this article. I am reorganizing this entry and adding references where I can. Thanks to all who helped add links and formatting to my text !! Luckymonkey 14:50, 14 December 2005 (UTC)

"How much stuff goes through your thing"? That doesn't seem like a very clear sentence..." --Scottbert who has yet to get a user account

It isn't very clear, but I came to wikipedia trying to understand this damn concept and that sentence was the best one I've heard - it all fell into place when I read that. I give it 2 thumbs up! 09:43, 3 November 2007 (UTC)

The quote from Bird, Stewart, Lightfoot is wrong. They define flux as "flow per unit area" (page 13, 2nd edition). —Preceding unsigned comment added by Liftaren (talk • contribs) 14:07, 7 July 2010 (UTC)

an archaic use in biology
Flux is also an archaic term for dysentery, but also seems to cover any extensive flow of fluid from the body, so there's bloody flux, etc.

Flux is Flux - there is only one type
There seems to be some confusion as to whether there are two types of "flux" or one. There is only one from a mathematical standpoint, the integral of a vector quantity over a surface. Of course, there are different "types of fluxes" in that the physical quanity integrated across the surface can vary but they are still the same in their core definition. The first definition in this article is the same as the second, it just uses synonyms. Someone added "Flux in this definition is a vector" -- but flux always has a direction, across the surface, and that direction is the surface normal -- which is only defined after integration as either the average surface normal or, in the case of a flat surface, the constant surface normal. (BTW I write fluid simulators for a living -- see my homepage for videos) --Ben Houston 03:02, 21 August 2006 (UTC)


 * Reading the discussion on this talk page has me doubting myself. I just checked Encarta's dictionary (see Encarta: Flux definition) and found only these two physics-related definitions, both of which reinforce my original belief that there is only one definition:
 * "3. physics rate of flow across area: the rate of flow of something such as energy, particles, or fluid volume across or onto a given area
 * 4. physics strength of field in particular area: the strength of a field such as a magnetic or electric field acting on a particular area, equal to the area size multiplied by the component of the field acting at right angles to the area"
 * --Ben Houston 03:07, 21 August 2006 (UTC)

In my understanding, the first definition breaksdown into measuring the rate of transport of something across a surface and is measured in "units of something" per metre per second. Whereas, in electromagnetism the things which are called 'flux' are rarely measured in this way. For example, current density (charge flux) and magnetic flux density (?) are flux by the [unit]/m^2.s definition, but are not refered to as flux rather as density. This leads me to consider that flux in EM means something different to it's meaning in transport phenomena.

While we're at it... for me, the first two biology definitions of flux fall under the transport phenomena defintion, i.e. the rate of transport of a quantity (molecules across a cell membrane, carbon through an ecosystem) across a surface or boundary. Journeyman 04:01, 21 August 2006 (UTC)


 * I was wrong, I see your point. The difference seems to be related to "flux" and "net flux" (the integral of flux on a surface).
 * Thus I think the current definitions are exactly backwards - in physics flux is a scalar, not a vector as it is currently stated. And in electromagnetism, flux is a vector quantity since it is the physics type of flux integrated across a surface to produce a "net flux" that has direction.  --Ben Houston 05:25, 21 August 2006 (UTC)


 * Still not sure about the vector issue. In membrane technology we don't tend to talk in terms of vectors. Yet the flux does have both quantity and direction. The direction is related to the net driving force: temperature, pressure, concentration, etc. Although this could just be considered 'sign'. I'm sure this has been resolved before, we just need the references. Journeyman 02:31, 28 August 2006 (UTC)

The flux over the surface of a sphere enclosing a charged particle is non-zero, and ‘flux’ comes from the latin for ‘flow’, so how come there is nothing flowing across the surface (there isn't - the Coulomb field of a charged particle does not flow away from it)? The point is that vector fields often represent movement, so the relevant integral does indeed give a flow (of e.g. energy, particles). But with E and B (the electric and magnetic fields) the direction at a point in the field does not represent an actual movement. It is like the steepness of a hill as opposed to, say, the blowing of the wind. So what we have are two different types of thing that vector fields represent, and so we do have a corresponding two different types of thing that the flux can represent (one with flow, one without). I do think flux is always a scalar though, since it is the surface integral of a vector field. The direction is implicit, since flux is flux over a specified surface.

