Line Integrals Are Simpler Than You Think

#maths #calculus #multivariable #multivariablecalculus #perspective #some #someπ #learn #learning #intuition #intuitive

In this video, I introduce the concept of a line integral from multivariable calculus by building it up intuitively from integration in single-variable calculus. Math is all about expanding previous knowledge into new domains, after all! I also try to present the intuition behind line integrals, and finally demonstrate a practical calculation of a line integral for an ordinary function: z = x^2 + y^2, over line segment y = x extending from (1,1) to (3,3).

Enjoy!

Chapters:
0:00 Intro
1:48 Prerequisites
2:09 Video Outline
2:33 Integration in Single-Variable Calculus
5:27 Line Integrals - Intuition
10:33 Line Integrals - How To Calculate
17:42 Line Integrals - Example Calculation
19:24 Side Note

Feel free to reach out to me at:

[rezaa]

A complicated concept is nicely explained. I loved the domain expansion :)

[abbasfadhil]

Where have u been when i first looked up line integrals this vid made the most sense, u earned a vote (:

[copywright]

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.

[alsfiend]

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

[EjayB]

Incredible video man. A million thank yous for going through those parametrisation steps so slowly and clearly!

[polarjsapkota]

This is the best video on line integrals in the Internet right now!

[ym-xxkj]

This video's editing was legendary. Please make more videos I beg you

[johnstuder]

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 YouTube). 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!

[Professional-Hater]

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 👏🏻

[monkeychicken]

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!

[saumitrachakravarty]

I wish I had you as my teacher when I took my calculus courses in school

[ZaeemAhmad]

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!

[caioesteves]

amazing video! what’s that app you used to write in your ipad??

[fireballman]

Ridiculously good video. I had been searching for exactly this

[AlbertTheGamer-gksn]

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.

[HamzaAli-hhub]

you just earned a subscriber dear friend

[alejrandom]

I just skimmed through the video but I think ur good at explaining stuff 😊

[danmiller]

Please make a video on surface integrals next!!! I love the way you explain

[cronos]

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

[antoniocicchella]

What app do you use for your notes?
However, good and clear video!