В спешке проходили криволинейные интегралы в прошлом семестре. Решил глянуть и добить гештальт от вас узнал больше чем на парах. То как вы провоцируете мышление четко обозначая проблему, тем подводя нас к выводу формулы это невероятно!! :)) И не думал, что корни вместо дифференциациалов это эхо теоремы Пифагора
Да ничего совковые преподы объяснить не могут. Я от индусов на ютубе больше узнал чем за весь прошлый учебный год. Сейчас ВУЗ в России это просто отсрочка от армии, а не источник знаний.
Thank you! You have the ability to make math concepts clear. Not sure what it is exactly…the explanation is clear, without too much math lingo. Others explain the material, but for some reason do not ‘connect the dots’ and generalize the concepts. I see this in all of your math videos - not sure if it is deliberate, but I think it really helps when the basic concept is generalized to the more interesting and powerful ‘global’ concept. Examples: in ‘Essence of Multivariable’ you show how many of the vector calculus concepts boil down to just one formula. In the line integral video, you take the concept of parameterization, and generalize it to n dimensions (which I think is novel on TH-cam). This is super helpful. I personally really like this approach. But your video titles don’t describe your videos…so those searching for these general descriptions won’t find them. The Essence and Line Integral titles should say something eluding to the generality lurking within, otherwise people think they are just regular videos - which they are not. They are very special and unique, and deserve way more views! I realize you name them this way on purpose - so you don’t scare some viewers off, but maybe create another channel with more general titles which link to the same videos, or maybe use keywords to attract a broader audience? Judging from the comments, many could benefit from your talents, and you deserve more hits! Best of luck to you!
With this treasure I summon, divine general Stokes! In all seriousness though, nice vid. Line integrals are weird, and it's hard for some to understand how you transform this formulation into a method suitable for integrating over vector fields.
Cool! Now, try a flow integral, which is defined as a line integral that involves a path through a vector field, which are usually marked as ∮ F ∙ dr, or ∫(F ∙ t)ds. Also, this is an example of a work integral for a force field. For the same force field, a similar formula exists for the magnitude of torque in a 2D vector field with ∫(F ∙ n)ds.
I think the basic line integral is a weighted sum of infinitesimal vectors. The result is again a vector. Taking the length of ds complicates matters, and generates a big discrepancy with line integrals in the complex analysis.
Thanks a lot for this intuitive explanation. By the way, would you mind sharing what app you're using to write the notes? It looks really clean. Thanks in advance!
I've tried multiple videos to introduce me to multi-variable calculus and line integrals and all have failed except for this one, not one single concept not understood. Great video honestly as a student who's still in highschool and addicted to math it's really hard to find good videos about topics this complicated, which I get because less and less people get interested as the topics get harder and harder because of unfamiliarity and the lack of will to learn new concepts. Thanks a lot I hope you can keep making videos like these even though they probably won't do great views wise!
There;s lots. But a lot are follow the recipe and shake and bake, to get the answer. A cat can do some of that. If you;re just looking for 'how to get the answers' and pass high school math... and that sort of stuff... basically monkeymatics... Khan Academy does a decent job. For a much better presentation...3Blue1Brown, Grant... (former Khan Academy presenter, and the very best part of it) does a better job. But you can spend a lot of time to get to the level of understanding presented there. If you like geometry, and 'visual learning, 3Blue1Brown is the master of it... and one of the first to do it. For a more analytical approach, Dr. Perun is good.
Great video! It has been a while since I took courses on calculus and this was a great way to refresh my memory. I don't really love infinitesimals lol it would also have been nice to see the proof using limits
Yup, exactly! r(t) is the vector-valued function that contains x(t), y(t), z(t), ... etc as vector components, and the notation says to replace all x's in f(x,y,z) with x(t), all y's with y(t), etc
Thanks for the video. IMO... line integrals over a curve in... for instance R^2 or R^3... are 'busy work" 99% of people... outside math class... needing to do this (and who does?)... They are going to be doing it numerically.
Very true! Aside from maybe the arc length formula, I don't think I've never done a line integral of a regular curve in a non-math-class setting. Though I do think thoroughly understanding them is really important for understanding the more practically useful topics ... particularly with line integrals over vector fields
You should define a versor tangent to the integration line, namely, for example, tau hat, and define d\vec{s}=ds • \hat{\tau} and in the integral you would integrate the scalar product of the field and this vectorial displacement
Bam ! Great work bro but I have a doubt what does line integral refers to ? Like integral in single variable refers to area under curve and 2D ones refer to volume what does this(line integral) refers to ? and yep they will be different for different curve what is the relation among them for all the curves ?
