Vector Calculus - Divergence

[@DayaCIDfan]

May I please come and bow down and kiss your feet then tell you that I am a guy who cannot proceed without intuitive understanding of anything. And often, that turns out to be a major obstacle and devours major time off my life. I could not have stepped forward had it not been for this video clearly demonstrating the concepts. Man, I owe you a lot and more than anything, I respect you an immense amount for giving so much to the society. I had tears in my eyes when I found this video because I had gotten desperate and I am not smart enough to figure this out on my own. Thanks a ton man!!! Really fucking genius stuff this is! You are awesome, you are the god of vector calculus! I wish you loads and loads of success!

[@oratilemarope2480]

nice.

[@pabloj.g.g.471]

Fuck you !!!

[@shu5011]

don't stop making videos. this is the first time i was able to understand divergence and curl

[@astorina]

Hello , I am engineer of 50 years and I was seeking on the net for some time some nice explanation on this issue... You treated it in a wonderfull and effective manner.... Thanks a lot, I am looking forward checking the other video BRAVO...

[@vector8310]

Kahn Academy ain't got shit on you, kid! Bravo!. Lucid, intuitive and rigorous. And thanks for pointing out when Div=0, a fact just about every single expositor on youtube omits.

[@larssider]

Holy shit, this is perfect! I am taking a course in vector calculus at the university, and this type of explanation is what makes me understand and enjoy the topic. I try to fish for this from student assistants and professors, but either they do not have an intuitive understanding of it or they think of it as so trivial that the explanations get really general and difficult to follow. Thumbs up!

[@rohitdhawan5636]

For such an explanation on probability, there is a book life hacks and the numbers.

[@soumikpal8631]

wow. instructors like you should be teaching in all institutions. I was aware of this term for about 4 years, and it is now when i finally understand this term.

[@koshkat6461]

I'm not sure if the observations around 2:05 are communicated quite as clearly as they should. To be clear, there is a distinction between the divergence at any *point* - which is given by dotting with the del operator - versus the *total* divergence (flux) over an area, which is given by the identity of the Divergence Theorem. In the case of Earth's gravitational field, the divergence at any point is space is actually nonzero (because the vectors are indeed growing in the direction towards the sink (Earth)). However, it is true that the total divergence, or flux, of any region that does *not* include the sink will be zero, as the author correctly pointed out. Please be sure to not conflate divergence in general with Divergence Theorem or at least call out these two related concepts explicitly for clarification.

[@ExplanationPoint]

Yukhol Tsien Thanks for the comment. The differential form of Gauss's law for gravity states that the divergence of the gravitational field at any point in space is 4*pi*G*p, where p is the mass density at that point in space. So every point in free space does indeed have zero divergence, as stated in the video. The fluid analogy for this would be a drain in the middle of a full bathtub. As the water gets closer to the drain, the water starts to move faster towards the drain. But the divergence is still zero at those points, because there is no source/sink for the water except at the location of the drain itself.

[@ishraq23]

Your video is awesome man! All my confusions are gone. Please keep up the good work.

[@tejasarlimatti8420]

I'm sorry your videos don't get more vires, they definitely deserve millions

[@keithlyons8931]

Hey, thanks for the video. I really appreciated how you focused on the fundamentals of what was occurring. I look forward to future videos!

[@janeharrowing8145]

Thanks - this is the best explanation I've found

[@alannolan3514]

thank you - analogies and theory mesh very well, writing and highlighting is easily the best in the maths/physicsyoutubeosphere (that's gotta be a vector field). I'm trying to redo quantum mechanics having done it 30 years ago - this has sped me along my way- obs you are a hose ! thank you

[@WholeNewLevel2018]

Excellent video ,so illustrative .... keep up....

[@SairamKaushik]

Excellent explanation. I understood clearly now. Thank you.

[@zach_stow]

Awesome video! Made everything very intuitive!

[@rohitdhawan5636]

Really nice explanation.💯💯💯

[@mamm_ad]

great job......excellent explanation

[@sanlistonmadzima]

Thank you for this excellent and very informing explanation. Keep it up!

[@andrewstuhr8626]

Great video. Nice job!

[@omarm.7068]

Great job! Subscribed.

[@gevarykrunchy7179]

You are amazing

[@astha192]

Simple and informative

[@luismanzano7355]

great video, pure gold, THX !!!

[@patipateeke]

can you please elaborate a little bit on 2:14 : why is the divergence in free space zero? It seems to me the vectors are converging, actually

[@superusr]

Thank you. Perfect explanation

[@tonymac674]

Thanks! this video was so helpful.