why is this not in simple words
the internet is meant to be simple not confusing —Preceding unsigned comment added by VMMK (talk • contribs)

Says who? And if you want simple, I suggest not reading articles about difficult concepts. 09:45, 3 November 2007 (UTC)

This lede is terrible.
This lede is terrible. It puts me right off reading the rest (and Im an engineer) trouble is, how to rewrite --TreeSmiler (talk) 00:38, 6 January 2008 (UTC)

Needs work; related articles
I think this needs work (the early paragraph, which is archaic/historical but doesn't currently say so), and I dug up related articles, but I'm out of time for the moment. Here's what I've got, raw notes:

http://en.wikipedia.org/wiki/History_of_calculus http://en.wikipedia.org/wiki/Method_of_Fluxions http://en.wikipedia.org/wiki/Derivative#Newton.27s_notation
 * $$\dot{y}$$ &emsp

$$\ddot{y}$$ http://en.wikipedia.org/wiki/Help:Formula

In physics and other fields, Newton's notation is used mostly for time derivatives, as opposed to slope or position derivatives.

That's just my raw notes, and part of that was copy-editing of the afore-mentioned history article.

If I have time I'll follow-through on the needed cleanup; if years go by, and I have not done so, someone else should consider doing so. Dougmerritt (talk) 04:45, 24 September 2008 (UTC)

Unit
Adamryanlee added a line saying, "Electric flux is measured in Coulombs per square meter (C/m2)." I'm fairly certain this is totally wrong, and Electric flux agrees with me. For now, I'm just removing the line. I'll leave it to someone else to put the correct units. Superm401 - Talk 04:10, 26 January 2009 (UTC)

Scalar vs vector quantities
The flux of a vector field is usually defined by a scalar product integral over a surface as in magnetic flux. The result is a scalar quantity. For a scalar field the flux would have to be defined in terms of an element of area normal to the stream lines. The dot product in the integral computes the area normal to the vector field. If the field vector and the surface vector are normal to each other the dot product is zero or, in other words, there is no flux through a surface when the field vector is tangent to the surface. If one considers "flux tubes" which are bounded by stream lines one has to use small elements of area to avoid the curvature of the surface. The fluxes at two points along the tube are not necessarily equal but if the flow is uniform and parallel they will be since the area at both ends is the same and there are no losses.

In discussing the flow of some quantity through a tube, say water, the amount of water, Q, is either measured in kilograms or liters. The flow or current is I = dQ/dt. The flux through the pipe would be I/A where A is the cross-sectional area of the pipe. This is an average flux. The value of the flux at different points along some cross-section may vary. Due to drag and viscosity the flow is higher in the center and zero at the walls of the pipe. Smaller pipes require larger fluxes for the same flow. The vector quantity associated with the flux would be the current density, J. If the quantity of water was measured in kg, the definition of current density would be J = &rho;v, the product of mass density and the velocity. Both the flux and the current density have the same units. In many cases involving flux one needs to be careful about the context and avoid confusion.

A natural law often can be written in both differential and integral forms and can be converted from one form to another using vector analysis. --Jbergquist (talk) 23:38, 2 August 2009 (UTC)


 * I may have inadvertantly made the mistake I was being critical of. In the case of the example of water flowing through a pipe the scalar product integral computes a rate of flow through the surface. The definition of flux as I/A would give an average flow. It could be that in going from scalar equations to vector equations some expressions were also transferred in the process and resulted in some confusion. Or it may be that terms were borrowed from Latin. flu·ō -ēre -xī -xum is the Latin verb for flow. The adjective is flux·us -a -um. One could literally translate a Latin word or give it a technical meaning. And language has a tendency to change over time especially when isolated groups use jargon and there is no effort to establish a consistent common language. --Jbergquist (talk) 17:24, 3 August 2009 (UTC)


 * Flow, flux and flux density can be confused. Flow seems to be used for the transport of some quantity. Flux is used for both the surface integral of a vector field and a surface density. It is inconsistent to use the same term or expression for a whole and its part so some qualification is needed to distinguish the two. More care should be used in definitions but precision can be sacrificed when the meaning is understood by its context. So one needs to ask if we are speaking in general terms or about some special usage and the article should resolve any ambiguity. --Jbergquist (talk) 18:54, 3 August 2009 (UTC)

