Divergence Theorem

In this video, I present the divergence theorem, which is a natural generalization of the fundamental theorem of calculus in 3 dimensions. It says that the surface integral of a vector field F is the triple integral of the divergence of F, and it provides a neat way of calculating surface integrals. Enjoy!

[@Galileosays]

Chapeau ! You've honoured Lagrange (1762), Gauss (1813), Green (1825) and Ostrogradski (1831), the introducer, the re-inventors, and the one who proved the theorem.

[@bluepurplepink]

I love how engaging you are and the expressiveness often helps to keep the focus on the material and not sound monotonous.

[@d3u5-2s]

dear professor, i owe you my analysis 2 exam,i spent a full day watching the vectorial calculus playlist and now i've almost got how to make this things work, thank you very much i love you so much thanks from italy!

[@drpeyam]

This is amazing, thanks you so much!!! 🙂

[@bobmike828]

I love.. you You're the best Dr. Peyam you make calc sound so childlike its actually fun watching your videos

[@etienneparcollet727]

Green-Ostrogradski to rule them all!

[@shayanmoosavi9139]

I just binge watched the whole Playlist and now have a much deeper understanding of line and surface integrals and know what each of them means. So the line integral of a vector field is the sum of the x shadow and the y shadow of the fence produced by the surface of f(x,y) along curve c which is the geometric interpretation of the line integral of a scaler function. I thought they were fundamentally different but it seems that it's not. Nice :) I have a question though. Our professor introduced us to a completely different method for calculating a surface integral. He said that you can calculate it with this formula : ∫∫f(x,y,z)(||∇g||/|∇g.p|)dA Where g is the equation of the surface, p is the normal vector of the plane which the surface's shadow is casted on (you can use xy,xz or yz planes for this purpose) and dA is the surface element of the region that shadow makes. What is its connection to the formal method that you provided? What I understood is that the gradient vector can act as the normal vector. I couldn't find any more connections and it would be nice if you could answer it.

[@drpeyam]

That method works if the surface is the graph of a function g. Not every surface is like that (like the sphere for ex), so my method is more general

[@shayanmoosavi9139]

@@drpeyam he calculated the sphere by calculating two hemispheres but as you said not every surface is like that. I get it now :) P.S : surface parameterizations can be painful sometimes and we didn't learn it in our course. Do you have a video on that?

[@jameswilson8270]

I'm stoked! ;)

[@drpeyam]

Hahaha, you’ll be Stoked on Friday :)

[@jameswilson8270]

@@drpeyam Nice work! I will say, however, it's actually not that hard to do this example without the divergence theorem. But the divergence theorem made it a little easier. I'm always stoked to see a video by Dr. Peyam.

[@sweetyvany99]

Good video but could you please explain again how to figure out the parametrization?

[@TheNachoesuncapo]

first! love it

[@XanderGouws]

Hey, do you think we could get a series for PDEs similar to what you're doing here for vector calculus? I know you're mainly making these because you're teaching it atm, but when could we get a PDE series done in a similar way?

[@drpeyam]

It’s already been sort of done, see my videos on Laplace, Heat, and Wave equations!

[@andres.robles6]

Cálculo vectorial 😍

[@dhunt6618]

yea! more E&M! (and other physics :)

[@okami4133]

I´ll never forget the jacobian of cylindricals coordenates and esphericals coordenates until my decease. I promise it for "La horda" xd

[@jonasdaverio9369]

Is vector calculus connected to differential geometry?

[@connorshea9085]

yes, very much so! The fields of multivariable/vector calculus, differential geometry, and partial differential equations are very closely related.

[@drpeyam]

Absolutely! Differential geometry provides a neat generalization to calculus! Unfortunately it’s beyond my understanding

[@jonasdaverio9369]

Oh, it's sad, I've heard many people being excited about it providing them a better understanding of many things in mathematics. I would have loved to see a series on it. Someone told me that it is not as difficult as people say. You should take a look at this branch, seeing how qualified you are (in our university, there's a course on it for graduate student, Master I in Europe, and he told me that he would have understood it when he was undergraduate)

[@Rundas69420]

I bet that the one who disliked this video believes that the divergence and rotation have swapped formulas and then got the wrong result out of their calculation. xD

[@drpeyam]

Hahahaha 😂

[@georgecooper7389]

:))

[@alanhiguera3484]

E

[@racimeexe9868]

I don't understand how did u found the bounds of the integral