The wikipedia article on the divergence theorem states that it was first discovered by Lagrange in 1762, Gauss in 1813, Ostrogradsky in 1826 - who also gave a proof of the general theorem, Green in 1828,etc. Gauss published special cases of the divergence theorem in his 1813 article on the attraction of homogenous ellipsoids, and a full version and proof of the theorem in 1840 in his main work on potential theory ("Allgemeine Lehrsatze"). According to p.205-207 of the book "A History of Analysis", Gauss's proof appeard at section 24 of his 1840 work, where Green's identities are implicit in the method of proof. However, Michael Ostrogradsky had successfully proved the theorem already in 1826 and with greater generality. What i want to understand is the subtelties of the rigorous proof of the divergence theorem. So my question is:

  • What does it mean to give a proof of the divergence theorem in greatest generality? i read in some places that despite that the divergence theorem can be made intuitively obvious by the sources and sinks analogy (total flux), a proof of the general case is suitable only for advanced mathematical courses (in contrast with the simpler versions that can be thought to first year students). So i'd like to know the meaning of "rigorous proof" in this case. At the same time (but this is a bonus: the more important question is the first one), i'd also like to get an explanation of Gauss's and Ostrogradsky's methods of proof and the ways in which they differ.

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