Maxwell on Intensities and Fluxes
In A Treatise on Electricity and Magnetism (1873) Maxwell makes the distinction between two kinds of forces. The first associated with lines are referred to as Intensities. The second associated with surfaces are referred to as Fluxes. Lines of force or lines of attraction are associated with work, a scalar quantity. With respect to heat Maxwell writes,


 * "...the flux of heat in any direction at any point of a solid body may be defined as the quantity of heat which crosses a small area drawn perpendicular to that direction divided by that area and by the time. Here the flux is defined with reference to an area."

And later on,


 * "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the Surface-integral of the flux. It represents the quantity which passes through the surface." --Jbergquist (talk) 23:35, 3 August 2009 (UTC)


 * It is unfortunate that no one has come up with a mutually acceptable expression for the surface integral of a vector field. It may have been Maxwell's privilege to do this and he just referred to it as a surface integral. I tried to think of something suitable and the first similar term that I found was moment of area but moments involve cross products so I deemed that unsuitable. I tried to come up with something involving a vector field and an area and the application of a force over an area is associated with an impression as in the impression left by ink and a block of type on a piece of paper. The impression that results depends on both force and area. Maybe someone could come up with something better and suggest it to the scientific community. But the word "impression" has its appeal. --Jbergquist (talk) 02:49, 4 August 2009 (UTC)


 * "Collective impression" which includes the idea of integration might be better still but it would work best if the vector field is a force. Even with a flux such as the action of a wind on a sail there is a response which could be called an "impression". It looks like Maxwell thought of the integral as mathematical operator acting on a field. That's probably the simplest way of looking at it. From Maxwell's prespective the result of the operation on a flux, current density, is a flow. In Maxwell's time quaternions were the primary tool of analysis and they included scalar and vector parts. What we study now is not quite the same as what was done back then. The term flux is used in connection with the action of a magnetic field on a coil of wire and in computing the voltage that results which was discovered by Faraday. So the use of flux may predate Maxwell who mathematized and improved on what Faraday did in connection with fields. There is a precision in Maxwell's language that is not found elsewhere. --Jbergquist (talk) 17:19, 4 August 2009 (UTC)


 * The first impression of the electromagnetic field was that it was mathematically similar to a flow field. Coulomb was the first to measure the effects of the electric and magnetic fields and showed that they followed an inverse square law. Faraday introduced the field approach and the idea of lines of force. Detailed information on the electromagnetic field is not found in his notebooks on electromagnetic induction. There is no indication of measurements but he notes the movement of magnetic needles. Maxwell's later approach is more sophisticated and he discusses tubes associated with the lines of force in An Elementary Treatise on Electricity. This is probably why they thought there was an ether but Michaelson's experiment indicated that there was no ether. Einstein's Relativity showed that the electric and magnetic fields were related and their magnitudes were affected by motion. There was no experimental evidence for magnetic monopoles. So the older perspective fell out of favor. If there is no ether is it correct to speak of a magnetic flux? Probably not. But there seems to be something missing. Perhaps this gap can be filled by the "impressions" mentioned above. They are observables and contain the idea of point of view and the electromagnetic field tensor is consistent with Relativity. It may just be incorrect to speak of a flux. --Jbergquist (talk) 17:36, 5 August 2009 (UTC)

Flux, Fields and Action-at-a-distance
Field theory resulted from a concern over action-at-a-distance. To avoid action-at-a-distance one has to hypothesize some medium which acts in place of the source of the field (mass in the case of gravity and charge in the case of electric fields). But the surface integrals of the forces bounded by "lines" of force suggest a "flux" which is constant. Since we can only measure forces there is no direct evidence for a flux. And the only evidence for a conservation of some "medium" is the inverse square law. One should also note that the force of bouyancy in water is due to a pressure difference and not the result of a flow. So there is a lot of uncertainty about the nature of any hypothetical medium.

