![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
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)
should use the substantial derivative:
May your describe the integral's form?*
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.
The revisions to the main statement of the equation don't make sense, or at least I don't understand them.
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)
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)
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 jm (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:
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.
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)
Hi again Sbyrnes321, for the integral form, you wrote the integrals for Σ and q as;
but shouldn't they be;
or (in cartesian coords);
(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.
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)
ALL: Sorry...
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)
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)
Next steps are:
-- F = q( E + v × B) 08:32, 14 December 2011 (UTC)
-- 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)
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)
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)
I plan to implement the new template for the double integral sign \oiint into the integral form of the continuity equation;
and in the blue box
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)
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)
I plan to make the following additions/changes:
{{
cite book}}
: CS1 maint: extra punctuation (
link)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)
\mathsf
, \mathcal
etc. as you say for more general quantities (but preferably not \mathbbf
which tends to be reserved for
sets and definitely not \mathfrak
!!... 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!\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:
\mathsf
and \mathrm
look completley different to me... caligraphic may be better to label objects like regions of space or flows or so on...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)
Concerning the last line, it is my understanding that to find the conserved charge from the conserved current the integrand should read and not . 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)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
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)
should use the substantial derivative:
May your describe the integral's form?*
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.
The revisions to the main statement of the equation don't make sense, or at least I don't understand them.
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)
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)
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 jm (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:
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.
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)
Hi again Sbyrnes321, for the integral form, you wrote the integrals for Σ and q as;
but shouldn't they be;
or (in cartesian coords);
(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.
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)
ALL: Sorry...
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)
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)
Next steps are:
-- F = q( E + v × B) 08:32, 14 December 2011 (UTC)
-- 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)
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)
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)
I plan to implement the new template for the double integral sign \oiint into the integral form of the continuity equation;
and in the blue box
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)
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)
I plan to make the following additions/changes:
{{
cite book}}
: CS1 maint: extra punctuation (
link)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)
\mathsf
, \mathcal
etc. as you say for more general quantities (but preferably not \mathbbf
which tends to be reserved for
sets and definitely not \mathfrak
!!... 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!\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:
\mathsf
and \mathrm
look completley different to me... caligraphic may be better to label objects like regions of space or flows or so on...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)
Concerning the last line, it is my understanding that to find the conserved charge from the conserved current the integrand should read and not . 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)