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Gauss's law in differential forms 
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#1
Jun2407, 03:55 AM

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Hi,
I'm seeing that many authors like Griffiths and Halliday/Resnick (I've not seen Jackson and Landau/Lif****z) are deriving the differential form of Gauss's law from the integral form (which is easily proven) by using the divergence theorem to convert both sides to volume integrals and then claiming that the integrands must be equal as the integrals are equal over all volumes. But this argument is flawed. I can change the value of any one integrand over any set of measure zero (such as a countable number of planes) without disturbing the equality of the integrals. The integrands will no more be equal, but the integrals will still be equal, over all volumes. The derivation by directly calculating the divergence from Coulomb's law also seems dubious. It hinges on writing the derivative of a function that is clearly not continuous, let alone smooth, using Dirac functions (which are of course not functions at all). How can the differential form be derived rigorously from Coulomb's law/integral form? Molu 


#2
Jun2407, 04:10 AM

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#3
Jun2407, 04:44 AM

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Secondly, I said why the argument was flawed in my previous post. Equality of ndimensional integrals, even over all possible nsurfaces, does not rule out pointwise inequality over a set of measure zero (n1surfaces, for example). Molu 


#4
Jun2407, 05:27 AM

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PF Gold
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Gauss's law in differential forms



#5
Jun2507, 07:18 AM

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Two CONTINUOUS functions that are equal almost everywhere are in fact equal everywhere.



#6
Jun2507, 08:28 AM

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Thanks. Molu 


#7
Jun2507, 08:35 AM

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A discontinuity in the electric field can only come from a singularity in the charge distribution. In your perfect conductor example there is a Dirac delta function like sheet of surface charge. In reality its not singular, just very concentrated, but it's a useful approximation.



#8
Jun2507, 01:55 PM

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Molu 


#9
Jun2507, 02:14 PM

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If you insist on complete rigor, the way to deal with a delta function is to represent it as a limit of continuous functions (representatives of delta functions). So a delta function can be represented arbitrarily well by a continuous function. As I said, a delta function is a nonphysical (but useful) idealization. Indeed when Heaviside invented the operational calculus it was attacked as being nonrigorous. But they can be dealt with rigorously as distributions.



#10
Jun2507, 07:42 PM

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You mean, of course, as a limit of the operators those continuous functions represent.



#11
Jun2507, 10:06 PM

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#12
Jun2807, 03:35 AM

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So, can you show/point out to me a mathematically valid derivation of the differential form of the Gauss' law? While I like physics far more than mathematics, I also like the mathematics in my physics to be real mathematics, instead of the watereddown handwaving version used by most physicists. Thanks!
Molu 


#13
Jun3007, 03:40 PM

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Anyone to help?



#14
Jun3007, 10:15 PM

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Before one can rigorously prove Gauss's law, one must precisely state Gauss's law. The proof you reject is a perfectly good proof of one statement of (the differential form of) Gauss's law. Explicitly state what you want to see proved, if that was not it.



#15
Jun3007, 11:15 PM

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At this point you don't seem to have any problem with continuous charge distributions. To extend to delta function type charge distributions just do what everyone else does and consider them as limits of continuous distributions.



#16
Jul107, 01:15 PM

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Molu 


#17
Jul107, 01:33 PM

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Some discussions only consider smooth vector fields, and the divergence operator you learned in your elementary calculus classes. That the integrands are equal is a easy consequence of continuity. Other contexts consider more general classes of vector fields, and even generalizations of that notion! They also use generalizations of the divergence operator. Without knowing what type of objects you are using for your fields, and what definition of divergence you are using, it is essentially impossible to provide a proof that would satisfy you! 


#18
Jul207, 03:25 AM

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http://hyperphysics.phyastr.gsu.edu...ic/gaulaw.html 


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