Curl or Circulation Density of a Vector Field // Vector Calculus

[@vijaysinghchauhan7079]

Your viewers like your videos for the following reasons: (1) Your Animation. (2) Precisely telling the goals. (3) Your derivations are very Intuitive. (4) We get everything in a short Lecture. (5) Your Enthusiasm and Energy excites us. (6) You reply to our comments. final words Dr. Trefor Bazett is valuable to us.

[@DrTrefor]

Thanks for that!

[@sugoi9680]

@@DrTrefor I appreciate you doc 😎

[@AlexLee-vu5bo]

this guy need more views

[@robertoberidojr.435]

Dr. Your passion is so contagious I gain mine back

[@royelhajj2612]

The way you visualize these mathematical concepts is truly amazing !

[@SultanInStem-t3i]

Honestly one of the best videos of multivar calculus out there. I would be willing to pay for a more detailed version of the course

[@louisgibson3305]

Could someone please explain why when deriving the circulation density at a specific point at around 6:05, why for the bottom and right he used the vector field at the beginning of the line( for the bottom line that goes (x,y) to (x+delta(x),y) he used F(x,y)) but for the top and left line he used the end point of the line (e.g. for the top line that goes from (x+delta(x),y+delta(y)) to (x,y+delta(y) he used F(x,y+delta(y)) and not F(x+delta(x),y+delta(y)) ) Is this because we are assuming delta(x) and delta(y) to be small, it doesn't matter which part of the line we take the vector field from? So doing this would make deriving the partial derivative thing later easier ? Apologies for the poor wording.

[@cedricsipakam2601]

Im stuck there too

[@j_rryplays8038]

Not 100% sure, but in the context of an infinitesimally small region, the choice of the point that you use is arbitrary and doesn't matter

[@lutherlessor4029]

I just want to say that, in addition to giving students or the generally curious person an intuition for things, these are GREAT videos for us instructors who can benefit from a conceptual reminder of what we're teaching now. 🙌

[@hikmatullahpakhtoon3694]

The things that i didn't understand in 3 to 4 days, just got that in 10:27 interval of time, thanks Dr.

[@DrTrefor]

Awesome!

[@BoZhaoengineering]

This is so coool. It tells us the reason the partial differential comes in to play.

[@DrTrefor]

Thank you!

[@szkoarodzenia5153]

i am just so grateful that i have free acces to so many high quality lectures pretty much about everything! Your channel is super helpful and even though i am a highschool student you explain it so plain that even an idiot like me can understand it. Appreciate the work!

[@derickblacido2267]

Thanks for the time you spend doing this awesome teaching Dr. Trefor. You enlighten us all. I was struggling with this concept. It was nightmare. I could handle the mathematical formulas but my intuition was utterly messy and was struggling. You are a genuine dataist. Greetings from Machu Picchu: ; )

[@richardneifeld7797]

Trevor, double thumbs.up, for this video!

[@TNT9182-j1e]

At 5:40 When going along the top why use F(x,y+ Δy) and not F(x+ Δx, y+ Δy)? Also for the left, why use F(x,y) instead of F(x,y+ Δt).

[@PratyushaRavi]

Well I was thinking the same too! @DrTrefor can you please let us know

[@matthewmccarthy9293]

Same

[@matthewmccarthy9293]

@@PratyushaRavi Found it look at @agh1750

[@vaiterius]

If my professor had animations/visuals like these, it would make this whole thing a whole lot easier! thanks

[@SHREYANAND-dn6jc]

I have a doubt, in 5:50 for the top integrals you should have to write F(x+delta x, y+delta y) instead of F(x,y+ delta y) and also for left integral it should be F(x, y+ delta y) instead of F(x,y). Is there anything i am missing?

