Talk:Continuity equation/Archive 1

Archive 1

Untitled

The continuity equation applies to more than just mass and charge. It applies to any conserved property. This is the form of the conservation law derived from Noether's theorem. IMO The article should be changed somewhat to reflect this. 63.205.41.128 03:23, 31 Jan 2004 (UTC)

${\frac {\partial \varphi }{\partial t}}+\nabla \cdot f=s$

should use the substantial derivative:${\frac {D\varphi }{Dt}}+\nabla \cdot f=s$

No! It should be the partial derivative, as on the Navier Stokes page and as in the derivation of the Maxwell case. I have changed this back. Paul Matthews (talk) 17:52, 1 February 2008 (UTC)

INTEGRAL'S FORM

May your describe the integral's form?*

Derivation

When the article talks about Ampère's law as one of Maxwell's equations, it should be noted that the formula given given includes Maxwell's correction. —Preceding unsigned comment added by Iain marcuson (talk • contribs) 19:31, 21 October 2007 (UTC)

Yeah I thought that Maxwell's correction was introduced into Ampere's Law specifically to satisfy the continuity equation. Which would mean that the continuity equation cannot be derived from Maxwell's equations at all, only it is consistent with them.

I changed "Ampere's law" to the more specific "Ampere's law with Maxwell correction". The term "Ampere's law" is vague, it can refer to either with or without the correction. Ampere's law with Maxwell correction is one of the Maxwell's equations, and the continuity equation for charge can be derived from Maxwell's equations, as proven in the article. :-) --Steve (talk) 21:23, 12 November 2010 (UTC)

q and A

The revisions to the main statement of the equation don't make sense, or at least I don't understand them.

• "q = quantity associated with φ" -- what does that mean? For example, if φ is mass density, then q is...what??
• "$A,\mathbf {\hat {n}} \,\!$" -- A is an area and n is its unit normal. But area of what? I guess a surface? Why is there a surface? Who put a surface there? Is it a real surface or an imaginary surface? Is it a closed surface or an open surface? Is it flat or curved?
• $\varphi ={\frac {\partial q}{\partial V}}\,\!$ -- There seems to be a very basic misunderstanding of calculus here. I guess V is the volume of some imaginary blob in space. When you increase the size of the blob, q increases or decreases, I guess. You can calculate the differential rate of change of q and of V, and divide one by the other, a derivative. But there are many ways of increasing the size of the blob: You can pull out a lobe, you can expand it out from one center or another. All will generally lead to different relative rates of change of q and V. Therefore the derivative ${\frac {\partial q}{\partial V}}\,\!$ does not seem to be mathematically meaningful.
• $\mathbf {f} =\mathbf {\hat {n}} {\frac {\partial ^{2}q}{\partial A\partial t}}\,\!$ -- according to this equation, the flux points in the direction $\mathbf {\hat {n}}$. In other words, the surface is always exactly normal to the local flux. I was assuming until now that this was an arbitrary imaginary surface that I can define however I want, but I guess not. It is a very specific surface that everywhere points normal to the flux. Hold on though, this surface may not be closed, so it may not have a volume V. What happens in this case? Then there is the other problem of what it means to differentiate with respect to A. Again, I don't think it can be mathematically defined.

Anyway, my confusion piles up. My best guess is, this is an attempt to state the integral form of the continuity equation, and not just the differential form. Is that right? That's a fine and reasonable thing to do...provided, of course, that it is done correctly and clearly! :-) I'm happy to do so myself. But as is, I suggest restoring the definition to the previous version. Thanks in advance! Looking forward to understanding better what's going on here. :-) --Steve (talk) 18:12, 13 July 2011 (UTC)

UPDATE: I went ahead and rewrote the section as I suggested above. I hope it is both clearer and more mathematically correct now, but I'm happy to hear what other people think :-) --Steve (talk) 04:37, 14 July 2011 (UTC)

Why table?

I propose deleting the summary table, it just restates what is in the previous three sections, but minus almost all the useful explanation and discussion of what the variables mean and when the equation can be used. I think the previous three sections are already enough. It's basically having a good explanation of an equation in one part of the article, and a mediocre explanation of the same equation in another part. It's better to just have the good explanation, so that readers spend their time looking at the good explanation instead of the bad one. What do other people think of deleting the summary table? --Steve (talk) 04:00, 14 July 2011 (UTC)

It makes comparison easier, so I see no reason to delete it.--Patrick (talk) 10:23, 14 July 2011 (UTC)

Table deleted, problematic explaination of q and A

The table has been deleted, I created it, and was responsible for the confusion stated earlier about q and A. As said, it is a mediocre repetition, as it ended up. The intension of creating the table was to summarize the equations to be used in conjunction with the other physics formulae articles; defining equation (physics), list of elementary physics formulae, and constitutive equation.