Great question! Line integrals for 3D functions still refer to areas under a curve, except now image the curve is squiggly instead of straight. I mention that briefly around 10:15 in the video, and I think the image there is very helpful. In dimensions higher than 3D (>2 input variables), you need 4+ dimensions to graph your inputs vs your output, so it's hard to visualize line integrals at this level ... I would say the relationship between all line integrals, regardless of the curve, is that they are all just infinite sums of function outputs taken over 1D shapes (lines), as opposed to higher-dimensional integrals like surface or volume integrals which are taken over 2D or 3D shapes.
@@FoolishChemist ty ❤ I also took multivariable class watched,trefor bazzet but still was not able to recall this 😂 I think now I will be able to recall "domain expansion " 😊 good job...you should have choose maths instead of chemistry 😅
Don’t be!! It takes some time to understand, but if you really dig deep and try to understand calculus on a fundamental level, it will come much more easily!
@@FoolishChemist Just want to say youre the goat man. Im kind of in a love hate relationship with math, but clear overviews of concepts always excite me, and the 4 videos of yours ive watched on multivar calc did do just that. Application is what scares me the most but ive got time. Hoping i pass my current calc courses to happily get to them tho 😭
I seen another version of line integral where the output of the function themselves are vector. In this the formula I see is this integral f(r(t)) dot r'(t) dt. Can u explain the difference?
I think you might be thinking of line integrals over vector fields-vector fields can be thought of as functions that output a vector to each point in space, and yes you can do line integrals over them! (That's the topic of the next video) In this video, I was working with the line integral of f(r(t)) * ||r'(t)|| dt (not dot, dot product would be for line integrals of vector fields). It's just the line integral of a ordinary function (that takes in scalar values and outputs a scalar value, nothing to do with vectors itself) over some curve C, and we find it is convenient to express C as a vector-valued function (NOT the same thing as a vector field). Note that f(r(t)) still outputs a scalar, and we multiply by the magnitude of r'(t), which is also a scalar.
@@khiemgomI think effectively, yes! It just may be somewhat unintuitive to write d||r||, since that would refer to an infinitesimal change in magnitude of a vector-valued function which isn't easy to interpret visually.
DOMAIN EXPANSION MULTIVARIABLE 🙏👹
🥶🥶🥶
nah, i'd integrate
@@AyushKumar-md9ut wanted ti say that too
That was sick
A complicated concept is nicely explained. I loved the domain expansion :)
Glad you liked it!
@@a.kofficial6140 What a fantastic reference!
Where have u been when i first looked up line integrals this vid made the most sense, u earned a vote (:
В спешке проходили криволинейные интегралы в прошлом семестре. Решил глянуть и добить гештальт от вас узнал больше чем на парах. То как вы провоцируете мышление четко обозначая проблему, тем подводя нас к выводу формулы это невероятно!! :))
И не думал, что корни вместо дифференциациалов это эхо теоремы Пифагора
Да ничего совковые преподы объяснить не могут. Я от индусов на ютубе больше узнал чем за весь прошлый учебный год. Сейчас ВУЗ в России это просто отсрочка от армии, а не источник знаний.
Incredible video man. A million thank yous for going through those parametrisation steps so slowly and clearly!
Thank you! You have the ability to make math concepts clear. Not sure what it is exactly…the explanation is clear, without too much math lingo. Others explain the material, but for some reason do not ‘connect the dots’ and generalize the concepts. I see this in all of your math videos - not sure if it is deliberate, but I think it really helps when the basic concept is generalized to the more interesting and powerful ‘global’ concept. Examples: in ‘Essence of Multivariable’ you show how many of the vector calculus concepts boil down to just one formula. In the line integral video, you take the concept of parameterization, and generalize it to n dimensions (which I think is novel on TH-cam). This is super helpful. I personally really like this approach. But your video titles don’t describe your videos…so those searching for these general descriptions won’t find them. The Essence and Line Integral titles should say something eluding to the generality lurking within, otherwise people think they are just regular videos - which they are not. They are very special and unique, and deserve way more views! I realize you name them this way on purpose - so you don’t scare some viewers off, but maybe create another channel with more general titles which link to the same videos, or maybe use keywords to attract a broader audience? Judging from the comments, many could benefit from your talents, and you deserve more hits! Best of luck to you!