[@ravenarc3652]

I am a simple student, i see Dr Bazet i like the video. As simple as that. ❤️

[@jamyllecarter3016]

Dr. Bazett, I am teaching Calc III (you would call it multivariable calculus and vector calculus) this semester for the first time in 13 years. Since we're fully online, I am referring my students to your phenomenal videos! Such passion, such clarity, such a brilliant use of color, such intuition behind the physical explanations--you're a wonderful educator! Please tell me the algebraic formula for the vector field you generated at the beginning of this video. Thanks so much!!

[@DrTrefor]

Hi Jamylle, thanks for letting me know, I'm always so happy when other educators choose to recommend my videos. And thank you for becoming a member, I really appreciate that! The field was something like F=sin(x+y)i +cos(x-y)j. I forget the exact choice of sin and cos, but fields of that general structure all lot pretty and "swirly".

[@jamyllecarter3016]

@@DrTrefor Absolutely! I told my entire department (and some other mathematician friends as well) about you. Be ready for lots of likes and maybe subscriptions from the mathematics department at Diablo Valley College in Pleasant Hill, CA (San Francisco Bay Area)! And thanks for that vector field.

[@simonribas4625]

Hi Dr. Trefor Bazett, I have a question regarding the derivation process.. When you are computing the line integral/circulation of the point and use C1-C4 to represent F(x,y), how come you start from the tail on bottom/right but the tip on top/left? Does this make a difference (6:10)? For example, left is given by F(x,y) dot -(change)y j^. Would this actually be a correct representation of the left side? Your answer would bring a lot of clarity, Thank you!

[@saharshbhansali6004]

I have the same question. I came back to this from the divergence video, and it's not making consistent sense to me now :/

[@lorentacla9740]

I realize this is an old comment, but thought I’d comment on it anyway in case you never got your answer, as I had the same question. You can do it from the tail for all four points and it works out the same. Really, we’re taking the limit as del x, del y approach zero…two partial derivatives arise, one with del x in the denominator and one with del y. For example, for path C3, Lim del y -> 0 of M(x+del x, y+del y) is exactly equal to M(x+del x, y), you just have to evaluate the limit.

[@roshanpradeep8532]

@@lorentacla9740 Thanks! But I still don't understand why we neglected del(x) there is lim del(y)->0.

[@lorentacla9740]

@@roshanpradeep8532 It has been awhile since I’ve watched this video or toyed with this math. I think the issue is that we are evaluating C3 as both dx and dy approach 0. The limits can be evaluated in any order, so lim dx,dy -> 0 = lim dy,dx-> 0. That means lim dx,dy -> 0 of M(x+dx, y+dy)/dx is equal to lim dx->0 M(x+dx,y)/dx which is the partial of M wrt x. This works out the same even if you applied the limits in the opposite order. I’ve worked this out myself using only the tails of each vector and it works out the same. If I can find that work, I’ll try to link it to you somehow

[@rushaidamahbub]

@@lorentacla9740 it doesn't work out the same if we do it from tail for all four points tho

[@redtree732]

AMAZING explanation. Best on KZbin!

[@agh1750]

Can you explain why at 5:40, for the top part of the rectangle its F(x,y+delta y) and not F(x+delta x, y+delta y) ? I'm confused because for the bottom and right sides, the vector field is being evaluated at the points labelled on the diagram, but for the top and left sides it appears that's no longer the case. Is this because in the limit as the deltas go to 0 it won't matter anyways, and writing it that way is more convenient for the purposes of the derivation?

[@DrTrefor]

Either would be totally fine as you say because it doesn't matter in the limit. I arbitrarily choose the formula just to make it fit into the derivative formulas with one less manipulation.

[@agh1750]

@@DrTrefor thanks!

[@carlosjacobfield-sierra3759]

Thanks for asking this

[@AgenciaEspMexicana]

Congratulations for this very good question! Most of us, just accept blindly what a teacher says.

[@Tclack]

Not quite getting this. if we choose as our top the upper right, we'd have -M(x+Δx,y+Δy)Δx, so when we add the top and bottom (and factor out the negative) we get: -[M(x+Δx,y+Δy) - M(x,y)]Δx . I can see that "multiplying top and bottom by Δy" will give us a near-textbook definition of a pertial derivative. But how do we justify letting Δx going to 0 inside but not outside the bracket?