The possible existance of flow fields gives an interesting interpretation to the word physics. --Jbergquist (talk) 21:02, 7 August 2009 (UTC)

Heaviside on fluxes
Oliver Heaviside in Electromagnetic Theory I writes about forces and fluxes,


 * §20 "...In the study of these mechanical forces we are led to the more abstract ideas of electric force and magnetic force, apart from electrification, or magnetisation, or electric current, to work upon and produce visible effects. The conception of fields of force naturally follows, with the mapping out of space with the lines or tubes of force definitely distributed. A further and very important step is the recognition that the two vectors, the electric force and the magnetic force, represent, or are capable of measuring, the actual physical state of the medium concerned, from the electromagnetic point of view, when taken in conjunction with other qualities experimentally recognisable as the properties of matter, showing that different substances are affected to different extents by the same intensity of electric and magnetic force. The electric force is then to be conceived as producing or being invariably associated with a flux, the electric displacement; and similarly the magnetic force as producing a second flux, the magnetic induction..."


 * "§21 All space must be conceived to be filled with a medium which can support displacement and induction...Away from matter (in the ordinary sense) the medium concerned is the ether..." --Jbergquist (talk) 02:04, 8 August 2009 (UTC)

Heaviside uses flux in both senses,


 * "§28 ...We do not, I think, need a number of new words to distinguish fluxes through a surface from fluxes per unit surface..." --Jbergquist (talk) 02:23, 8 August 2009 (UTC)

Conflation of flux and flux density.
The SI distinguishes between flux and flux density, and so does everyone else who knows what they are talking about. For example, this article blatantly contradicts the SI definition of radiative flux. The correct unit for energy flux is the watt. The correct SI derived unit for energy flux density is watt per meter squared. Magnetic flux (weber) is not the same as magnetic flux density (tesla = weber per meter squared).

Flux is a scalar, and flux density is a vector. Flux is only defined for an arbitrary given surface. Flux is a property of a surface, while flux density is a property of a point, i.e. it is a field quantity.

"flow rate through a surface", is not the same as "flow rate thru a unit surface". The first is flux, the second is flux density.

In many fields it is common to abbreviate "flux density" to just "flux" when it won't lead to confusion. This does not imply that experts in those fields are conflating or equating the two concepts. In Wikipedia, this shorthand must either be avoided, or explicitly noted.

The definition of flux density must be added to Wikipedia, either here or in it's own article.


 * Yes, the relicts from the past; I tumbled over the same thing here. Somebody noted the (outdated) use of flux as a density and made it to the main description. I already did the split and a little bit of edit, but flux density is still a stub and this article still needs clean-up. Also the descritption in the lead-in is unnecessary heavy. -- Tomdo08 (talk) 01:25, 28 December 2010 (UTC)

An Integrated Approach
I have been reading many sources over the last year or more about the use of the term flux. This article is useful, but it seems to rely almost entirely on the Stauffer 2006 reference (cited in the further readings at the end). It seems to me that the difference in usage between the transport community and the EM community is largely historic and perhaps unnecessary. Precedent is not relevant here... If that were important, then we would still be using the original quaternion-based form of Maxwell's equations. Vector calculus was invented after the first transport laws (e.g. Fourier's law). It provides a comprehensive and generalized view of the relationship between forces and flows. Maxwell's publications emphasize this generalization. Onsager went much further to formalize a general view of forces and flows. That contribution is especially important for understanding the modern usage of flux. It would also help if this article included a discussions of how flux is used in fluid dynamics and in the advection equation. In both cases, the usage of flux seems much closer to the usage in the EM community. In other words, flux is an intensive scalar quantity defined relative to a surface. --MuTau (talk) 16:21, 22 February 2011 (UTC)

An example of this integrated approach is provided by the generic scalar transport equation, which is described at the following wiki page http://en.wikipedia.org/wiki/Generic_scalar_transport_equation. --MuTau (talk) 20:18, 27 February 2011 (UTC)

>> Stauffer, July 2011   If you look at Bird - Stewart and Lightfoot "Transport Phenomenon" they clearly define all fluxes as flow per unit area per time, a vector. — Preceding unsigned comment added by Luckymonkey (talk • contribs) 19:43, 13 July 2011 (UTC)

