The Divergence of a Vector Field: Sources and Sinks

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This video introduces the divergence operator from vector calculus, which takes a vector field (like the fluid flow of air in a room) and returns a scalar field quantifying how much the vector field is locally expanding or contracting at every point. The divergence is a fundamental building block in vector calculus.

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This video was produced at the University of Washington

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0:00 Introduction & Overview
3:30 The Divergence is a Linear Operator
4:41 Example of Positive Divergence
8:05 Example of Negative Divergence
10:25 Example of Zero Divergence
13:58 Vector Field is a Differential Equation
16:17 Recap
17:20 Divergence of a Gradient is the Laplacian
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Thank you for these videos! This is the best explanation that I’ve ever seen for divergence. I’ve been struggling for years to really understand vector calculus but this finally made it click. Thanks for all of the time and effort that you put into your videos!

glennr.fisher
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Now I know the secret of Steve being an amazing teacher. He is interested in soooo many things and those things are important to him. Cheers! Love your book and the video companion. you have revolutionized learning.

phafid
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I am here so I can help my son who is 1st year undergard in NIT. This is undoubtedly the best video on this topics before we start solving PYQ for better CGPA

haldersubrata
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I am a plasma physics student and we come across equation with divergence curl and sinks in MHD theory.Steve has done a great job in making high quality video explaining the concept methodolgically.

bharatrawat
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Just brilliant . This is pedagogy.
Transform something complex to something understandable.

almirbravin
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Feels like watching a TV series, can't wait to see the next episode

Pradipnpk
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hello from Oxford Uni! Professor Brunton, your explaination in my opinion is better than the lecture notes here. I loved seeing the concept in multiple ways instead of staring at definitions.

c.l.
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Nice and generous breaking down. Thanks a lot.

taherjijel
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Really nice lecture. Looking forward to watching the next ones!!

pablocb
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Thanks, sir. I really needed these videos

firstofallbasics
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Eagerly waiting for your next video on curl and the examples are awesome

dineshvagicharla
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Really nice ! One lesson a week is an excellent rhythm, it gives us time to investigate.
Thank you.

bodormarcel
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These videos are so good, Steve. Thanks so much. Clarity and concision!

hdheuejhzbsnnaj
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Besides rotation, constant fields would also have divergence of zero, like f(x, y)= i+j ; Its a constant flow, so there's no expansion or contraction, but its not rotating either.

DerekWoolverton
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Iam from Ethiopian thanks for you interesting videos

SemieAzie
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Hi. It's a big ask but... Could you explain Maxwell's equations with this clarity? Could you explain the speed of light. I would be so grateful! Lee

leepatrick
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Interesting and brilliantly explained.

rajendramisir
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Laplacian is the divergence of the gradient (of any potential field). Mind = blown. Why was I not taught this before?

RahulMadhavan
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i did not quite get what he said at 14:00. Can someone explain?

parampatel
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Really nice! Looking forward to the next lesson. Thank you so much for this valuable content.

fabioantonini