[@wargreymon2024]

This video and this channel are GOLD!

[@ayushjha6894]

U TEACH EXCELLENT . I COULD REALLY FEEL THE MATH EQUATIONS AFTER U EXPLAIN. CAN U PROVIDE NOTES ALSO OF THESE AS THE WILL BE REALLY USEFUL FOR QUICK REFERENCES ? By the way , u are owesome teacher and i recommend your vedios to all my friends. thank u for ur efforts

[@eee_bangla]

tnx a lot ...this January completed bsc in EEE..when i was a student i alwyas wanted to know the basic concept of those vector calculas but i did not get any proper explanation any where..now it is giving me the proper idea..tnk u a lot Sir.

[@noahbarrow7979]

another incredibly lucid video. Bravo!

[@DrTrefor]

Glad you liked it!

[@davidhanson9987]

You are absolutely brilliant. Is your approach explaining Green's Theorem your own? or does a text you use explain it that way? The text I use goes immediately to the math without giving any intuition.

[@DrTrefor]

My approach is certainly not new, but I did spent some effort trying to give the clearest possible explanation for the idea

[@sounakroy1933]

Hello sir. Hope you are fine. I had a question. At 5:59 mark, the line integral of left side has vector field defined by F(x,y). Shouldn't it be F(x,y+dy)?

[@ramazan-2424]

I think too that this video is completely false because of this

[@ndydiy9963]

Happened to come across this. Amazing material and presentation.

[@schizoframia4874]

That’s a nice way to think about it

[@mathanimation7563]

so useful video , my concept got clear

[@DrTrefor]

Thank you!

[@Bechear72]

he give the impression that the subject is more simpler that it looks like ❤❤

[@saudyassin5352]

Thank you so much for taking the time to share your knowledge with us. I will be majoring in Physics this fall and Your videos are a great resource as I like to self-study mathematics to understand physics, especially Maxwell Equations.

[@mathanimation7563]

why did not take -(x+dX,y+dy)dx for c3

[@DrTrefor]

That would be fine too, it doesn't ultimately matter as dx and dy are all being sent to zero, but I write this way so as to make it line up nicely with the limit definition.

[@mathanimation7563]

@@DrTrefor ok thanks

[@EdwardSnowden125]

​@@DrTreforwouldnt the entire thing tend to 0

[@apoorvmishra6992]

sir at 5:30 should you not integrate the dot products too because the circulation is the integral of dot products

[@tim-701cca]

9:56 I have a question. The integral on LHS is the same as int_ F dot T ds , which is defined in the video "Flow Integrals and Circulation"?

[@pawankumarpandey6551]

@9:55 the video states the circulation density to be kth component of the curl. But actually, it seems to be the negative of the kth component of curl, if I am wrong. please correct me if I am wrong.

[@DJ-yj1vg]

What a brilliant explanation. 👏

[@continnum_radhe-radhe]

Thank you very much sir.... I don't know how i express my feeling..... ❤️🔥🙏

[@soldireofsolomon9411]

Bro You are really an under-rated guy.........😭 May Allah bless you....... Keep it up.I am really grateful to you. You see,I wanna learn maxwell's electromagnetic equations.So,I need to learn vector calculus first........And it's really much more helpful......due to the visual and geometric interpretation beside you.🥰🥰

[@playitback-os7mh]

Superb explanation

[@roshanpradeep8532]

I still don't understand why we neglected del(x) there is lim del(y)->0. @6:34

[@yacoubaibrahimatraore1982]

8:31why don't we have the integral symbol as well as in the right hand side ?

[@ewanfraser4014]

Oh my I think it jus clicked!!! I'm a geography student with minimal maths and physics background trying to learn calculus for a meteorology course I aspire to complete

[@archismandas7760]

5:42 how did you get the expression for the Top part ?