Quantum mechanics
I think the quantum mechanics part on the probability current is incorrect, as it states "the number of particles passing through a perpendicular unit of area per unit time", though it is NOT the number of particles (note, that $$|\psi|^2$$ is normalized, or that $$\mathbf{J}$$ is complex). The linked probability current article is OK on the subject. Jafar271 (talk) 12:55, 2 February 2012 (UTC)

Derivatives of functions or densities of measures?
Dear colleagues, the definition
 * $$j = \frac{{\rm d}^2 q}{{\rm d}A{\rm d}t}$$

is very unusual and misleading. It is written as a derivative of a function q(A,t) but this is wrong. The flux j is a density of the time derivative of the amount of physical quantity q that is transported through the surface. If you like, you can define the density as a derivative of a measure (time derivative of the transported amount) over a measure (the oriented area of the surface), i.e. the Radon–Nikodym derivative. Another definition (if you like) may be a derivative of another measure, the transported amount of q, over the product of measures $${\rm d}A{\rm d}t$$ but it is misleading to join the usual time derivative d/dt and Radon–Nikodym derivative in one formula like $$\frac{{\rm d}^2 q}{{\rm d}A{\rm d}t}$$. It is better to avoid such a mixture and do not use the Radon–Nikodym derivative in the introduction of the flux. Formally, you can always use this derivative in the definition of any density but people happily avoid such an "overformalization". I have never met such a notation like $$\frac{{\rm d}^2 q}{{\rm d}A{\rm d}t}$$. I guess that this is a homemade notation. If I am wrong, then please give a reference to a reliable source._Agor153 (talk) 10:53, 23 June 2012 (UTC)


 * You may be right that this presentation is homemade, I don't know, but what you wrote in the article is mathematical nonsense. If $$\scriptstyle j = {\rm {d}q}/{(\rm {d}A\rm {d}t)}$$ then $$\scriptstyle j \rm {d}A = {\rm {d}q}/\rm {d}t$$ is implied. For any well-behaved real-life case $$\scriptstyle \rm {d}q / \rm {d}t$$ is finite, and since $$\scriptstyle \rm {d}A$$ is an infinitessimal, $$j$$ is implied to be infinite.  Spinning  Spark  13:37, 23 June 2012 (UTC)
 * What did I write? Are you sure? (I did not write anything there, somebody else did.) All these notations are misleading. I propose to clean these formulas carefully._Agor153 (talk) 15:12, 23 June 2012 (UTC)
 * And now a couple of words about sense and nonsense: $$\Delta q = j \Delta S \Delta t$$. This is the first (correct!) formula in the article). You can see that there are the differences (not the second differences). The second differential in this context is nonsense. Somebody wrote: "In the limit Δq, Δt and A become infinitesimally small, the algebraic definition becomes a calculus definition:..." This is a trivial mistake because nether q nor $$\scriptstyle \rm {d}q / \rm {d}t$$ is a function of A. (They are measures on a surface with the area measure A). Somebody confused the introduction of density (Radone-Nikodym derivative - I dare to recommend for your attention the well written article Radon–Nikodym theorem) with the calculus derivative of the differentiable function. A simpler example gives us just the definition of density in 3D space: $$\Delta q = \rho (\Delta x)^3$$ but nobody writes for density $$\rho={\rm d}^3q/{\rm d} x^3$$. There is no such a thing here as $${\rm d}^3q$$ or $${\rm d}^2q$$. All the formulas with $${\rm d}^2q$$ in the paper are complete nonsense._Agor153 (talk) 15:36, 23 June 2012 (UTC)
 * Sorry for implying that you were the author. You posted with a redlinked account soon after I reverted the IP.  I assumed you were the same person.  It seems we were talking at cross purposes.  Spinning  Spark  16:56, 23 June 2012 (UTC)

The notation is not supposed to imply derivatives, but ratios of differential quantities. It is correct and self-consistent.

