I remember reading this during calculus. My basic understanding is that it is useful to explain conservation of a quantity. Can you explain the physical/ historical significance of this theorem?
It (along with Stokes’ theorem (they’re actually the same theorem in different dimensions)) helps yield Maxwell’s equations; specifically, if you want to change the flux of the electric field through a surface (right hand side), you need to change the amount of charge it contains (the source of the divergence on the left hand side). In other words, if you have the same charge contained by a surface, it will have the same flux going through it, which means you can change the surface however you wish and the math will still be the same. Physicists use this to reduce some complex problems into problems on a sphere or a box—objects with nice, easily calculable symmetries.
Stokes theorem in disguise