[@nehalkalita]

6:08 Why the "a" values in F(a) are different from the ones mentioned in diagram? (Can anyone explain)

[@rodionraskolnikov6989]

Nice video! A question: at minute 8:49, the circulation is only approximately equal to the circulation density times Delta x Delta y, right?

[@ductanle2806]

The reason for setting the bottom position change +delta x and top position change - delta x is because the horizontal conponent of the vector field at an infinitestimal scale is the same, so 2 segments intergrated from opposite directions will be of opposite sign?

[@DrTrefor]

Exactly

[@pochotom2834]

on the top it should be: F(x+Ax, y +Ay)*(-Ax i) and on the left it should be: F(x , y +Ay)*(-Ay j) or not? (Note "A" is delta).

[@karthicksk8793]

Wow thanks a lot!

[@agrajyadav2951]

FANTASTISCH!!!

[@adityak7144]

Very well explained videos! I really hope you get hired as a mathematics professor at MIT or equivalent 👍🏽👍🏽 Edit: It is because of you that I have any hope left for myself in my multivar-calculus course.

[@jelleoudega116]

Dear Mr. Trefor, could you perhaps explain why you chose a rectangle and not a circle, is there perhaps a way to prove that the definition of curl is irrespective of the chosen enclosed curve? Thank you!

[@DrTrefor]

Mainly it’s that in the Cartesian coordinate system, a square is the natural object because each side is fixed in one coordinate. With a circle you have to deal with both coordinates changing, not impossible, but have to be more careful

[@jelleoudega116]

@@DrTrefor Thank you for your quick reply and clear answer, this is a rather subtle subject which I couldn't find much about in most textbooks, so this video along with your comments definitely help. Again, thank you.

[@suleymanergin9499]

I imagine that the term circulation density is nothing but the z-component of the curl vector. The curl is actually a 3D operation. If we think the description of curl in terms of (Nabla x Vector field), we obtain another vector as a result. And I think we can describe each of the components of this resultant vector as circulation density in direction of x-, y- and, z-axis respectively. And the curl of a 2D vector field will always be in z-direction. Am I right?

[@DrTrefor]

Exactly.

[@suleymanergin9499]

@@DrTrefor thank you for your reply.

[@khanredoy591]

Sir, what is the graphical representation of a function of three independent variables in 4D? Is that necessary to realize the surface integral?

[@DrTrefor]

Any time the graph is four dimensional it is just hard to visualize completely. We can use tricks like color or changes in time to represent one of the dimensions or in the case of a vector field from two to two dimensions there is that trick or plotting it we see here

[@khanredoy591]

Thank you sir.

[@sfs8730]

Bro wtf, this is so good

[@sfs8730]

😍

[@MishaShvartsman]

Just wonderful!!

[@metalicanabach9206]

thank you.

[@youngtreebig1]

Dr. Trefor Bazett, Could you pls change this video's auto-generate caption language?

[@momen8839]

In physics ..the electric or magnetic flux is definded as number of lines of elctric or magnetic field ...my question is how this is right however we can draw infinite vectors with increase density? and take in mind numbers as .1,.2,.3,...etc and other question is how to express vector field in lines instead of vectors on axis?

[@DrTrefor]

If you impose a fixed standard then the gaps between lines mean something. For instance on a topographical map with 100m spacing between lines, areas of more dense lines are steeper.

[@carultch]

The "number of lines of electric or magnetic field" is an oversimplified statement, because as you've stated, it is arbitrary how many lines you draw to represent the force field. You'd need to draw the force field with a standard line density, for flux of the field to "equal" the number of lines. What flux really is, is a surface integral of a certain section of the vector field.

[@tanle7164]

If all the points put into the line integrals are (x,y), for example, how would you manipulate for the partial derivatives to show up?

