Multivariable Calculus | When the divergence theorem doesn't apply.

We classify all surface integrals over a special vector field, including a case when the divergence theorem doesn't apply.
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[@sergey7375]

I remember things like that popping up in my electromagnetism class. We treated these singularities like true physicists with lousy delta functions and no rigor whatsoever. It's nice to see this more rigid explanation.

[@hoodedR]

Wow. That was amazing. Only towards the end did i realize that the vector field is the inverse square-central vector field. And what you had derived was basically gauss law. That's beautiful Also how do you reason the fact that the integral diverges under case 1?

[@Alysio]

Because you're integrating over a singularity. (0,0,0) \in S and F diverges very badly at (0,0,0), too much so for it to be integrable at that point basically. Nice job catching Gauss's law! More precisely you could see it as "the 1/r² field only has a source at the origin", and Gauss's theorem tells you that you only need to measure the "strength" (in physics, this would be mass or charge depending on the force you're looking at) of the sources *within* the volume E to know the value of the integral. :D You could also link it to Laplace's equation, in a very similar way: F is the negative gradient of a 1/r potential, which is the fundamental solution of Laplace's equation, so by construction divF=0 everywhere but at the origin! :D

[@MichaelPennMath]

It has been a long time since I took E&M and I didn't even realize thats what I was getting at!! I should have mentioned it.

[@riadhhocinebouseder5776]

really , Great job .

[@GabrielPohl]

Every sentence in this video is a poem.

[@iyadindia862]

At 11.40 How the orientation of the inner surface of sphere inwards?...I think I need more clarification on this.

[@peternorth7737]

Hello Mr Michael. Could you kindly put number 1, 2, 3,... on each of your videos in sequence.. Im afraid i miss your videos.