The intension of introducing the other variables q and A were to illustrate the common theme between mass m, charge Q, energy E and probability P continuity equations.

- The general quantity q ("assoicated with φ") can be any of m, Q, E, or P.
- φ would be the volume density of each; respectivley they are ρ (mass density), ρ (charge density), u (energy density), ρ (probability density function).
- The general flux vector f would be current densities of each, respectively they are j_m (mass current density), J (charge current density), and heat flux density q and probability density j.

There wasn't a misunderstanding of calculus though - I simply didn't make it clear. The volume V is not for some arbitary blob in space, real or imaginary where the transport occurs, its the volume occupied by the conserved quantity, so that a density can be defined, here I used φ. Think about how charge density or mass density etc is calculated. The volume density throughout the quantity (not charge) q can vary through space (possibly time) so cannot simply divide by volume, at a point the density is the infinitesimal amount of property dq per unit infinitesimal volume element dV, so:

$\varphi ={\frac {\mathrm {d} q}{\mathrm {d} V}}.\,\!$

How else can a density function be defined for a continuum? This is consistent since integrating (w.r.t. V) would obtain the total amount of property.

$\int \varphi \mathrm {d} V=\int {\frac {\mathrm {d} q}{\mathrm {d} V}}\mathrm {d} V\,\!$
$q=\int \varphi \mathrm {d} V\,\!$

The area A is an arbitary surface - real or imaginary, through which the quantity flows through, to obtain the current density or flux density f etc; whatever its called, it the rate of transfer of q (AKA current) which is dq/dt, per unit infinitesimal area dA. It doesn't matter if the surface is open or closed, its a consistent definition since integration (w.r.t. A) gives the total current dq/dt across the surface.

The reason for using partial derivatives is because in the case that q varies with respect to other variables for any reason at all, partial derivatives automatically narrow down the calculation to differentaiting with respect to volume, or time then cross-sectional area. There is no loss of continuity in using partial derivatives instead of ordinary derivatives since the partial differentiation w.r.t. required variable/s is exactly the same as for ordinary.

It may as well be forgotten. This isn't a request for writing all that again, if it was misleading and unneccersary, only at least an explaination has now been provided.

Maschen (talk) 23:55, 17 July 2011 (UTC)

Hello, that helps, I think I understand better what you were going for.

• I would have no big problems if you had written

"$\varphi =dq/dV$, where $dq$ is the infinitesimal amount of q in the infinitesimal volume dV, and $\varphi$ is the volume density of q"

That's what you were going for, but the problems are #1: A differential/infinitesimal quantity is never written $\partial x$, only $dx$. This is a universal convention. You could say "In a short time dt, the ball rolls a distance dx=v dt...", but you will never see anyone say "In a short time $\partial t$, the ball rolls a distance $\partial x=v\partial t$...". #2: You want dq/dV to be the ratio of two infinitesimal quantities, but anyone looking at it will think (wrongly) that it's a plain old derivative. This is exacerbated, of course, by the lack of explanation (e.g. a reader will not understand what "q = quantity associated with φ" means!) and by the use of ∂ instead of d that I mentioned above.

• You just said "A is an arbitrary surface", but again, you wrote the equation $\mathbf {f} =\mathbf {\hat {n}} {\frac {\partial ^{2}q}{\partial A\partial t}}\,\!$ which requires that the flux be normal to A.
• Based on your explanation, the $\partial q/\partial t$ in the flux equation is, once again, not a derivative, but a ratio of two infinitesimal quantities. You need to say instead:

"$dq/dt$, where dq is an infinitesimal amount of q that passes through the surface in the time dt"

I hope you understand the difference between a derivative and a ratio of two infinitesimal quantities. The latter is much more general. The former usually requires that the numerator is a dependent variable and the denominator is the associated independent variable with the obvious definition and relation. If you ask someone "Is q a function of t?", they'll say yes, but what they mean is the obvious "q (associated with a given fixed volume) is a function of t (the time coordinate)". They certainly would never think of the weird definition "q (meaning the amount of q transferred through a certain surface) is a function of t (the time duration over which you integrate the flow)".