[@ileanadominguez6055]

Thank you very much :)

[@fourthsideofthecoin1636]

thanks

[@hanmi4735]

Hi Dr.Bazett, I have a questions abou the definition of curl here. I am reading the textbook written by James Stewart, and the definiton he gave in the book, which is "Curl F = (dR/dy - dQ/dz)*i + (dP/dz - dR/dx)*j + (dQ/dx - dP/dy)*k" where F = Pi + Qj + Rk, looks different from what you have here (though yours is in 2-D and his is in 3-D). How can we relate these two expression together?

[@carultch]

Trefor uses the letter trio of M, N, P, avoiding O for obvious reasons. Jimmy Stewart uses the letter trio P, Q, R for the components of the vector field. Trefor also included the area element Δx*Δy, that shows the relationship between curl and closed path line integrals. Curl itself omits those terms.

[@ricardoalvarez4976]

1:54 shouldn't the propellor spin clockwise?

[@yasirmahsud4129]

No, because the velocity below is greater than the above. The high velocity vector below will push the propeller anti-clock wise.

[@zhenlanwang1760]

Shouldn't the top be -M(x+dx,y+dy)*dx as we evaluate F at (x+dx,y+dy) i.e. top-right corner? Similarly for left -N(x,y+dy)*dy

[@zhenlanwang1760]

I thought more about this. I think the mis-alignment will go away, if we pick the point of interest to be the center of square and evaluate the F at the mid point of the sides of the square.

[@munozali1]

I like the video, but too are too many advertisements for a 10:27 min long video.

[@jhanolaer8286]

how to generate this?

[@zookush1]

Can you explain this , in the derivation you only took dot product for each curve(C1,C2...) and added them up to be equal to the circulation but shouldnt you also integrate dot product of each curve as circulation is the sum of line integrals not just dot products? Also for each curve , like C2(right curve) you took dot product as F(x+delta x,y).(delta y) but this force isnt const throughout the entire C2 so shouldnt an integral also come in the picture? Except these doubts the video was really amazing! :)

[@apoorvmishra6992]

i have the same doubt did you not get it yet?

[@zookush1]

@@apoorvmishra6992 no

[@bigbluewhale3957]

I also have doubts about this, but I have a feeling that because the rectangle we are dealing with is infinitesimal, we can think of each side as being infinitesimal and so we just take the dot product of the side vector and vector field. I realise this comment is old, have you arrived at an answer to your questions?

[@MechEngin3er26]

Would be great if you had thrown in the cross product version of the formula lol. Unless you covered that in another video.

[@TALCOHOME]

Why doesn't N(x,y) matter for the dot product in 5:15?, Amazing playlist

[@carultch]

Because line integrals are a dot product, and ignore the perpendicular component of the vector field to the local section of the path. When travelling parallel or antiparallel to the x-axis, only the x-component of the vector field [which he calls M(x, y)] contributes to the line integral result. When travelling parallel or antiparallel to the y-axis, only the y-component of the vector field [which he calls N(x, y)] contributes to the line integral result

[@kalvincochran9505]

^ this is a good explanation! Perhaps another way one could conceive such a concept could be to consider that the path along c only has x components so when you take the dot product it is x components of first times x components of second plus y components of first time y components of second. So in this case the N is multiplied by a 0 because in the path c1 the y component is 0. Thats why the “bottom” c1 is N times 0 plus M component times delta Y component.

[@j.o.5957]

Beautiful!

[@mohamedirshaathm32123]

Me watching this and finally got the green's theorem proved in silent in this video

[@Ranbir.Bhardwaj]

Why captions auto generated in Korean? It would be helpful if they were in english

[@andrewvictor1489]

The same problem for me

[@duckymomo7935]

what is your field of research? (you have a PhD)

[@DrTrefor]

Algebraic Topology (and a bit of math ed)

[@GameinTheSkin]

@@DrTrefor You're putting out insanely high quality lectures man, along with visualizations. You have no idea how much I appreciate it.

[@mnada72]

Is the circulation at a point a scaler quantity 🤔

[@JosephTakach]

In 2 dimensions yes, but in 3 dimensions it is a vector

[@andrewvictor1489]

l want the captions auto-generated in english

[@albertyeung5787]

excellent