Divergence Theorem example: Flux across unit cube // Vector Calculus

In this example we use the divergence theorem to compute the flux of a vector field across the unit cube. Instead of computing six surface integral, the divergence theorem let's us compute one triple integral of the divergences of the vector field over this unit cube.
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[@Sasuke-Uchiha]

bro, you just came in real clutch for my final later today

[@DrTrefor]

hey nice! Hope it went well:)

[@alfredandens7911]

Thank you brother, the whole serie have helped me a lot.

[@michaelwinder7879]

You are a legend! These videos have carried me through my Calc 3 class

[@nat6429]

This was really helpful and clear, thanks so much!

[@ManojKumar-cj7oj]

vocabulary 'flux' and 'circulation' makes the questions more interesting and simple

[@张鑫-e9o]

The handwriting of z is a little bit confusing.

[@softwarephil1709]

Yes. I thought some z were 2 (squared).

[@tiomemo8958]

same here lol@@softwarephil1709

[@utsavdesai3451]

Really Helpful for finals next week!!!

[@TheDeluxeBacon]

Thank you for making these, I have my vector calculus final tomorrow lol

[@DrTrefor]

Good luck!!

[@chloemirisch8159]

Did you not take the partial derivatives correctly? I thought the partial derivative in respect to x,y,z would be . very confused as to why you added z to each of them.

[@Brenticus12]

the field equation is (xyz, xyz, xyz) not (xy^2, xy^2, xyz)

[@j.o.5957]

Nice, good stuff. Question to self: how do I solve this using standard flux calculations? I frankly don't know, don't think I've been through that. Should look into it later.

[@dantepillon]

Hi this is a great video! But I was wondering when you were talking about the deformation theorem, that is; when divF = 0, any 2 closed boundaries share the same flux. What I cannot seem to get is, if divF = 0, should the total flux for any closed boundary also = 0??

[@DrTrefor]

The total flux over all boundaries have to zero, any specific closed region might not be though

[@dantepillon]

@@DrTrefor @Dr. Trefor Bazett thank you very much for replying sir, but I guess what I still don't get is, can you call any specific closed region its own region. And then apply the divergence theorem on that region alone to say it has 0 flux?

[@firstlegend5105]

​@@dantepillon You are correct that if divF = 0, the total flux for any closed boundary should be 0. In other words, the flux entering the boundary must be equal to the flux leaving the boundary. This is actually a consequence of the divergence theorem: if divF = 0, then the integral of F over any closed surface should be 0. The theorem should read: when divF = 0, any 2 closed boundaries share the same amount of flux and the total flux for any closed boundary is 0.

[@CaseyKlein]

Thank you!!

[@rodaneblackwood1079]

Thanks!

[@brianofarrell1977]

F*** me, my final was yesterday... never understood the concept and now i do!

[@DrTrefor]

Haha great timing😂

[@ShubhamPatel-gx7pb]

If F= yz i + zx j + xy k and S is the part of the surface of the sphere x^2 + y^2 + z^2 = 1 , which lies in first octant , the the value of double integral F•n ds is Sir why we can't use Divergence theorem in this question?

[@DrTrefor]

Divergence theorem ONLY applies to a closed surface, not part of one.

[@ShubhamPatel-gx7pb]

@@DrTrefor Thank you Sir

[@carultch]

@@ShubhamPatel-gx7pb You can often use the divergence theorem as a shortcut to evaluate the flux over a surface that isn't closed, when symmetry works in your favor. Like if it is easier to find the flux over a closed surface using the divergence theorem, and the open surface in question is just one octant of 8 identical octants, that would form a closed surface if you connected them.

[@leondavidpedraza8618]

What happened with the x that comes out of the first integration? I am sorry, I missed it, it was too fast for me to be too slow! thanks.

[@swayamkumarpatro776]

Sir, can you please attach a list of problems which are very important in this vector calculus course ?

[@NelsonMbigili-es6pw]

💜

[@emilykang4354]

Thank you for such a clear statement

[@im_KiraL]

How did you get yz+xz+xy from this F

[@zeeeeeeeeeev6493]

good vid

[@aashsyed1277]

What is flux core welding and fabrication near me a lot of

[@carultch]

It's an unrelated meaning of the word flux.

[@ironsideeve2955]

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