A more general note: I object in the strongest possible way to writing equations without properly explaining them. For example, you wrote "V=volume". Volume of what?? If you've ever taught physics you know exactly what happens when you teach people an equation out of context. For example, let's say you explain an equation like this:

"E=mc^2, where E=energy and m=mass and c=the speed of light".

Now here's a question: A car has mass M and velocity V. What is its kinetic energy K?" Oh it's easy, K is an energy, so K=E. And M is a mass, so M=m. So the answer is: K=Mc^2. Oops, wrong answer. If you've ever taught anyone physics, the #1 most common mistake people make by far is plugging a quantity into a symbol that means something else. So when I read what you wrote, "V=volume", I was literally cringing. We physics teachers try so hard to stop students from plugging quantities into symbols that mean something else! But here, they can't understand what they can legally plug into V (volume of what??) even if they want to!!  :-(

By the same token, I despise the lists of equations in those three articles you linked to at the top. I can barely even look at them. Grrr. --Steve (talk) 11:53, 18 July 2011 (UTC)

Integral form - Σ and q typo

Hi again Sbyrnes321, for the integral form, you wrote the integrals for Σ and q as;

$q\equiv \iiint _{V}\varphi ,\quad \Sigma \equiv \iiint _{V}s$

but shouldn't they be;

$q\equiv \iiint _{V}\varphi {\rm {d}}V,\quad \Sigma \equiv \iiint _{V}s{\rm {d}}V$

or (in cartesian coords);

$q\equiv \iiint _{V}\varphi {\rm {d}}z{\rm {d}}y{\rm {d}}x,\quad \Sigma \equiv \iiint _{V}s{\rm {d}}z{\rm {d}}y{\rm {d}}x$

(etc for whatever coord system used) if they are triple integrals ?

Just checking, they seem incomplete, you have to integrate w.r.t. something... I know at that time I never completley clarified the definitions either - so I have no room to talk.

Thanks for your many positive corrections - Maschen (talk) 21:06, 3 September 2011 (UTC)

P.S. hope you don't mind the blue box around your integral equation, to parallel the highlight with the diff form.

OK. No I don't mind the blue boxes :-) --Steve (talk) 21:48, 3 September 2011 (UTC)

QM applications

I cut and pasted the derivation for the probability continuity equation from probability current - it should really be here than there. Superficially there seems to be too much emphasis on QM, but i'll extend the scope of the article in other areas soon...

F=q(E+v^B) (talk) 22:00, 18 November 2011 (UTC)

Article is crazy

ALL: Sorry...

• Apologies for subsequently making the article blind the viewer with maths - I’ll try simplifying it throughout at some point. It was better before I modified the example sections.
• If it were possible to render a closed triple integral in LaTeX then I would have done that (double can be just about done $\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset$, already in some articles including this one). But at a first glance this $\oint _{V}$ appears to be a contour integral along a curve - or something (even with the V = volume of region indication). Its non standard to use this symbol in this way for a closed volume integral, and probably less familiar, so it'll be changed to $\iiint _{V}$. As long as it is known that the volume is closed its fine. All surface integrals are fine as they are.
• Also I’ll rid the article of the hidden-code derivations in the example sections - there is no need since the derivation is identical to the general equation (the whole point of including that was to remove the need of deriving every continuity equation again) Only derivations from other laws (Maxwell's and Schrodinger's equations) should be be there.
• The article could do with the more elementary treatment of the continuity equation in the form (+ mentioning equivalents):

$\rho _{1}\mathbf {u} _{1}\cdot \mathbf {S} _{1}=\rho _{2}\mathbf {u} _{2}\cdot \mathbf {S} _{2}$

since a reader will understand this far better than the general PDE (which should come after) but i'll leave that for now.

By the way - this link may be helpful. I created images of the integrals which can't be rendered in LaTeX while editing this article.

--F=q(E+v^B) (talk) 14:43, 12 December 2011 (UTC)

There is no reason to ever have a circle in a 3D volume integral sign. The circle means "without a boundary". For example, a curve that makes a loop with no beginning or end, or a surface that wraps into a sphere. There is no 3D analog to this except for 3D surfaces that curve around within 4-dimensional space. (The word "closed" is used sometimes, but "closed" means several different things, I think you're recalling an irrelevant definition.) In other words, if the circle should be used for some 3D integrals but not others, then what is the criterion?? I think it should never be used. (Except for 3D manifolds in 4-dimensional space, i.e. not in elementary physics.) --Steve (talk) 15:11, 12 December 2011 (UTC)

Hmm, I would say that the derivation of continuity eqn from Maxwell's eqns and the derivation of continuity eqn from Schrodinger eqn are two good candidates for putting in show/hide boxes, in that they're relevant but technical and unnecessary to understand other aspects of the main topic. (For those who don't know what a show/hide box is, see for example Faraday's law of induction#Proof of Faraday's law.) They would also be OK candidates for deleting altogether. --Steve (talk) 16:57, 12 December 2011 (UTC)

I see what you mean by the circle and triple integral - that was the whole point of reverting yesterday since its nonsense.