The current or "rate of flow of quantity q" in terms of average changes by:


 * $$I = \frac{\Delta q}{\Delta t}$$

because this is just the change in q per unit time - the current, so for infinitesimal changes dq and dt instead of Δq and Δt, we have:


 * $$I = \frac{dq}{dt}$$

Agreed? So the flux (current per unit area) is likewise the ratio of infinitesimal current dI per unit infinitesimal area dA:


 * $$j = \frac{dI}{dA} = \frac{d\left(\frac{dq}{dt}\right)}{dA} = \frac{d^2q}{dAdt} $$

which is consistently used notation - for both derivatives and differentials.

You can multiply back by dA:


 * $$jdA = dI = d \left(\frac{dq}{dt}\right) = dA\frac{d^2q}{dAdt}= dA\frac{d \left(\frac{dq}{dt}\right)}{dA}$$

then integrate wrt A:


 * $$\int jdA = \int dI = \int d \left(\frac{dq}{dt}\right) = \int dA\frac{d^2q}{dAdt} = \int \frac{d\left(\frac{dq}{dt}\right)}{dA}dA$$


 * $$I = \frac{dq}{dt} $$

which is correct. I have no clue what this Radon–Nikodym theorem/derivative has to do with it. Again: these are not derivatives, they ratios of differentials.

See also
 * Radiance,
 * talk:Continuity equation (where a similar dialogue occurred),
 * these photometry links.

Unfortunately I don't have sources which explicitly state these flux formulae in complete generality, but pretty much any form of current density/flux (as some form of current per unit area) takes this form. Hope this helps. z = z²  + c 14:44, 18 July 2012 (UTC)
 * Your $$j = \frac{\Delta I}{A}$$ is definitely not a clarification. It is not intuitive nor mathematically correct. First, some intuitively appealing formula has to be presented, such as $$j = \lim\frac{I}{A}$$ for $A → 0$, where such areas are coplanar (or lie on the same smooth surface). Then, the article should contain more strict definition, probably along the lines of Agor153's approach. "These are not derivatives - they are ratios of differential quantities" is some hyphen-bearing obscurity where the author's point I even cannot realize. Incnis Mrsi (talk) 18:23, 18 July 2012 (UTC)


 * Hrm, had a niggling feeling my $$j = \frac{\Delta I}{A}$$ was stupid, so that should be replaced by your $$j = \lim \limits_{A \rightarrow 0}\frac{I}{A}$$, but I'm sorry I don't really understand what Agor153 is proposing to define. Would it be better to just use the integral definition:


 * $$q=\int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}{\rm d}A{\rm d}t $$


 * judging from his formula $$\Delta q = j \Delta S \Delta t$$? I asked on his talkpage what to advise.


 * Also why is "ratio of two differential/infinitesimal quantities" wrong? I thought that was the correct way to define densities/fluxes (well fluxes need limits like you just said)...
 * Thanks for feedback anway Incnis Mrsi, I'll at least update to the correct changes for now. z = z²  + c 19:52, 18 July 2012 (UTC)


 * Tried to fix... z = z²  + c 20:15, 18 July 2012 (UTC)


 * Dear colleagues. The density (for example, the flux density) is NOT a derivative in the sense of calculus. Also, there are no differentials. The formula
 * $$j = \lim \limits_{A \rightarrow 0}\frac{I}{A}=\frac{dI}{dA}$$ is wrong because there is no differentiable function $$J(A)$$ and differentials have sense for such functions only. I ask to follow the rules of Wikipedia and do not organize a discussion about the mathematical notions. It is necessary to go to a respectable source instead. Could you please give any reference to any respectable paper or a textbook where these $$\frac{d^2q}{dAdt}$$ (and other "differentials") are published?-Agor153 (talk) 20:52, 20 July 2012 (UTC)
 * PS. "* In calculus, the differential represents a change in the linearization of a function." - see Differential (mathematics).-Agor153 (talk) 20:55, 20 July 2012 (UTC)
 * And, of course, you are right when you propose to use the integral definition $$q=\int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}{\rm d}A{\rm d}t $$. This is exactly the definition of a measure through measures, as the Radon-Nikodym derivative. (One additional difficulty for simple understanding is that there is a usual derivative too, the time derivative of q).Agor153 (talk) 21:01, 20 July 2012 (UTC)
 * I did not assert that there is a certain function $I$ depending on $A$. Of course, both depend on the choice of 2-dimensional piece (say, $Σ$ ) of a plane or a surface those area $A$ is. Also, if the flux is not constant, then "$$\lim \limits_{A_\Sigma \rightarrow 0}$$" is certainly not correct. The correct filter for this limit must also restrict these $Σ$ to belong to increasingly small neighbourhoods of the point where the flux to be measured. I did not say $$j = \lim\limits_{A_\Sigma \rightarrow 0} \frac{I_\Sigma}{A_\Sigma}$$ is correct. I said it is appealing, but not strictly correct, there must be some clauses to make it correct. On the other hand, Wikipedia is not a formal theory – one should not reduce an article to existing articles such as Radon–Nikodym theorem. Indeed, an article should try to explain the notion by elementary means. Incnis Mrsi (talk) 21:33, 20 July 2012 (UTC)