Thanks for kind words and the suggestion! I'll try implementing this and see how it improves viewership
With this treasure I summon, divine general Stokes!
In all seriousness though, nice vid. Line integrals are weird, and it's hard for some to understand how you transform this formulation into a method suitable for integrating over vector fields.
This video's editing was legendary. Please make more videos I beg you
In my first year of ug rn and the gid was soo good that I was able to keep up until the vector valued functions! Great video 👏🏻
This is the best video on line integrals in the Internet right now!
from Morocco thank you very much
This is so easy to understand!!!
I wish I had you as my teacher when I took my calculus courses in school
Ridiculously good video. I had been searching for exactly this
Awesome video! Thank you!!!
Cool! Now, try a flow integral, which is defined as a line integral that involves a path through a vector field, which are usually marked as ∮ F ∙ dr, or ∫(F ∙ t)ds. Also, this is an example of a work integral for a force field. For the same force field, a similar formula exists for the magnitude of torque in a 2D vector field with ∫(F ∙ n)ds.
Next video already in the making 🫡
Which one to choose itegral or differential.
amazing video! what’s that app you used to write in your ipad??
Penbook!
I think the basic line integral is a weighted sum of infinitesimal vectors. The result is again a vector. Taking the length of ds complicates matters, and generates a big discrepancy with line integrals in the complex analysis.
incredible explanation.
Pain with extra steps; I love it 👍🏼
Thanks a lot for this intuitive explanation. By the way, would you mind sharing what app you're using to write the notes? It looks really clean. Thanks in advance!
Penbook on iPad!
I've tried multiple videos to introduce me to multi-variable calculus and line integrals and all have failed except for this one, not one single concept not understood. Great video honestly as a student who's still in highschool and addicted to math it's really hard to find good videos about topics this complicated, which I get because less and less people get interested as the topics get harder and harder because of unfamiliarity and the lack of will to learn new concepts. Thanks a lot I hope you can keep making videos like these even though they probably won't do great views wise!
There;s lots. But a lot are follow the recipe and shake and bake, to get the answer. A cat can do some of that. If you;re just looking for 'how to get the answers' and pass high school math... and that sort of stuff... basically monkeymatics... Khan Academy does a decent job. For a much better presentation...3Blue1Brown, Grant... (former Khan Academy presenter, and the very best part of it) does a better job. But you can spend a lot of time to get to the level of understanding presented there. If you like geometry, and 'visual learning, 3Blue1Brown is the master of it... and one of the first to do it. For a more analytical approach, Dr. Perun is good.
Dr Trefor Bazett made a playlist on vector calculus.
I enjoyed watching it.
What app do you use for your notes?
However, good and clear video!
Pen book on iPad!
@@FoolishChemist Thank you!
you just earned a subscriber dear friend
Please make a video on surface integrals next!!! I love the way you explain
Love your video style. Keep it up!
this guy is good
Great video! It has been a while since I took courses on calculus and this was a great way to refresh my memory. I don't really love infinitesimals lol it would also have been nice to see the proof using limits
7:10 mind blown.
ikr???
I just skimmed through the video but I think ur good at explaining stuff 😊
Please do path integrals next :)
A path integral is not a distinct thing.
@@epicchocolate1866 you mean Feynman's path integral in quantum field theory is not distinct?
How did you get ti + t^3j from y=x^3? How is y=x ti+tj and not -ti+tj?
Bro use the scaler method of S(dQ/dx-dP/dy)D integration limits are 1 to 3 choose x or y direction
Bro you rocked
That HDR effect😂 shined through my eyes
To expand the domain wouldn't you need f(x,y) not f(x)?
Genius.
Great video keep it up man
Hello, what app do you use for notes?
Penbook on iPad!
Officially new favorite channel. Wide Putin 😂
Easy and clear sir,,,,will you please also clear doubts on surface integral and volume integral
Will do! Those are coming up soon ✍
You deserve more views 👏
underrated
Hello, someone say me which software have used in the video to write the mathematical expressions?
I love multivariable calculus!
Loved the video thanks!
thanks bro
just curious, when you have f(r(t)) written, does that simply just equal f(x(t),y(t),z(t),…) for all variables involved?
Yup, exactly! r(t) is the vector-valued function that contains x(t), y(t), z(t), ... etc as vector components, and the notation says to replace all x's in f(x,y,z) with x(t), all y's with y(t), etc
@@FoolishChemist There are also vector fields such as F(x, y) = .
