2025-10-12

[public] 296K views, 16.5K likes, dislikes audio only

3Blue1Brown

Visualizing the most important tool for differential equations.

Previous chapter: /youtube/video/-j8PzkZ70Lg

Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support

An equally valuable form of support is to simply share the videos.

Home page: https://www.3blue1brown.com

Artwork by Kurt Bruns

Engine animation borrowed with permission from this (excellent) blog: https://ciechanow.ski/internal-combustion-engine/

Timestamps:

0:00 - Understanding the engine

1:16 - Key background ideas

5:41 - Definition and intuition

10:43 - Complex integration

20:43 - Analytic continuation

23:52 - The transform of exponentials

26:15 - A deep look at cos(t)

32:59 - What’s coming next

------------------

These animations are largely made using a custom Python library, manim. See the FAQ comments here:

https://3b1b.co/faq#manim

Music by Vincent Rubinetti.

https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

------------------

3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly.

Mailing list: https://3blue1brown.substack.com

Twitter: https://twitter.com/3blue1brown

Bluesky: https://bsky.app/profile/3blue1brown.com

Instagram: https://www.instagram.com/3blue1brown

Reddit: https://www.reddit.com/r/3blue1brown

Facebook: https://www.facebook.com/3blue1brown

Patreon: https://patreon.com/3blue1brown

Website: https://www.3blue1brown.com