But I certainly don't agree with "deleting altogether" the derivations from Maxwell's and Schrodinger's equations - those are worth keeping since they don't follow the general derivation, and show that other laws of physics also lead to continuity equations in their own form. (Btw I know how to use show/hide boxes, but good for you to point out for others who actually don't). I'll leave it for anyone to decide weather or not they place the said derivations in show/hide boxes - if you want then go for it.

-- F = q(E + v × B) 22:11, 13 December 2011 (UTC)

On second thought there are couple of obvious cases where the triple integral + circle could occur in this article - you know the answer already of course. Not in elementary physics, are 4d integral conservation laws from special/general relativity, for conservation of 4-momentum and 4-currents (4-currents are already included). For the 4-momentum, according to Gravitation Misner, Thorne, Wheeler (admittedly taken almost letter for letter),

• $P^{\mu }=\oint _{V}T^{\mu \nu }{\rm {d}}^{3}\Sigma _{\nu }=0$ (components),
• $\mathbf {P} =\oint _{V}\mathbf {T} \cdot {\rm {d}}^{3}{\boldsymbol {\Sigma }}=0$ (4-vectors, coord. independent),

where:

• V = surface 3-volume of the 4-space region,
• T = stress-energy tensor,
• d³Σ is a surface 3-volume element.

Also linked here. Its tempting to add this to the article but I shouldn't and will not: I don't understand tensor calculus or differential topology (yet).-- F = q(E + v × B) 16:14, 14 December 2011 (UTC)

Next steps are:

• Eliminate useless subsections "derivations from 1st principles" (but preserve refs as further reading)
• Make derivations shorter, less elaborate, more concise

-- F = q(E + v × B) 08:32, 14 December 2011 (UTC)

• Missed out one part of int form - changed $\iiint _{V}{\frac {\partial \varphi }{\partial t}}{\rm {d}}V\rightarrow {\frac {{\rm {d}}q}{{\rm {d}}t}}$
• Add show/hide tabs for the derivations
• Removed useless section on thermodynamics - doesn't lead anywhere. Its such a short section that if this is to be extended it can be done from scratch anyway.

-- F = q(E + v × B) 10:51, 15 December 2011 (UTC)

You have done it again havn't you? Traced you to here by following your'e edit history. LEAVE the fucking Navboxes open so they can be seeeeen. How is a reader supposed to know whats in them?--Maschen (talk) 17:38, 15 December 2011 (UTC)

It is often reasonable to have a show/hide box that defaults to hide. You want this in a circumstance that (1) If a reader never opens up the box it doesn't significantly detract from their conceptual understanding of the topic, (2) The section-heading and text makes it clear what's inside the box and how to open it, (3) When the box is open there is such a flood of equations and details that it would distract and scare away readers, and take up space in the article far out of proportion to the box's importance in the bigger picture. In this case I would prefer closed by default, but open is OK. My ideal preference would be deletion from this article....

For example, understanding how to prove the QM probability continuity equation starting from the Schrodinger Equation doesn't help you conceptually understand the QM probability continuity equation, let alone continuity equations in general. It slightly helps you understand Schrodinger Equation. Mainly it reassures you that Schrodinger Equation is consistent with your other expectations about QM. This proof is worth putting in a textbook, or I guess in a very specific article Continuity equation (quantum mechanics), but not this one. Likewise the EM one: The concept that charge is conserved is more intuitive, and even more fundamental, than Maxwell's equations. To "prove" charge conservation from Maxwell's equations adds nothing to understanding, it's just part of the enterprise of double-checking that different EM equations are all consistent with each other. This is a worthwhile enterprise as part of a course or textbook, but not too important for understanding. You can completely understand each of two different equations without ever seeing the proof that the two equations are mathematically consistent with each other.

