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Visualizing the most important tool for differential equations. Previous chapter: https://youtu.be/-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 Pi creature car 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

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