I agree we should use the integral definition, since integration has the very direct intuitive meaning of the continuous limit of summation (as you all obviously know of course) to get the quantity q from j or I, and is less ambiguous than differential notation (derivative or differential? what type? Fréchet, Gâteaux, or as already stated above Radon–Nikodym?).

I'm partly responsible for this use of the above differential notation on WP so will replace all instances of this with the integral definition in other articles. Maschen (talk) 07:46, 13 September 2012 (UTC)
 * Thank you a lot. BTW, due to this discussion I understood a gap in the standard calculus modules: in the standard textbooks the notion of density is not introduced and it appears in a mathematically correct form after the Lebesgue integration only.-Agor153 (talk) 10:12, 13 September 2012 (UTC)


 * No problem, I cleared up most of my mess but it may still remain in other places (from other people?...). Feel free to convert to integrals also if you see any.
 * The reason I used to use it was because of the above presentation of differentials, and that I have seen "d2" for "second differential" in places, such as the links above, and even in MTW (p.127) who write a differential equation as:


 * $$(a^2-x^2)\mathrm{d}y^2 + 2xy\mathrm{d}y\mathrm{d}x + (a^2-y^2)\mathrm{d}x^2=0$$


 * As you say, the Lebesgue integral has the interpretation of n-volume with (n − 1)-boundary, appropriate for defining densities (although are apparently differential forms nicer for that too independent of coordinates and Jacobian factors due to the exterior product...). Maybe this should be clarified in the article in a physical context? Maschen (talk) 10:49, 13 September 2012 (UTC)

plexus flux
network flow — Preceding unsigned comment added by 112.207.16.81 (talk) 06:57, 17 June 2013 (UTC)

Dot product Expression in the First Figure
I re-wrote the caption of the first figure, but it is appropriate to change the capital 'F' in the dot product expression (F· dS) in the picture with the lowercase 'f' (for flow), i think. Whether in maths or physics or any other field, the term flux has inherently a consistent mathematical meaning: integration of a vector field over a surface. Therefore, flux gives the net/total strength of the field over the surface that is subjected to the field.

A vector field spans over a certain space and assigns a vector (i.e. a magnitude and a direction) to each location in that space. Depending on the quantity associated with the vector field, be it an electric field around a charge configuration, magnetic field around a current or a permanent magnet, displacement field or velocity field of a flowing liquid, the flux means the net change of the associated quantity throughout the surface under investigation, i.e. the net effect of the field over the surface. This can be the net electric field penetrating a certain surface area, net velocity change throughout a track, net amount of mass transported in and out of a certain surface area. Any vector quantity is a candidate to form a vector field, and any vector field is a candidate for a flux calculation.

If any vector is considered as a flow, flux means the net flow. I left it as "flow per unit area", but this is only from the perspective of mass transport phenomena; it is more appropriate to consider flux as "flow over the total area", as a compound unit. mA·h is a compound unit used for total electric charge stored in batteries, more commonly referred to as charge capacity since it correlates electric charge and current. Just like it is more appropriate to refer to mA·h as "flow over total battery life" instead of "flow per unit time", flux is the resultant vector over total surface area (whatever that vector represents). That is, in terms of dimensional analysis, the second unit (compounded with the first unit) for flux is area (A) instead of hour (h). 85.110.186.9 (talk) 22:43, 7 November 2013 (UTC)


 * Both terms, Flux as flow rate per unit area, or the Flux as a surface integral, are used for flux.