Love ur vids❤
this is cool
Thanks for the video. IMO... line integrals over a curve in... for instance R^2 or R^3... are 'busy work" 99% of people... outside math class... needing to do this (and who does?)... They are going to be doing it numerically.
Very true! Aside from maybe the arc length formula, I don't think I've never done a line integral of a regular curve in a non-math-class setting. Though I do think thoroughly understanding them is really important for understanding the more practically useful topics ... particularly with line integrals over vector fields
Multivariable > Malevolent Shrine imo
Why only you can be clear 😭😭
12:46 i feared youd say parametrisation…
Such a good video 😂
Bro gimme that music at the domain expansion
But what do we do in a case with F à vector field?
You should define a versor tangent to the integration line, namely, for example, tau hat, and define d\vec{s}=ds • \hat{\tau} and in the integral you would integrate the scalar product of the field and this vectorial displacement
The function x^2 + y^2 would be f(x,y), not f(x).
Bam ! Great work bro but I have a doubt what does line integral refers to ? Like integral in single variable refers to area under curve and 2D ones refer to volume what does this(line integral) refers to ? and yep they will be different for different curve what is the relation among them for all the curves ?
Great question! Line integrals for 3D functions still refer to areas under a curve, except now image the curve is squiggly instead of straight. I mention that briefly around 10:15 in the video, and I think the image there is very helpful. In dimensions higher than 3D (>2 input variables), you need 4+ dimensions to graph your inputs vs your output, so it's hard to visualize line integrals at this level ... I would say the relationship between all line integrals, regardless of the curve, is that they are all just infinite sums of function outputs taken over 1D shapes (lines), as opposed to higher-dimensional integrals like surface or volume integrals which are taken over 2D or 3D shapes.
@@FoolishChemist ty ❤ I also took multivariable class watched,trefor bazzet but still was not able to recall this 😂 I think now I will be able to recall "domain expansion " 😊 good job...you should have choose maths instead of chemistry 😅
Bro precalculus is a different thing from multivariable calculus what u on??
You do not see f(x,y) in highschool bro
Seems the Mandela effect got to me… 😂
nah bro, new sub
david schwimmer looks so young here
Haven't gotten this one before but thanks! 😊
0:33 That hurt me 😢
Ill be doing these in 3 months approx and im scared shitless tf 😭
Don’t be!! It takes some time to understand, but if you really dig deep and try to understand calculus on a fundamental level, it will come much more easily!
@@FoolishChemist Just want to say youre the goat man. Im kind of in a love hate relationship with math, but clear overviews of concepts always excite me, and the 4 videos of yours ive watched on multivar calc did do just that. Application is what scares me the most but ive got time. Hoping i pass my current calc courses to happily get to them tho 😭
me watching videos abt like integrals when i dont even have practice with the rules for 1D integration👁👄👁
Awesome bro, great vid.
Your humor annoyed me tbh, but its s good video
GAS
Nah I'd win😂
u cooked
Actually y is just a constant, since your function, f(x), is just a function of one independent variable x.
😂😂😂😂THESE SHOW UR LEVEL....FIRST THEOREM...ACTUAL MATH KAA AISA HII HOTA😂😂
You kinda lost me after the minute 16:00 got too hard
sir please make a video on surface integral
just a clown
Too much circus. I understand, you are very young, but a lot of time is spent on exhibitions.
Boomer detected, opinion rejected
I seen another version of line integral where the output of the function themselves are vector. In this the formula I see is this integral f(r(t)) dot r'(t) dt. Can u explain the difference?
I think you might be thinking of line integrals over vector fields-vector fields can be thought of as functions that output a vector to each point in space, and yes you can do line integrals over them! (That's the topic of the next video) In this video, I was working with the line integral of f(r(t)) * ||r'(t)|| dt (not dot, dot product would be for line integrals of vector fields). It's just the line integral of a ordinary function (that takes in scalar values and outputs a scalar value, nothing to do with vectors itself) over some curve C, and we find it is convenient to express C as a vector-valued function (NOT the same thing as a vector field). Note that f(r(t)) still outputs a scalar, and we multiply by the magnitude of r'(t), which is also a scalar.
@@FoolishChemist so this is actually f(r) d||r|| right
@@khiemgomI think effectively, yes! It just may be somewhat unintuitive to write d||r||, since that would refer to an infinitesimal change in magnitude of a vector-valued function which isn't easy to interpret visually.
@@FoolishChemist actually ||dr|| i think, my mistake, but yeah, it help to learn the difference
Love your video style. Keep it up!