What are double integrals? What are they for? |

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Hi! This is the video that took me one month to create. I'm so proud of it. Forgive me if you encounter any grammatical errors, this is my first video completely in English :(
#3b1b launched this challenge and I wanted to throw myself into explaining one of the most beautiful topics that fascinated me: double integrals.
I hope I have given them the honor they deserve, given their extreme beauty and practicality.
For any questions and advices, feel free to write me on the comments!
This is the official submission for the contest Summer of Math Exposition (SoME1)
Contacts: [e-mail] / [e-mail]
Code: github.com/EserClub/3b1b-cont...
0:00 Introduction - a curious practical problem
0:34 Introduction - What is a Double Integral?
3:31 Subdivision of Domains
4:00 Rectangular Domain
5:38 1st numerical example
7:22 Domain bounded by 2 functions
8:20 2nd numerical example
9:14 Change of Coordinates
11:25 The Jacobian
12:52 The answer to the practical problem
15:04 Greetings

[@3blue1brown]

This is such a good introduction to double integrals, with really well-chosen examples. I can see this being helpful to many students.

[@ClearMath1]

Thank you so much. This feedback is really valuable to me, i will do my best to continue these calculus lessons!

[@mohitpandey4655]

Sir big fan 🖤

[@bigaaron]

Its sweet you give other channels feedback. Good on you man

[@beatboxanimationnandan]

Big fan

[@paufranquet1906]

I searched "multiple integrals 3blue1brown" and came across this video. It was what I was looking for.

[@Death-zc7de]

I'm glad I found the Mario that teaches calculus

[@sleepycritical6950]

I learned how to square a circle

[@sinasoltan.m4859]

Please keep making such videos in English so a wider range of learners can use your videos

[@ClearMath1]

Thanks! This feedback Is very valuable to me! My project is to bring a full multivariable calculus course made with Manim, and the next video will probably be about optimization.

[@guyedwards22]

I also find it super helpful to think of common Jacobians geometrically so you don't have to remember it strictly all of the time. For example, "dxdy" is the area of an infinitesimal rectangle since the x and y bases are unit and orthogonal everywhere on the plane. "rdrdtheta" is the area of an infinitesimal sector of annulus since "dr" is the same size no matter where you are in the plane, but "dtheta" scales linearly with distance from the center, so its actual contribution to the area is "rdtheta". I remember learning this approach in an Electrodynamics textbook that treated infinitesimal Area and Volume elements as "dl1dl2dl3..." and figuring out what usually comes out of analyzing Jacobians amounted to figuring out how each of the "dl" length elements related to each of the coordinates geometrically. Calculating the Jacobian by hand is needed for more complicated or nonstandard transformations for sure, but a lot of the time, it's a little quicker to just draw how everything scales with the coordinates in your head.

[@ClearMath1]

DISCLAIMER: i just noticed that at 13:36 i said the density is a _-"depending-position density"-_ , while i obviously meant *_"position-dependent density"_* . Sorry for my bad english; as you have probably noticed from the accent, i'm italian, and this is my first video completely in english. I promise that i will improve my vocabulary to bring you increasingly clear and quality content. If there are other words or phrases unclear, feel free to ask me anything on the comments!

[@RisetotheEquation]

No need to apologize; your English is stellar, just like the video. Congratulations on this excellent work. Check out my entry where I attack the Basel Problem using Cauchy's method.

[@ClearMath1]

@@RisetotheEquation thank you!

[@qzwxecrvtbynumi256]

I think there is a mistake on 12:50, the dtheta and drho should be swapped. Keep up the good work)

[@philipbarthelma1903]

This was incredible! I really liked the pacing as well as the animations. Your Visuals really helped to get an intuitive feel for double integrals.

[@MasterHigure]

Awesome video! Love that you pointed out that multiple integrals are often better thought of as repeated / nested integrals, taking care of one dimension at a time. That's something I wish I had been told when I started out with integration in higher dimensions.

[@real_michael]

I'm so thankful I found this channel before I start cal 3. This video was awesome!

[@TheActualTed]

Wow, this is extremely well done. Great job!

[@kim15742]

Yes! Please make a video on the Jacobian. This is a great way to learn!

[@tamp_]

Thank you for such a comprehensive and easy to understand video!

