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What are the differences and advantages of Gauss's Integral and Differential Equations? Thank you so very much!

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marked as duplicate by Aaron Stevens, Thomas Fritsch, Jon Custer, John Rennie electromagnetism Aug 2 at 10:20

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They are equivalent, and one can be derived from the other using the divergence theorem, except when the E-field is not well-behaved, in which case the integral form might be slightly more general. As an example, the divergence of the E-field due to a point charge is proportional to the Dirac delta function, which often requires more care than ordinary functions. I'm sure someone can comment if there are even more pathological cases. In most situations, if you don't have a problem working with Dirac delta functions, you can use either form.

Some advantages of the integral form:

  • Doesn't require vector differential operators, so the integral form is frequently used in more introductory treatments.
  • Can be readily used to calculate fields by exploiting symmetries in the problem (spherically symmetric charge distribution, uniformly charged infinite sheet, uniformly charged infinite line, etc.).

Some advantages of the differential form:

  • More compact.
  • Doesn't require the introduction of an arbitrary volume whose boundary you integrate over.
  • Relates the field and charge density at a point rather than over a volume. This is often more convenient if the problem doesn't lend itself to the use of the integral form because of symmetries.
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  • $\begingroup$ I want to stress Puk’s final point. The differential form makes clear that electromagnetism is a local field theory. $\endgroup$ – G. Smith Aug 1 at 23:50
  • $\begingroup$ Numerical solutions of the integral form are inherently conservative, while differential forms are much harder to guarantee conservation (at least, this is true for Navier-Stokes -- not sure how often discontinuities arise in E&M). $\endgroup$ – tpg2114 Aug 2 at 0:04

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