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In Purcell's, Electricity and Magnetism, 61, is stated that the following limit : $$\lim_{V_{i} \to 0}\frac{\int_{S_{i}} \overrightarrow{F}.\overrightarrow{ds}}{V_{i}}$$

[ Where $\overrightarrow{F}$ is a vector field, $S_{i}$ a small closed surface containing the point $i$ and $V_{i}$ the volume inside $S_{i}\,$] , which is the definition of $\,\mathrm{div}F\,$, is independent of the shape of $S_{i}$. Why is it this true? Why do these two shapes:

enter image description here

have the same limit defined above?

P.S. the second shape is essentially the first one completed

ruohola
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Hilbert
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1 Answers1

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The numerator in the limit may be rewritten as $\iint_{\partial S_i}\mathbf F\cdot\mathbf n\,dS_i$, which by the divergence theorem is equivalent to $\iiint_{V_i}(\nabla\cdot\mathbf F)\,dV_i$. Thus the limit gives the mean value of $\nabla\cdot\mathbf F$ in $V_i$, and the limit as $V_i\to0$ is just $(\nabla\cdot\mathbf F)(i)$, which is independent of $S_i$.

Parcly Taxel
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  • There is a problem with the proof, it uses a result of the definition of divergence, and this result is what we're trying to proove. We cannot proove the limit above is independent of $S_{i}$ by using the known result that the divergence is independent of $S_{i}$ because the divergence IS defined as the limit above. – Hilbert Aug 07 '19 at 09:27
  • @Hilbert As I was taught in Multivariable Calculus, divergence is not defined as that limit, but rather as \nabla\cdot\mathbf F$ using coordinates. Thus I can use the divergence theorem. Please reaccept my answer. – Parcly Taxel Aug 07 '19 at 09:41
  • $\mathrm{div} F = \nabla . F$ is nothing but a consequence of the definition above. – Hilbert Aug 07 '19 at 10:25
  • @Hilbert No, it's the other way around. – Parcly Taxel Aug 07 '19 at 10:26
  • https://math.stackexchange.com/questions/300024/formal-definition-of-the-divergence-of-a-vector-field – Hilbert Aug 07 '19 at 10:30
  • @Parcly Taxel, You are mistaken, this is really the definition of divergence. ${\rm div}{\bf F} =\nabla\cdot{\bf F}$ is the consequence. – Stefan Octavian Jan 19 '21 at 06:34
  • @Stefan Then how do I fix it? – Parcly Taxel Jan 19 '21 at 06:36
  • Give a useful answer. This one is useless – Stefan Octavian Jan 19 '21 at 06:37
  • @StefanOctavian Please suggest how I can make it useful. – Parcly Taxel Jan 19 '21 at 06:37
  • Well, I've been searching for an answer to this question for days. I though I might finally find the answer here but it seems that's not the case. I will continue to try to write a proof and when I make it I'll post an aswer to this question myself – Stefan Octavian Jan 19 '21 at 06:39
  • @StefanOctavian If you are writing an answer, why don't you remove the downvote? And also your downvote on the other answer you downvoted. I have edited my answer here. – Parcly Taxel Jan 19 '21 at 07:10
  • Your new answer only reflects an idea, it's not rigurous. Since it's a good enough intuitive explanation, I will remove my downvote, but this really is a limit of nets actually, and that is an important detail to be noted. Moreover, your argument doesn't apply to this limit since it's not a limit of the form $p\to p_0$ with $p, p_0$ points in space – Stefan Octavian Jan 19 '21 at 10:21
  • Btw, not even this is good enough since it doesn't explain why this limit exists in the first place (which is the question). – Stefan Octavian Jan 19 '21 at 10:26
  • Parcly Taxel, your other answer was quite ok, it seems. One can define intuitively divergence as the flux density, then use the case with rectangles to find $\nabla\cdot{\bf F}$, then actually define divergence as it, then prove divergence formula and Green's Theorem using this definition, then prove the intuitional definition must be valid (even though trying to prove that from the start is hard as hell). I gotta admit this way is easier and perhaps safer for a good understanding of the subject, even though I live for the day I will see a proof without, that would be truly sexy. – Stefan Octavian Jan 25 '21 at 13:19
  • That being said, I present my sincere apologies and I think a rollback on the edit would be the better option for the good of this Q&A community. – Stefan Octavian Jan 25 '21 at 13:22
  • @StefanOctavian I've rolled it back. – Parcly Taxel Jan 25 '21 at 13:25