Divergence and curl: The language of Maxwell's Imaginary Numbers Are Real [Part 2 days ago   19:59

Intuitions for divergence and curl, and where they come up in physics.
Thoughts on going sponsor-free: https://www.patreon.com/posts/19586800

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My former work on divergence and curl at Khan Academy:

For more fun fluid-flow illustrations, which heavily influenced how I animated this video, I think you'll really enjoy this site:

Music by Vincent Rubinetti:


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Comments 1649 Comments

I put this in the description, but let me just link this field flow interactive here because I think you all will like it so much:
Steven Yoon
oh my god...
Vipul Ram
you are a master of mathematical visualization. As an engineer I am in awe of the graphics you are creating to explain these topics.
Kilian Dietrich
I was craving for something like this during my phyisics studies... made it through anyways, but I never understood it as you did. YOU ARE A GOD, and I am not saying this easily :/ Kudos!
Marcos Pimentel
I wish i could understand all of that but im just in the first year of highschool😅
Ignacio Córdova
Who is this man?
Martin Johnsons
You clearly deserve a Nobel.
Ahmed Zribi
Your very smart. Very clear. I'm doing my masters in applied science/electrical engineering, photonics and EM and this video made my vision much more clear. good job.
Michael Wang
Where is the next video?
Anna Roddick
We were taught this in our uni and when we were taking down the notes, everything (thanks to our professor) sounded like greek (which I don't know). Took a long time for us to get accustomed to the concept. Wish this video was there back then. :(
bhuvan vemula
how do you do this type of video i mean the animation.
looks like Windyty
Great that ypu removed sponsor advertisung, and great how you visualize what you explain. Every student should see such animations from their teacher.
Madina Atymkhanova
THANKYOU very much!!
love this channel!
Div F curl F grad F
Etam Levy
it is amazing how many professors dont rly understand these things well enough in order to explain it simply like u do.
not that i underestimate my professors. it is just the i have a lot of respect to what you do.
in my Uni everybody watches your videos (we are not from america ;) ), everyone that does elgebra, calculus, physics etc in different lvls.
the name "3blue1brown" is known in the halls.

in addition for that i highly appreciate how candid you are. im a poor stundent (arent we all? ) who ows you a lot and the "donation" (i would like to call it a very cheap price for a mountain of knowledge) doesnt represent how grateful i am for these explanations.

if you cant explain it simply enough that means you dont know it well enough - some very smart guy ;)

well sir you do know it well enough.

thx from students all around the world!

ps - i passed in good grades some of my exams thx to you and i will not forget it. when the time comes that i will be able to i will "donate" more. i know thats not the point of your videos but that is the only way i can express my gratitude.
Vittorio D
I love your channel!! Which software are you using to make all these stuff i'd like to know it to play around with it
Daniel Arco
It's so interesting.Thanks
Elie Touboul
When the next video ?
Great content
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Imaginary Numbers Are Real [Part Divergence and curl: The language of Maxwell's 2 days ago   05:47

More information and resources: http://www.welchlabs.com

Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.

Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces

Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.