[@233conte]

Great video, thanks for the help

[@tamizhtamizh412]

Awesome bro.. Thanks a lot..

[@suyashsharma5027]

Hello, Thanks for the great video explanation. You say that free space has no divergence of the vector field (3:00) which means that they may look like they are getting strengthened when moving closer to the earth but in reality , they are only getting concentrated due to the area becoming smaller and smaller. This implies that as we move closer to the earth, the Gravitational field concentration is increased while the strength of the field remaining the same. But on the other hand, we know that the gravity is dependent on the distance from the surface or the center of the earth, and so is the electromagnetic force, so that means that actually there is an implicit relationship between the vector field concentration and the height from the center to define the strength of the gravitational field. Is it possible that actually the rate of concentration of the gravitational vector field as a function of the height from the surface is fairly negligible and that they change independently, i.e the change in gravity due to change in height and change in gravity due to the change in the concentration of the vector field are disproportionate and cannot be related??

[@ajabbar3336]

Great

[@shoobum]

it was really helpful! thank you

[@PRASHANTSRIVASTAVABME]

Very informative video, ty.

[@mrgaiusbonus]

great video thanks! i wonder what hard and software you use to make such videos?

[@adosar7261]

do we need to define a direction (right or left) from the point that vectors increase or decrease ?

[@thebobbieLee]

it is really easy to understand.

[@pixooo2808]

plllzzz pllllzzzzzzzzzzzzz sir im in 4 months searching why at 2:11( why ther is no divergance in free space even if ther is a point charge or a gravitational feild liie you showed ,and why in simulation when i write a vector field 1/r the divergance iz zero in free space however in 1/r^2 the divergance exist in free spaceee ) ??????????????????????????? sir i really appriciate if you will answer

[@nafriavijay]

I wanted to brush up a bit, your explanation is way better then khan academy , but instead of "increasing w.r.t. x (or y)" use "varying w.r.t." ... great work though

[@BatistaR0X]

Honestly good stuff

[@nitinbansal1600]

can you please explain physical meaning of the gauss divergence theorm

[@tajamulislam7423]

Best explantion

[@kornelijekovac9793]

4:23 Doesn"t the gradient do the same thing? Why didn't we just calculate the magnitude of gradient vector?

[@ExplanationPoint]

Divergence takes a vector field (in this example, water flow) and creates a scalar field (amount of source/drain at that point). In contrast, Gradient takes a scalar field (like a function for the height of a mountain at any x, y) and creates a vector field (the vectors would point up the mountain at every point). Its inputs and outputs are opposite. So we can't use gradient in this circumstance because you can't take the gradient of a vector field.

[@kornelijekovac9793]

That's exactly what I'm asking. Someone intentionally said that you can't calculate gradient of a vector field, but why is that? I understand that gradient of a vector field wouldn't have the same physical explanation as for a scalar field, but that wouldn't be the first time we use an absurd intermediate mathematical number, to get a meaningful outcome out of it afterwards.

[@ExplanationPoint]

I think I get what you're asking, and I probably don't have a satisfying answer for you. Could we define some operator that could operate on vector fields, and ouputs a vector field, where the magnitude at any point was equal to the divergence? Yeah, we could. But how would we define the direction of the vectors, and what information could they give us? If we're trying to find divergence, why add an intermediary step when it's unnecessary? As things are currently defined in vector calculus, we have a single operator, del, which acts like a vector. And del(f), del * F, and del X F are all useful quantities that mesh perfectly with the normal rules of vector multiplication (scalar, dot, and cross products). Mathematicians could have defined vector multiplication in different ways, sure, but these ways are the most useful.

[@macrogiants]

thanks for explanation , ♥♥♥

[@vortexblue12]

Question for the final answer. What happened to the -5y?

[@prashantvashisht6651]

keep up the good work:)

[@DUJSH]

very nice.

[@tanvirkaisar7245]

If divergence of a vector field is zero then it is called solenoidal,right? But why is it called solenoidal?

[@RamuSingh-mq6my]

Good morning sir please tell me best book of calculus belong higher mathematics thanks

[@saisrisai9649]

Thank u sooo muchhhhh

[@majidhesami4812]

perfect

[@advait8783]

The equation you formed is of Gradient not Divergence

[@MichaelDeschoolmeester]

McGregor Kid No, but the two are alike. The dot product of the Dell operator with a vector field results in the divergence. The dot product of the Dell operator with a scalar field results in the gradient.

[@داعسعلىرأسالاسدمنالجدإلىالولد]

👌

[@Fukenstein]

Thanks - This is also the best explanation I've find