The Divergence Theorem, a visual explanation

This video talks about the divergence theorem, one of the fundamental theorems of multivariable calculus. The divergence theorem relates a flux integral to a triple integral.

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[drrajendrap]

At 12:14, The Gauss divergence statement should be ∇ · F instead of ∇ X F.

[TheTKPizza]

I do have to say, the quality of your videos is quite amazing, close to 3blue1brown. Keep on delivering quality like that and this channel is going to grow fast (as you can already see with a few of your vids). The visualization of Green's and Divergence Theorems really helped me a lot with actually understanding them (and not only being able to calculate on their basis). Thanks a lot, dude!

[stevekeller]

I really enjoyed learning this some decades ago. Here it's better explained, and without the exams.

[ElZenom]

I'm surprised that I only stumbled upon this channel when I misclicked, thinking that it's 3blue1brown's video. One of the best mistake of my life.


One suggestion I have is for you to slow down in some transitional parts. For example, when you're calculating 2D Flux integral for F=[xy+x, x+y], you can show the third step in which you input in the {xy+x} portions before inputting the boundary values (cos, sin). Those small inputting steps might mean little once you know it but will help more in visualization if you show it.

[adityagiri]

Great job buddy, u really explain it in depth

[darinarieko]

This came up on my recommended page 3 years after graduating college. I am not complaining.

[MrThemastermind]

At 5:20 there's a mistake in the formula: in the left hand side you have the line integral of a vector field F over a curve C with a parametrization r, that is the "work" integral. On the right hand side you have the expansion of the line integral over a curve C of a SCALAR field f, in which you multiply f evaluated at r(t) by the magnitud of the derivative of r(t).

The reason why you need the formula of the right hand side (the expansion of the line integral over C of a scalar field) is because the dot product of the vector field F times the n hat vector is in itself an scalar field.

Sorry for any spelling mistakes, and great videos man. Keep it up, will subscribe

[mathOgenius]

Hi, Can you please tell me which software are you using to make these awesome videos, Please !?

[aadhuu]

Just discovered your channel today! Absolutely amazing! How did you learn all the partial differentiation, divergence and stuff?

[alanioth]

At 5:11 you say that you're rotating the tangential vector by 90deg. Then you show an expression in radians that includes 2*pi. How does this represent rotation by pi/2?

[bmet]

At 6:37, surely in a linear flow field the divergence is zero? Advection into and out of the region F are identical, no? Would love to know why it is grad.F > 0

[Jirayu.Kaewprateep]

In case of Electric flux, that is not only the electric field BUT the random high potential electric discharge ( Vander Graff ).
What if it is magnetic field, in imbalance shape magnetic force is stronger near by the magnet or at the pointing area? ( Spherical shape, average force is reasonable )

[nikitakipriyanov]

In Russia we know this theorem as Ostrogradsky-Gauss theorem.

For me, this better serves to explain what divergence is, rather than to "explain a theorem".

[OlliFritz]

This is funny, I subscribed when you were making cubing content, and now there's advanced math videos that are relevant in my University courses.
What do you study?

[RealLifeKyurem]

At 6:34, the divergence is 0, since the flux going in the circle/surface is equal to the flux going out the circle/surface. So ∇ · F = 0, not ∇ · F > 0.

[pbj]

In a line, the amount of "fluid" flowing out of a volume is equal to the fluid flowing out of its surface if it has a closed surface.

Edit: As Nikita Kipriyanov has pointed out below, the amount of imaginary fluid flowing out of the volume is equal to the amount entering it PLUS what is created/sucked inside/into it

[MrJdcirbo]

vcubingx: to get normal vector you take the tangential vector and rotate it by 90°.

Cross product: Am I a joke to you?

[ThomasHPuzia]

@12:16 it should read nabla dot F on the right

[dlmacbr]

it should be the div F (diverence of vector field F) in the triple (volume) integration instead of the rot F (rotational of vector field F), Thus, divergence theorem. Otherwise, great video.

[gaaraofddarkness]

10:35 12:22 in one its divergence, another its curls?