Divergence and curl: The language of Maxwell's Imaginary Numbers Are Real [Part 5 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:
Impossible Physix
The first Example looks for me like the Bend of space time because of matter
Very cool video!
Warning fro beginners: the divergence is intuitively described as being +1 at an electric charge; more accurately that is the flux through the little sphere; this is constant for each radius, and by Gauss' law is the integral of the divergence, but the volume goes to 0 as the radius decreases, whence the div AT that point is infinite. It's like a Dirac delta, a point mass.
Houssam Assila
The use of bold and normal characters to specify difference between vector and scalar is confusing!!! who's the lunatic autistic who thought of that?! please make arrows great again.
Mario De Sio
Nice video, I'm looking forward to meet you in real person!
Thiago Lorente
What is the name of the program that you use to simulate math equations and get vector images?
Your videos are amazing, really respect how simple you make complex mathematical concepts. Basically gravity is nothing but the the result of electric and electrostatic forces acting on larger scales(just like inter-atomic forces that are stronger because from our point of view they are applied on a much smaller scale). I feel like scale is something that is not taken into consideration when comparing time dilation of atomic particles and inter atomic forces.
Fulvio Cerlesi
Bravo! Complimenti per la chiarezza e soprattutto per il fine educativo. Good job indeed! Congratulations on clarity and, above all, for the educational purpose of this work.
Tuaigets DJ
Great video👏👏
vivek Mahajan
Very nice video thank you for the help
veera J
Thanks for your amazing videos...where is next video?
Little Narwhal
As soon as im more independent with my money i will become a patron. As a relatively young amateur mathematician your videos help me so much, and even though I didn't fully understand this one, I love your content and can recognize that you are great at explaining math. I wish that one day ill understand all these topics as well as you so that I may solve my own problems as well!
Granite Harbor Merchants
Although https://anvaka.github.io/fieldplay/ is a beautiful page, It could not have been used to make these animations. Specifically how do you animate what the vector field is doing to particles?
Such clear explanation and great graphics, subbed!
Suyash K Kishore
why is this video not monetized???
Joshua Tilley
How about a follow up on differential forms, and how they unify these notions?
Lemon Dragon
Hey 3B1B, love the videos! I was just wondering what you meant by “incompressible”? Thank you!
You know what, i don't know about all other ads but i pretty much bought everything you advertised about, 'art of learning', 'zen, the art of motorcycle maintenance' and 'brilliant'
Louis Thielman
Where is the next video? Thanks for the great content!
Colin Byerly
Thank you !
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Imaginary Numbers Are Real [Part Divergence and curl: The language of Maxwell's 5 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.