[@junkaccount4926]

What a beautiful video! A beautiful, intuitive explanation combined with a comprehensible rigor makes this video awesome. Keep up the excellent work, I look forward to more excellent math videos!

[@ClearMath1]

thank you so much! stay tuned for the next video about optimization!

[@NiranjanND]

Perfect video with perfect visuals! informative

[@punditgi]

Beautiful explanation!

[@zaydmohammed6805]

This is super helpful especially for someone like me who wants to learn multivariable calc but can't choose that course in college, thank u for this

[@DynestiGTI]

Love the animations, love the topic, love the accent. Instant sub

[@alessandrofrancescato2877]

Veramente bello, sei un grande!!!

[@ClearMath1]

grazie mille! Sono contento ti sia piaciuto. Ci ho messo veramente tanto impegno, e ora sto usando queste e tante altre animazioni che ho fatto per creare corsi di Analisi 1 e 2. Non vedo l'ora di pubblicarli e farveli vedere!

[@leventnergiz]

Such a beautiful explanation with great visuals! I looked for other videos in English. Unfortunately there are not much.

[@nathansidibakari]

Excellent video, from the mathematical approach (spatial volumes) to the more physical view (the disc example). It is really useful as an introduction to double integrals as it gives a solid and easy to understand foundation. I used it as a supplement to my undergrad maths course and it made everything clearer thanks to the animations and visuals. Thank you very much and once again, excellent work!

[@ClearMath1]

Wow i'm glad it was useful! I'll surely bring more contents and animations to this channel, stay tuned!

[@prasadborse2161]

Hats off 🙌 that's the most beautiful video on double integrals on youtube

[@ClearMath1]

wow thank you! glad you enjoyed it!

[@thebread8918]

amazing work, I love it, please keep uploading this videos :)

[@ClearMath1]

Thanks! I uploaded much more videos like this, even though they are in Italian. You can still turn subtitles on and enjoy the animations! Here's the playlist: kzbin.info/aero/PL4tHcCynIz4BfVqvciaKTRHd4GHfEkMAD

[@omanshsharma6796]

thanks a lot for such a beautiful video :>

[@davidemoretti7008]

Very good explaination. You are the paladin of math students!

[@ClearMath1]

This video is the lay on hands of calculus

[@vik0_052]

I LOVE YOU BROOOOOOOOO😭😭

[@shubhamg9495]

My first video to learn calc 2 starts from you. Nice!

[@ClearMath1]

I'm glad to hear that! If you are curious about It, your second video may be the last i uploaded, about Optimization, Gradient and Hessian

[@Derpuwolf]

Amazing video. I understood everything so well and my furthest math is early Calc 2. I've never integrated for volumes before and it seems so beautiful.

[@ClearMath1]

Thank you so much for this comment! This is exactly what i want to do with this video, to show how beautiful multivariable calculus Is and tho help dealing with your exercises. Stay tooned for the next video that will arrive very soon

[@Derpuwolf]

@@ClearMath1 I can't wait. I'm so exited!

[@donlansdonlans3363]

You've earned a new subscriber :D

[@amritawasthi7030]

This is really a very good video as an introduction. Also I'd like to you to do a vid on jacobian as soon as possible because I can not wait to watch it in a summary as u did for double integrals. More success to you 🥃

[@ClearMath1]

thanks! just yesterday i uploaded a video about Optimization. If you are curious about it, take a look. Among the next videos I will definitely do one on a Jacobian. Thanks again!

[@youtubepooppismo5284]

Bellissima spiegazione

[@fryingpanm6977]

thanks man it help me a lot

[@siddhantpathak9405]

11:24 is Pure MathGasm.......... Thank You so much for the brilliant video !! Lots of Love from India

[@ClearMath1]

ahahahahah thank you!

[@physicspeople6163]

Amazing video.

[@amitshoval7653]

Great video, also your voice is relaxing

[@nalayak862]

wow, this is super helpful

[@SimonTheDankOne]

This is very cool.

[@valeriodestefano3784]

Italian accent is so much better than indian accent! (#joke)

[@satyampanchal-1016]

brilliant

[@greatfate]

keep it up man! Love your Italian accent

[@ClearMath1]

Thank you! the world needs a super mario that explains calculus

[@carlosraventosprieto2065]

incredibly nice video!!!! thank you so much

[@beatboxanimationnandan]

Your videos will be very popular

[@awesomeacademy4454]

that’s a super explanation and some serious manim skills! Is it possible for you to publish your source codes for other manim learners?

