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!
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.
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.
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
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.
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.
Stellar lecture, looking forward to the following ones. Just wondering how the video is recorded, is Prof. Brunton really writing from right to left ? If so, mad respect
It's been 20+ years since I studied this, and this review series is exactly what I've been looking for. My only frustration is that I am too impatient to wait for your next installment. Can you please clone yourself and put these out faster? It is causing me physical pain to know it will be several more weeks to get to the really good stuff.
So, is there a connection between divergence of a vector field and the determinant of a matrix used to describe that vector field? Both seem to describe how a portion of space grows or shrinks.
This is just a small attempt to answer the question, it might not be enough. Determinant amplifies the area in one direction and may shrink in the other. Not all the vectors in a matrix are expanded by the factor of determinant of a transition matrix. Some vectors which do not change, are the eigen vectors of that transition matrix. Divergence here is causing expansion/contraction in all directions, the drop of oil will expand in all directions, may not uniformly.
@@arvindp551 hi again! I appreciate your answer; that being said, I am not sure I understand. The way I learned to understand determinants was as a scaling of a unit area, and all unit areas being scaled by the same degree. This is different for vectors, because as you mention, there are eigenvectors. I say all this knowing that there are many ways to understand these concepts, and I may simply not be familiar with your framework, or I may just be wrong.
In Acoustics, there is something that I didn't understand. The acoustics power can be measured on any surface all around a source. The sound decreases with distance, but the divergence of the Acoutics pressure is nul. It seems that it is a conservative law (or something like this). How is it possible to have a conservative law or divergence = 0, if the sound pressure decreases with the distance ?! It's very mysterious...
I'm not sure but isn't it just because the energy spreads out over a spherical region around the source, and this region gets larger as the sound travels a longer distance? No sink/source needed for this
@@gooblepls3985 W=I*S. W is the power, I the intensity and S the surface. If S is a spherical surface, then S = 4*pi.r**2. But if the divergence of the intensity is equal to zero, why the Intensity decreases with r ?!
In case where both vectors are filled with just numbers and not functions - yes, you're just adding up a bunch of products of numbers and you get a number. But in a situation where two vectors are filled with functions you may still get a number (for example if they're orthogonal you're always getting 0), but you may also get a function as a result. Although that function would be a "scalar function", which means that it takes inputs (could be one input, could be multiple) and outputs a single number. This also need clarification because it applies to a "dot product" of two vectors, which is different from a "vector product". Vector product takes two vectors and produces a new vector.
No problem! Have you heard of matrix multiplication? If not, its a very useful way to write down a bunch of related equations at once. The way it works for the example he's written down is this: first, he multiplies the [ x ; y ] column with the first row of the matrix, [ 1 0 ]. The x gets multiplied by 1, the y gets multiplied by 0, and then the results get added together. In this case, the result is just x. Then, he multiplies the [ x ; y ] column with the bottom row, [ 0 1]. The x gets multiplied by 0, and the y gets multiplied by 1, and the results get added together, equaling y. What's left now is d[ x ; y]/dt = [ x ; y ]. This is just a way of writing the two equations: dx/dt = x, and dy/dt = y. If you've ever seen a differential equation, you may recognize these. You can solve them by separating variables: dx/x = dt and dy/y = dt. When these equations are solved, they become ln(x) = t and ln(y) = t. Raising e to both sides, and you get x = e^t and y = e^t .
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](http://i.ytimg.com/vi/lKXW7DRyyro/mqdefault.jpg)
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!
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.
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.
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
Glad it was helpful!
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.
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.
Just brilliant . This is pedagogy.
Transform something complex to something understandable.
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
ua-cam.com/video/gx-OfbNzHK0/v-deo.html it is spanish, but may be can works. Just turn on cc in english
yeeeeeeeeeeeeeeeeeeeeeees
Yes
Feels like watching a TV series, can't wait to see the next episode
Laplacian is the divergence of the gradient (of any potential field). Mind = blown. Why was I not taught this before?
11:58 STEVE, NO!
"erika starts playing in the background"
😂😭😭
iykyk
Stellar lecture, looking forward to the following ones. Just wondering how the video is recorded, is Prof. Brunton really writing from right to left ? If so, mad respect
he reverses the footage
These videos are beautifully clear.
Really nice ! One lesson a week is an excellent rhythm, it gives us time to investigate.
Thank you.
Eagerly waiting for your next video on curl and the examples are awesome
These videos are so good, Steve. Thanks so much. Clarity and concision!
huge service to humanity!
Thanks, sir. I really needed these videos
Really nice lecture. Looking forward to watching the next ones!!
Dr Brunton you are a phenomenal teacher. Your videos make me wish that I did graduate school at the University of Washington 😂😭
It's been 20+ years since I studied this, and this review series is exactly what I've been looking for. My only frustration is that I am too impatient to wait for your next installment. Can you please clone yourself and put these out faster? It is causing me physical pain to know it will be several more weeks to get to the really good stuff.
Interesting and brilliantly explained.
