Here there are Maxwell equations https://en.wikipedia.org/wiki/Maxwell%27s_equations . Let's take the first in particular, Gauss's law. In its integral form, the left side is equal to the integral over the volume $\Omega$ of $\nabla \cdot E$ (because of Stokes's theorem). So I have a triple integral of $\nabla \cdot E$ on the left side, and a triple integral (over the same volume) of $\frac{\rho}{\epsilon_0}$ on the other side. I can't see what they mean by "differential equations". This equation in particoular becomes $\nabla \cdot E= \frac{\rho}{\epsilon_0}$. I just don't understand what this means. It seems like that if two integrals are equal, then their integrands are also equal, which is obviously false.


Let's break this down into pieces.

A differential equation just means an equation that uses differentiation. ∇⋅E=ρ/ϵ0 is a differential equation because it involves a gradient.

You're mostly right. Both sides become a triple integral over the same volume (for any volume) which implies that the integrands of both are the same thing. You'll notice that the differential equation is simply stating this.

  • $\begingroup$ To make things even more clear. The main point of the answer is the "for any volume" part. $\endgroup$ – Victor Palea Oct 4 '17 at 18:57
  • $\begingroup$ ^^ Thanks, I probably should have stressed this more $\endgroup$ – George Oct 4 '17 at 19:15
  • $\begingroup$ So, you are saying that those are equal because the equality holds for every volume, also "infinitesimal" ones. I think I got it now. The thing that puzzled me was that it looked like "integrals are the same, so integrands are the same too" $\endgroup$ – tommy1996q Oct 4 '17 at 19:50
  • $\begingroup$ Yes, exactly. If your integral equation works for all volumes, it works for differential volumes and hence the integrand may form a differential equation. $\endgroup$ – George Oct 4 '17 at 20:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.