[@ClearMath1]

thanks! i would gladly share it but i realized i don't know how to do it on reddit. Do you know how can i do it, aside copy it and paste it on a post?

[@awesomeacademy4454]

@@ClearMath1 hey, I don't even know whether you can publish something on reddit other than as posts :D, is it possible for you to publish your codes on github? It will be more meaningful to do it on github when you want to keep making such great videos. You can then paste the link of the github site below the videos so others can find it :).

[@ClearMath1]

@@awesomeacademy4454 thank you! i created a github account and now you can find the code on the description. If you need anything else, let me know!

[@aame6643]

I wish I had this video when I was taking calc 2

[@NovaWarrior77]

very nice!

[@ClearMath1]

Thanks!

[@-o-1793]

Using centrifugal casting a disc/cylinders of varying density can be casted. Premise: Every particle in the centrifuge will experience centrifugal force [ (dm)rw^2 ] i.e., particles away from the center will experience more force compared to the particles which are nearer to the center and because of this reason density of the disc will be more in the outer periphery, hence obtaining a non homogeneous disc of varying density as shown in the video.

[@jursamaj]

Liquids don't typically change that much in density. Rather, You will end up with a curved surface, with more of the mass piled against the outer wall of the centrifuge.

[@01k]

Good video

[@sivateja8632]

Thank u veru much sir ❤

[@rown3499]

Bellissimo

[@ClearMath1]

Grazie!

[@AmitKumar-xw5gp]

Quite good explanation.. And brilliant animation. Must have taken ages to render in manim.!! Could you please share the code.

[@ClearMath1]

Hi! Sorry if i write to you only now. Thanks for your compliments! I'm still on vacation and i left the computer home, so i am unable to share the code. When i return home, i'll share It to the Manin community. But i warn you..it's very, veeery messy

[@AmitKumar-xw5gp]

@@ClearMath1 I'm glad you are willing to share your code.

[@ClearMath1]

@@AmitKumar-xw5gp hi! i just wanted to inform you that you can now find the code in the link on the description. if you need anything else, let me know!

[@miguelramirez187]

i love you

[@Dziaji]

If you want to transform a circle into something easy to integrate over, then use a triangle of height R and base 2piR. The middle of the circle translates to the top of the triangle, and the base of the triangle is the circumference of the circle. If you do that, and treat the relative mass as relative height at each point on the triangle, the answer can be reached by a super simple double integral. The shape of the solid is such a geometrically simple, that i bet there is even a simple formula for it. It would basically be an irregular, square-based-pyramid.

[@Dziaji]

Oh nevermind, the shape would be different. I was thinking as if the center was the least dense. It would actually be an irregular tetrahedron... i think. Something like that anyway.

[@Dziaji]

The point is that it is easy to integrate over compared to a circle or polar coordinate rectangle.

[@JokerSpeaker]

Could you please tell me what software is used to make those animations

[@ClearMath1]

Hi! i'm using manim, an engine designed for creating explanatory math videos. You can find any information to begin coding with this tool on the reddit communiy (search r/manim) or by watching the channel "Theorem Of Beethoven"'s tutorial. I am super satisfied with this program, if you want to make math videos, it's the program you are looking for!

[@donaldhobson8873]

How do you really find the volumes? Programming.

[@kimsanov]

What app are you use for such beautiful animations?

[@ClearMath1]

Hi! i'm using manim, an engine designed for creating explanatory math videos. You can find any information to begin coding with this tool on the reddit communiy (search r/manim) or by watching the channel "Theorem Of Beethoven"'s tutorial. I am super satisfied with this program, if you want to make math videos, it's the program you are looking for!

[@kimsanov]

@@ClearMath1 Amazing! Thanks a lot for clarifying)

[@spiritmoon3457]

How you create such beautiful videos?

[@tulliolevicivita4443]

How have you learnt Manim?? Great video!!

[@ClearMath1]

thanks! i suggest to watch TheoremOfBeethoven's youtube tutorials. i learnt from them. Also have a look on the subreddit r/manim, it's super useful. Thanks againdor this comment. If you are curious about it, you may want to take a look to the video i uploaded yesterday, about Optimization.

