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Classical Physics
Electromagnetism
Arriving at the differential forms of Maxwell's equations
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[QUOTE="vanhees71, post: 6303416, member: 260864"] You can also think of the differential operators div and curl being defined as limits of integrals, which has the advantage that the definitions are manifestly covariant, i.e., independent on the coordinates choosen to evaluate them. Then your example becomes a triviality, because assuming that Gauss's Law in integral form holds for all volumes and boundaries of these volumes by just shrinking a volume to a point leads to the differential form of the equation. As a theoretical physicist I'd also consider the differential (local) form of Maxwell's equations the fundamental laws. The integral forms are much more delicate to handle right, which is shown by the great confusion they provide due to sloppy treatment in many (otherwise maybe good) textbooks. The prime example for this is Faraday's Law, which is mostly quoted for the special case of areas and boundaries at rest and then applied without further thinking to moving areas and boundaries, leading to a lot of confusion. The resolution of this confusion BTW finally leads to the only adequate framework for electrodynamics, which is (special) relativity. [/QUOTE]
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Physics
Classical Physics
Electromagnetism
Arriving at the differential forms of Maxwell's equations
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