Love all these videos!
He is left handed 🎉 so magic happens and yes great teacher
Nice! Thanks Prof! Looking for next lectures!
A jewel! Super lecture. Thank you
Really nice! Looking forward to the next lesson. Thank you so much for this valuable content.
Love this, what an awesome video
love this perspective :)
This helps with computer vision! Thanks!
Great explanation...thank you so much.
Thank you Steve
11:50 what is he drawing?
A symbol of good fortune in asian cultures
😂 😅 bro i was asking the same thing
Best build up to teach DE
Parabéns pela Aula !
Great lecture!
“I hope you find it interesting” - no more than Feynman’s Lectures on Physics nor less.
you are a legend
tnx 4 video 👍☺📚 very useful .
Hope to s33 next one you have a huge fan herr
Love it! ❤ Learning a lot! 😂
Thank you Sir!
just one question, nabla squared isn't the Hessian matrix rather than the Laplacian (19:10 in the video) ?
Do div-free fields have to rotate? I could imagine not, they could just be standing still. If they rotate, then in which direction and why?
thank so much sir
i did not quite get what he said at 14:00. Can someone explain?
So, is there a connection between divergence of a vector field and the determinant of a matrix used to describe that vector field? Both seem to describe how a portion of space grows or shrinks.
This is just a small attempt to answer the question, it might not be enough. Determinant amplifies the area in one direction and may shrink in the other. Not all the vectors in a matrix are expanded by the factor of determinant of a transition matrix. Some vectors which do not change, are the eigen vectors of that transition matrix. Divergence here is causing expansion/contraction in all directions, the drop of oil will expand in all directions, may not uniformly.
@@arvindp551 hi again! I appreciate your answer; that being said, I am not sure I understand. The way I learned to understand determinants was as a scaling of a unit area, and all unit areas being scaled by the same degree. This is different for vectors, because as you mention, there are eigenvectors.
I say all this knowing that there are many ways to understand these concepts, and I may simply not be familiar with your framework, or I may just be wrong.
@@garekbushnell3454 ua-cam.com/video/Ip3X9LOh2dk/v-deo.html
Is this guy behind glass writing backwards or is there some other devilry going on here?
thank you
At 17:40 it doesn't look right, f is scalar and doesn't have f1 f2 components. (never mind it was fixed immediately)
In Acoustics, there is something that I didn't understand. The acoustics power can be measured on any surface all around a source.
The sound decreases with distance, but the divergence of the Acoutics pressure is nul. It seems that it is a conservative law (or something like this).
How is it possible to have a conservative law or divergence = 0, if the sound pressure decreases with the distance ?!
It's very mysterious...
I'm not sure but isn't it just because the energy spreads out over a spherical region around the source, and this region gets larger as the sound travels a longer distance? No sink/source needed for this
@@gooblepls3985 W=I*S. W is the power, I the intensity and S the surface.
If S is a spherical surface, then S = 4*pi.r**2. But if the divergence of the intensity is equal to zero, why the Intensity decreases with r ?!
Can anyone explain how we get e^t in the matrix? Thanks
The net zero divergence was tense 😅😅😅...
best
Are we always get scalers from 2 vector products?
In case where both vectors are filled with just numbers and not functions - yes, you're just adding up a bunch of products of numbers and you get a number.
But in a situation where two vectors are filled with functions you may still get a number (for example if they're orthogonal you're always getting 0), but you may also get a function as a result. Although that function would be a "scalar function", which means that it takes inputs (could be one input, could be multiple) and outputs a single number.
This also need clarification because it applies to a "dot product" of two vectors, which is different from a "vector product". Vector product takes two vectors and produces a new vector.
14:30 I Didn't Understand. Please Somebody Help!
Me too...Why there is a matrix [1 0 0 1] and it becomes e^t?
Me too
No problem! Have you heard of matrix multiplication? If not, its a very useful way to write down a bunch of related equations at once. The way it works for the example he's written down is this: first, he multiplies the [ x ; y ] column with the first row of the matrix, [ 1 0 ]. The x gets multiplied by 1, the y gets multiplied by 0, and then the results get added together. In this case, the result is just x. Then, he multiplies the [ x ; y ] column with the bottom row, [ 0 1]. The x gets multiplied by 0, and the y gets multiplied by 1, and the results get added together, equaling y.
What's left now is d[ x ; y]/dt = [ x ; y ]. This is just a way of writing the two equations: dx/dt = x, and dy/dt = y. If you've ever seen a differential equation, you may recognize these. You can solve them by separating variables: dx/x = dt and dy/y = dt. When these equations are solved, they become ln(x) = t and ln(y) = t. Raising e to both sides, and you get x = e^t and y = e^t .
@@garekbushnell3454 Thank You So So Much! Really appreciate the way you have explained every piece of it.
@@arvindp551 happy to help! Good luck in your math studies!
13.30 yes.... for 2D
Only 55 comments?!
Why you guys just don't flip horizontally the video while editing?
you drew a swastika lol