Divergence and curl: The language of Maxwell's equations, fluid flow, and more

Visualizing two core operations in calculus. (Small error correction below)
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Timestamps
0:00 - Vector fields
2:15 - What is divergence
4:31 - What is curl
5:47 - Maxwell's equations
7:36 - Dynamic systems
10:37 - Explaining the notation
13:28 - No more sponsor messages

Correction at 4:55, the narration should say "counterclockwise rotation gives positive curl, clockwise rotation gives negative curl". The diagram is correct, though.

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:

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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[amitbenjam]

The fact that the same concept can explain both electromagnetic waves and foxes eating rabbits is one of the main reasons I have chosen to pursue mathematic studies.

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I'm currently applying to phd programs for pure mathematics now. This is in no small part to your videos over the last few years. They inspire me to see math in a new light. I watch this video having used divergence and curl for many semesters and I forget, or never realized how beautiful they are.

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Mr Sanderson,
You are an absolute joy and delight to this world. I am so glad to have the fortune of discovering your channel and learning from you. You brighten my day when I watch your video and truly begin to understand what math is about. For that, I am grateful. Please don’t stop!!!

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Sir,
I had a teacher (in a small town in India) who explained things as you do. He taught physics. Very passionate. Lived it. Had youtube mobiles been at his time, we would have been fortunate to listen to him even today.

[potatomo]

Small mistake: for curl it’s counter clockwise = positive, and clockwise = negative

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I am a german math student in 2nd semester and I am SO greatful for every single one of the videos I watch. I always keep recommending this channel to my peers and believe it to be the most... beautiful and informative one on maths on youtube. I love the colorful animations, your voice (it's very soft and motivating!) and - well, in fact, all of it.
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15:15 "What matters more than wanting something is to actually align incentives". Dropping math knowledge and generalised wisdom in the same video, very nice.

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Man I am a mechanical engineering student, but I must admit I learnt it for the first time in my life. Even though I took multivariable calculus in my college first year, I couldn't understand even a single thing. But you have encapsulated everything so beautifully and intuitively that I must say it's better to learn science and mathematics from you rather than going to college.

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You, sir, are a gift to humanity. Only if we acknowledged that.

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I am getting envious - as a "dinosaur" - how brilliant ways and tools You have TODAY to explain these things so _fluently_ that even I can grasp them finally. When I had some of that stuff over 30 years ago in school, it remained horribly abstract and left me clueless, almost. Especially the question how poeple like Maxwell, having not even an idea of digital computers or visualizations, could "make" those fundamental equations ... so I always stood kinda stuck in between: neither be a total ignorant for math, but also not even close to mathematicians, who always remained as one of the strangest species on earth to me (no offence intended =), as they would thrive in their abstract world called Math, and leave everyone else behind, not even understanding or being interested to understand, why non-mathematicians do not understand them and the matter which is "so obvious" to them. That mind gap is just astronomically great.
For me, it is these visualizations being the KEY to grasping. Unfortunately, too many people, even if educated, still mix up grasping with memorizing ...
Thanks for this video, most excellent!

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Grant, you have absolutely no idea how valuable you and your way of understanding and explaining things are. I mean it when i say i wish all teachers would teach like you. I'm certain if they did the world would be a better place. Thank you for your insights.