Multivariable Calculus | The Divergence Theorem

[@aks9545]

Thanks for your videos, keep up the good work.

[@bebarshossny5148]

Man i wish you could do the proof of the general case

[@A_Helder16]

You want blood xD

[@ingobojak5666]

For an idea of the general proof, imagine the volume chopped up regularly into many small cubes of the same size. Take a look at two cubes side by side, touching, i.e., they actually share a side (no gap). Since their surface normals are outward-oriented, their normal vectors point exactly opposite to each other on the shared side. This means that if we now try to sum the flow through all the sides of both cubes, then the contribution of the shared side will cancel, leaving us only with the flow through the other sides (the outer surface of the rectangular box they form together). But for each of the two cubes we already have the divergence theorem, so we get a sum of two divergence volume integrals as equal, which is simply the divergence integral over their combined volume. The same is true for any shared sides of any of the cubes. They all cancel the flow on the shared side, leaving net only that through the outer combined surface, whereas the volumes simply sum up. So finally, make the cubes small enough to approximate the actual volume arbitrarily well, and that will become a good place to stop...

[@pablorestrepodiaz8520]

Good video

[@Quved]

OMG.