What are double integrals? What are they for? | #SoME1 #3b1b

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)

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

[bluebrown]

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

[Death-zcde]

I'm glad I found the Mario that teaches calculus

[ClearMath]

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!

[sinasoltan.m]

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

[guyedwards]

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.

[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.

[prasadborse]

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

[philipbarthelma]

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.

[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.

[siddhantpathak]

11:24 is Pure Thank You so much for the brilliant video !!

Lots of Love from India <3 <3

[-o-]

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.

[awesomeacademy]

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

[kim]

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

[junkaccount]

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!

[real_michael]

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

[davidemoretti]

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

[zaydmohammed]

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

[NiranjanND]

Perfect video with perfect visuals! informative

[amritawasthi]

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 🥃

[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!