Divergence and curl: The language of Maxwell's Imaginary Numbers Are Real [Part 2 months 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

Special thanks to these supporters:

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:
Gabriel Frankford
I came for just a fun reminder of things I already knew but end up learning something about fluid flow.
Ray Xi
can you make a donation link? I don't know how to use patreon, it will be easier for the viewers if you have a simpler way of donation.
Ananthanarayanan R
Had this lecture at math class recently . Really wanted to understand better.Thank you for the video , always helps much :)
Donnie Dorko
Dude so good on you for making sure your visuals work at low resolutions. Thank you
Wahyu Guntara
Khalid Khattak
I really like the way you speak, i hate off for your knowledge on the subject. I am proud to be the part of your channel. Thanks bro
I love the insights and "tying things together" of your stuff. And the animated graphics. But at 10:00 you say "the [electric field] fluid is flowing incompressibly." If you look at your diagram, you will see it is actually becoming much less dense as it "flows" away from the positive charge, and it becomes more dense (compresses) as it "flows" toward the negative charge. Water, being incompressible, could not flow like that.
Aneesh Prasobhan
The amount of work that went into each of these videos are mind boggling. You videos will help many present and future engineers and scientists to understand the concept.
James M
Thank you so much for this video! I’m going through an online college course for vector calculus (I’m a high school senior) and this has helped me understand this so much better.
You are a hero sir. Know this.
Lars Hoernle
Shamim Yeasir Saion
Dude, you might just be the greatest explainer since Feynman!!
El Físico Sí Importa
Great explanation. Thank you!
Joe Deglman
The ether medium is literally a EM fluid. Everything acts just like it is moving through an EM fluid. That explains gravity, transformers, magnetic fields, etc.. Magnetic fields act just like a fluid around moving charged objects. "The earth is like a charged metal ball moving through space." "There is an ether medium that pervades the universe, like particles of air."
So if you have a marked increase of current flow onto a star, it cause an outflow of magnetic field from the star, thus stripping the gases off and away from the core, like a supernova. A double layer then is caused by opposite vortex flows due to differently charged bodies moving the same direction through space, thus providing a force keeping an electron out of the nucleus.
paul van Lingen
Thank you for this video!
Jeremy Mettler
Or a fluid. Makes the eye where magnatizim needs I to become eye to go around kinda the bunny and the rabbit equal the hair in and of a bugs chasing a porky is a pigof noise combined with a coyote and see a runner they become as eye of in I are self Know self hear true see just beaty in ears of truths understanding bunny rabbit making the hare jump the coyote Dodge the bird as a many in one Rabbit hare some cancel function of or add to run with the pig of porky listening for them while he hunts bunny
Wisam Doghoz
so i have been thinking ..... we can think of the positive charges as the sources of the electric field and the negative charges as the sinks ... and when the field sinks in one negative charge it does not vanish from existance but it reapears from a diffrent chrage a positive one and because of conservation of charge we could say that there is some spooky action that connct all electic charges in the whole universe and as if it transers the information instantly. but then I start confusing myself with another question ...
what if we use a reciprocal llatice to describe everytthing, the negative charges might be thought of as sources instead ... right? what would our laws lok like if we reverse the dimesions ?
Kamaljeet Singh
A big thank.
Zubair Khan
Really a very great job done by 3Blue1Brown...
One day this will the top channel on YouTube
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Imaginary Numbers Are Real [Part Divergence and curl: The language of Maxwell's 2 months 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.