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The proof of the Gauss's law for gravity provided by Wikipedia takes use of the divergence theorem.

  • Is it possible to arrive at the integral form of the Gauss's law in a way which doesn't require the use of divergence?

I'd like to derive it from the Newton's law. The main idea is: Assume that two points masses $m_1,m_2$ act on each other with force $F=\frac{Gm_1m_2}{r^2}$. Do something. Arrive at the integral form of the Gauss's law.

So it doesn't have much in common with the "duplicates"

I had one semester of analysis, so don't know anything about the Delta Dirac function and family

marmistrz
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  • I am not sure I follow. Gauss' Law is a general mathematical property of fields that has nothing to do with physics. Which of Newton's laws do you want to use to derive Gauss? The three laws of motion don't say anything about fields and his law of universal gravitation is an ad-hoc force law, which, strictly speaking, also doesn't require us to think of fields. – CuriousOne Jan 06 '16 at 10:18
  • I mean the law of universal gravitation. About the divergence: I simply want to avoid the mathematical concept of divergences, since I'm not familiar with them – marmistrz Jan 06 '16 at 10:36
  • Newton's law of universal gravitation is an ad-hoc force law. It doesn't say anything about a field. You can, if you like, define the gravitational potential as a field and then check that it is a special case of Gauss' Law, but you can't derive a general law from a special case. We are talking about this, right? https://en.wikipedia.org/wiki/Divergence_theorem – CuriousOne Jan 06 '16 at 10:40
  • Yes, but we define the field magnitude as $E = F/m = \frac{GM}{r^2}$, where the $m$ is the mass acted on. And the Gauss law combines the mass $M$ with the field magnitude $E$ (so force too). Yes, I'd like to avoid the divergence theorem linked – marmistrz Jan 06 '16 at 10:43
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    Duplicate of http://physics.stackexchange.com/q/73028/5739 which itself is flagged as a duplicate. – garyp Jan 06 '16 at 14:20
  • @CuriousOne Gauss's Law can be inferred from Coulomb's Law. See the question marked as a duplicate. (Or Jackson) – garyp Jan 06 '16 at 14:21
  • Ah! the OP wants to avoid the divergence theorem. I missed that. Sorry. – garyp Jan 06 '16 at 14:24
  • @garyp: Yes, indeed, but at the very least Coulomb is already a field interpretation, whereas Newton, in my mind, is not. Maybe I am wrong about that... in terms of science history. I am willing to relent on that notion if someone can give me a hint that Newton was already thinking of fields. I have never seen Gauss as anything else than a general mathematical theorem, anyway. One should really prove it independently of all physics, should one not? After all, physics is about observation, math is about proofs. – CuriousOne Jan 07 '16 at 00:22
  • @CuriousOne I see. My interpretation of the question was that the OP didn't care about how Newton thought of the concept, but rather how we today think about it. But as far as I know (not a historian) the field concept came much later (Faraday, perhaps?) so as you say Gauss's Law wouldn't make much sense applied to Newton's way of thinking. Some clarification from the OP would be helpful. Does the cited duplicate answer his question, or is he after something else. – garyp Jan 07 '16 at 14:19
  • Well, it's not what I mean. The main idea is: Assume that two points masses $m_1, m_2$ act on each other with force $F = \frac {Gm_1m_2}{r^2}$. Do something. Arrive at the integral form of the Gauss's law. – marmistrz Jan 16 '16 at 15:14

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