I don't feel that strongly either way. I'm not planning to delete those sections or even default-hide them. I'm just adding my own opinion. :-) --Steve (talk) 18:58, 15 December 2011 (UTC)

I also said its better to leave them closed and did leave them that way, for the very reason's you point out. Maschen seems to charge into articles and change them, then usually regrets it afterwards becuase of the problems/issues which arise, he/she is the one who has done it again (not that it matters in this case though).....

Not so sure about possibilities of very specific articles such as a continuity equation article for each part of physics (e.g. QM), they would probably become stubs which isn't very good. The point I made about how Maxwell + Scrho eqns lead to the continuity equations was statement of consistency, not to prove anything because they are just derivations. I would say the current article is fine as it is. Any additions/subtractions are essentially optional.

-- F = q(E + v × B) 12:30, 16 December 2011 (UTC)

On second thought, perhaps the article is not so complete as it could be. Right now it has all these derivations yet lacks to really explain what the continuity equation means. The new section I proposed earlier will be added now, with additional explaination. In light of this, I think we have the order backwards, people normally learn the seperate forms of the continuity equation for mass and charge,

$\rho _{1}\mathbf {v} _{1}\cdot \mathbf {A} _{1}=\rho _{2}\mathbf {v} _{2}\cdot \mathbf {A} _{2}$

and in doing so the general structure of this version of the equation, then differential and integral forms. While at it the boxes may as well be defualt hidden again, irrelavant of Maschen's crazy opionions... -- F = q(E + v × B) 00:13, 18 December 2011 (UTC)

\oiint

I plan to implement the new template for the double integral sign \oiint into the integral form of the continuity equation;

${\frac {{\rm {d}}q}{{\rm {d}}t}}+$ $\scriptstyle S$ $\mathbf {f} \cdot {\rm {d}}\mathbf {S} =\Sigma$

and in the blue box

${\frac {{\rm {d}}q}{{\rm {d}}t}}+$ $\scriptstyle S$ $\mathbf {f} \cdot {\rm {d}}\mathbf {S} =\Sigma$

not because I'm the initial author of the \oiint template and implementing them for sake of vanity, but more importantly other editors have worked extremely hard to render them the quality templates they are now. Any objections? =| -- F = q(E + v × B) 00:21, 30 January 2012 (UTC)

Looks splendid! I'll just add it.--Maschen (talk) 10:31, 1 February 2012 (UTC)

Thanks very much to you and the others for producing that. I changed "derivation from" to "consistency with" in the NavFrames, more accurate title. --Maschen (talk) 10:56, 1 February 2012 (UTC)

Just to clarify the hidden note...

It was not intended to be hostile - I didn't say people can't change it... Never mind... Maschen (talk) 15:52, 2 September 2012 (UTC)

Extend scope?

I plan to make the following additions/changes:

• It says $\mathbf {J} =(J^{\alpha })=(c\rho ,\mathbf {j} )$ can be any 4-current, not just for EM, so make the formulation using the 4-current: $\partial _{\alpha }J^{\alpha }=\sigma$, (same σ = generation rate per unit volume in the current article), a level-2 section in its own right.

This may be expanded to include differential forms later (i.e. mention the general 4-current in the language of diff forms, possibly also that Maxwell's equations automatically lead to ${\rm {d}}\mathbf {J} =0$ though not essential).

• Include a brief (level-2) section on general relativity and the stress-energy tensor $T^{\alpha \beta }$ since:

1. The 4-divergence of it is a continuity equation itself: ${\frac {\partial T^{\alpha \beta }}{\partial x^{\beta }}}=0$.
2. The equation is the divergence of a tensor field to yield a vector field (energy-momentum, hence energy-momentum is conserved), while the form in this article is the divergence of a vector field to obtain a scalar field (zero field for conservation obviously) - which breaks out of the dominating theme of this article.
3. It is a crucial constraint on the form of the Einstein field equations: $G^{\alpha \beta }=8\pi T^{\alpha \beta }$, this is easily sourced.

As we know this was mentioned above (in integral form).

• Add another (level-2) section which generalizes further the form of the RHS (i.e. what the divergence $\partial _{\alpha }J^{\alpha }=-\cdots$ actually can be for non-conserved currents) in the following way:

RHS = radial fields R (gravity/EM fields which act on the mass/charge elements forming the fluid)

+ surface effects Σ (shear stress, chemical/thermal conduction/diffusion through the surface of the region in consideration)

+ volume effects D (turbulence, viscosity, diffusive currents in the volume of the region)

+ generation rate G (production/removal of turbulent mass/energy, entropy, momentum)

where R, Σ, D, G can be tensors of any required order (including scalars), so in integral form the continuity equation would be:

$\int _{V}\left({\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} \right)dV=\int _{V}\left(R+D+G\right)+\int _{A}\Sigma dA$

as sourced from

J.R. Tyldesley, (1975). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. p. 35. ISBN 0-582-44355-5.{{cite book}}: CS1 maint: extra punctuation (link)

Here I used different letters from the book and article to prevent mutual conflicts, e.x. like changing S for vector area to A, use G for generation, Σ for surface effects etc. ... we may need to rename more letters later including f to j...).