[@Anujkumar-my1wi]

Hey ,I've been meaning you ask about derivative , derivative are best linear approximation around local neigbourhood of x (x+dx,x-dx) ,I want to ask what is the derivative at where these local linearities meet I mean if derivatives are ,as you said ,linear approximations then ,If we take two consecutive derivative as best linear approximation then there should be a point where both linearites meet and it could at only one point (as two linerities with different can meet each other at most 1 point) thus I want to know what is the derivative at the poin t of intersection and that point of intersection should be at function as if we have derivative as linear approximation around a point and the two consecuitive lineaarites must meet at the intersection of two neigbourhood, Thus I think that deriative should be piecewise at the point of intersection but it isn't why? To put it simply as derivatives as describes as linear approximation around the neigbhourhood of points ,I want to know what is derivative at the point of intersection of these two neighbourhood of points ,I mean sure some neigbhourhood must have equal linear approximation but some neighbourhood of point in continuous function can have different linear approximation ,then shouldn't be the second derivative of the functon at the point of the intersection be undefined?

[@ClearMath1]

hi! i prefer to discuss about things that concerns the topic of the video while we are on the comment section, please send me an email to the address you can find on the description, and i will do my best to answer you!

[@Anujkumar-my1wi]

@@ClearMath1 Ok ,thanks for support.

[@Anujkumar-my1wi]

Hey, Nice explanation ,I am slightly confused with the definition of limits ,I mean in general its pretty clear but my confusion is why x=a case isn't considered in the definition of limit ,I mean ,does the value of function at x=a doesn't affect the limit value ,If so then can you explain why the value of function at x=a doesn't affect the value of limit as x approaches a?

[@ClearMath1]

Hi! I'm not sure i understood what you are asking. Could you tell me what minute of the video you are referring to? I don't remember talking about limits on this video.

[@ClearMath1]

if your question is a general doubt about another topic, i assure you i'll explain limits in another video. Stay tooned!

[@Anujkumar-my1wi]

@@ClearMath1 I appreciate it but you see that I have to understand this concept of limits in this week due to a test, I would appreciate if you can clear my doubt ,I want to know why the the case x = a in limit is not considered, By not considered ,I mean that in the definition of definition we evaluate the value of limit as x approaches a but we never consider the value of function at x = a?

[@asrafialmamun5767]

Would you please inform which software you are using for animation?

[@ClearMath1]

hi! i'm using Manim, a python library created by 3b1b. There's a whole community where you can ask anything about this program, i've learnt anything i know from them!

[@syedarslanalishah6905]

How much knowledge of programming (especially python) should one know to produce such animations?

[@ClearMath1]

Not much really, i had a very Little knowledge of C when i started programming with Manim and i didn't even know what python was. I began to use it the first time about 6 months ago and i uploadedthis video 2 months ago. So in 4 months you can have pretty decent results. I strongly recommend to Watch TheoremofBeethoven's tutorial videos to learn how Manim works, nd of course 3b1b videos to get more creative with the animations.

[@subtlethingsinlife]

My god …. Why didn’t I know all of this earlier

[@ClearMath1]

Thanks! these comments are the fuel to create more contents! You may want to have a look to my other video, that i uploaded yesterday. It's about Optimization, gradient and Hessian. Hope you like it!

[@duckymomo7935]

@1:23 I think there is a notation error? lol

[@21.wildanyuniartha22]

what editing software do you used?

[@wanmanchannel189]

What program did you create?

[@flix7280]

Please keep making videos In English

[@ClearMath1]

Thanks! I Will!

[@arpitgaur4310]

But how that graph was 2d because it seems to be 3d

[@SuperMaDBrothers]

Nice job! But I don’t know anyone who would want to learn about double integrals and doesn’t already know about double integrals

[@alitalalhaidar6488]

In some parts of the video, it shows a 3D diagram without rotating it. Please rotate every 3D diagram to show it from different perspectives before you start explaining it

[@trenza2566]

i bet my ass that you are italian

[@cosimo7770]

Disgraceful that there is constant background 'music' i.e. noise. The reason is that the presenter regards his audience with disdain, thinks they they are stupid, and that they must be kept alert by background noise. The sure signature of an inferior teacher.

[@heavymetalslime]

Did someone's calculus exam didn't go well? I understand the frustration, but blaming it on the teacher is not always right, especially when you blame it on the teacher's background music. His video is just fine, and you know it. Good luck on your next exam in September.