 * Your edits to the caption of the figure are in good faith but have made it far, far too long for a caption. This discussion is already in the article should be there, not stuffed into a caption. I'm have reverted your edits for now, although by all means feel free to make edits in the article.


 * There isn't that much of a reason to change F to f for "flow". I'm considering later a tweak to the figure to make the tilted area clearer to see, and could change letters then. M&and;Ŝc2ħεИτlk 06:35, 8 November 2013 (UTC)

What is flux?
Flux is flow. In optics, it refers to the flow of radiant energy across a space and has units of energy per unit time. (In hydrology it could be the flow of a fluid in mass per unit time or volume per unit time. In photometry it would be the flow of luminous flux per unit time.) Typically one speaks of a defined beam of radiation flowing in a direction, in which case the radiant flux would be the energy transported in that beam through an imaginary surface of finite and defined area per unit time. The beam's cross-sectional boundary could be circular, elliptical, rectangular, or any other closed curve, even including a shape with a cross-over point, like having a figure of eight shape. The point is that flux is the quantity of whatever is flowing, commonly given the symbol Φ, that passes through a defined surface area per unit time. For descriptions of this quantity using differential and integral calculus, an infinitesimally small element of area, such as da could be specified, in which case the infinitesimally small element of flux passing through da could be designated dΦ.

In the case, for example, of a finite cylindrical beam of radiation centered on the Z axis and flowing in the positive Z direction in an orthogonal Cartesian coordinate system, the edge of such a beam in the plane perpendicular to the Z axis at z=0, i.e. the X-Y plane, would have the shape of a circle centered on the Z axis and having radius R, if the cylinder boundary of the beam is circular in nature. (Some definitions accept use of the word "cylinder" for an elliptical cross-section tube and some even permit the term "cylinder" to apply to an oblique version of such a surface, wherein one end of a right circular or right elliptical cylinder is translated parallel to the other end, thereby distorting the tube, giving it an elliptical cross-section of greater eccentricity when the cross-section is perpendicular to the center line or axis of the oblique cylinder than that of the circular or elliptical ends of the cylinder which remain parallel to the X-Y plane of this example.) The flow of energy in the cylindrical beam would have units of energy per unit time {Joules per second in the International System of Units (SI) called the "metric system."}

The rate of flow of energy within a beam does not have to be constant over any given cross-section of that beam. It could be higher in the center of the beam than at its edges, for example, or have any other lateral distribution spatially. If we define a new term, the spatial density of the flow, to be the flux per unit area passing through a point in a cross-sectional area of the beam, the units of this flow density would be energy per unit time and per unit area. In the SI system, the unit of such density flow would be Joules per second and per square meter. In the field of radiometry, this flux density quantity is called the [ital]irradiance[ital]. Since in the SI system, the Joule per second is also called the Watt, irradiance has units of Watts per square meter. The total radiant flux in a beam is the spatial integral of the irradiance across the cross-sectional area of the beam:

Φ = ∫AE(x,y) da

where A is the cross-sectional area of the beam, da is an infinitesimal element of that area, and E(x,y) is the irradiance of the beam at a point having coordinates (x,y) at z=0 in the beam. The integration process shown in the above equation "adds up" or sums all the flux in all the infinitesemal areas da of the beam, mathematically, giving the total flux in the beam across the area A of the beam's cross-section.

Drmccluney (talk) 16:01, 2 April 2014 (UTC) Ross McCluney, Ph.D. Drmccluney (talk) 16:01, 2 April 2014 (UTC) Research Physicist Drmccluney (talk) 16:01, 2 April 2014 (UTC) Author of Introduction to Radiometry and Photometry, Artech House, Boston, 1994

Magnetic Flux
I tried making an edit yesterday to the definition of magnetic flux, which currently involves a closed double integral, to use an open double integral instead, because Gauss's law for magnetism says the net magnetic flux through a closed surface is zero. Why is that misleading notation still present? — Preceding unsigned comment added by 18.96.7.167 (talk) 14:31, 31 May 2014 (UTC)


 * I reverted your edit because the template you invoked Iint does not exist and you left the page broken.  Spinning Spark  19:30, 31 May 2014 (UTC)