• I know we have been through all this before... anyway...

1. blend all the specialized sections (EM, fluid mech, thermodynamics, QM) into one level-2 section. Right now, each section isn't much of a section with just a list of quantities and the equation all in the identical general form. IMO they can, and should, be summarized by that outcast table I added last year (cut and pasted from Laws of science to here), with
2. the paragraphs of text before the table, and
3. boxes of derivations after the table.

Forget this... not only does the attempt fail - some links would become broken (in particular to the QM section).

• Not include the convection–diffusion equation, Boltzmann transport equation, and Navier-Stokes equations, which are linked to in the lead.

Yes - I'm sorry being a nuisance to this article yet again, but in this way - it will have more scope. Favour or oppose? Maschen (talk) 00:07, 7 September 2012 (UTC)

It's easy to wrongly assume that $\mathbf {J}$ is a regular vector, not a 4-vector. Perhaps you should use a different font. For example, ${\mathfrak {J}}$ ,${\mathcal {J}}$ or $\mathbb {J}$ .--老陳 (talk) 02:35, 7 September 2012 (UTC)

Usually, bold letters are used for 4-vectors (here J = 4-current) while lower case bold for 3-vectors (here j = 3-current).

This is an easy but important issue to consider; to use \mathsf ${\mathsf {J}},{\boldsymbol {\mathsf {J}}}$, \mathcal ${\mathcal {J}},{\boldsymbol {\mathcal {J}}}$ etc. as you say for more general quantities (but preferably not \mathbbf $\mathbb {J} ,{\boldsymbol {\mathbb {J} }}$ which tends to be reserved for sets and definitely not \mathfrak ${\mathfrak {J}},{\boldsymbol {\mathfrak {J}}}$ !!... IMO looks very horrible and is difficult to reproduce by hand). My first choice would be \mathsf but you (and anyone) are welcome to disagree!

Thank you for raising the point. If there is favour to this change we can decide on all the new notation here along these lines before actually making the change. Maschen (talk) 08:30, 7 September 2012 (UTC)

\mathsf looks quite like \mathrm. It would be nice if we can use \mathcal, but the font can not handle lower case letters. By the way, where did you find the convention that bold letters are used for 4-vectors? I have not found such a convention used in many popular textbooks that I have read. For example, in Griffiths' "Introduction to Electrodynamics", the 4-vector potential is expressed as follows:

$A^{\mu }=(V/c,A_{x},A_{y},A_{z})$

--LaoChen (talk)12:56, 17 September 2012 (UTC)

Fonts: ..... \mathsf and \mathrm look completley different to me... caligraphic may be better to label objects like regions of space or flows or so on...

Letter-case convention: Ok the convention is not that common but at the top of my head e.x. J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley. ISBN 978 0 470 01460 8. uses it, same with our article on four-vectors. As long as it's clear from the context it doesn't matter what letter-case is used.

You may have noticed the changes are essentially made now anyway, and the notation is self-consistent (or at least clear from context), with the exception of the non-conserved currents section above but that can wait till more refs are found for sufficient material. Maschen (talk) 15:13, 17 September 2012 (UTC)

I'm drafting things in a sandbox for now. It will not be added of course unless there is clear consensus for "yes" (or if no one opposes the changes within the next month or so). Some things may differ to the original plan (such as the table). Maschen (talk) 11:45, 7 September 2012 (UTC)

Conserved Charge

Concerning the last line, it is my understanding that to find the conserved charge from the conserved current the integrand should read ${\frac {J^{0}}{c}}$ and not $J^{\mu }$. As it is written now, the indices on the left and right side of the equation do not line up, which is nonsensical. For the life of me I cannot find a source for this, of course I do not own a dedicated QFT book. 174.78.149.150 (talk) 16:43, 29 March 2013 (UTC)

Yes - J⁰/c is the density. Typo well-spotted, it's fixed now. M∧Ŝc²ħεИ_τlk 03:59, 30 March 2013 (UTC)