@@20031bibi The joke is that nobel didn't made a nobel for math because a mathematician was fucking his wife. Tom is heavily flirting with a married man. Therefore, a joke. But he flirts with everyone, after this christimas season I'd say he had an only fans, but who am I kidding, this is youtube, everyone has an onlyfans.
Maybe you guys haven't seen it or it has been a while, but I would check out Feynman's Lectures on Physics. In volume 2 there is a neat section on the method of images for electrostatic potentials!
@@leif1075 those would be work with any walls along which the potential flow is 0 so perfectly stationary fluid walls with infinite mass would definitely work and solid walls would as well. Not sure about other stationary fluids with finite mass
Definitely one of the coolest ideas in fluids! One of my favorites is if you have a source at (1, 0) and 2 walls leaving from the origin at slopes of +30 degrees and -30 degrees, then you can replace it with 6 sources at the 6 roots of unity (hexagonal symmetry).
Your comment made me start thinking. Have you realized you can generalize this method? If you have a similar setup but with an angle of 2pi/2k between upper wall and x axis you can construct a solution with k charges on the vertixes of a poligon with k edges. And if k goes to infinity? I think in some sense it converges to the infinite channel example
I didn't know Tom had his own channel! I just saw the drag equation video on the Numberphile channel and the recommended video was this one. Thank you algorithm!
Its today's computer,, internet and technology, which paralyzed our mind and creativity. we are so much dependent on computers that we even don't try to imagine things, we search youtube for animations etc, which feels super easy to grab the things but in long term our brain gets lazy. the time when there were no computer machines, all computations were supposed to be done in the brain, as a matter of fact, the more you use the brain the more it gets trained and powerful. and then curiosity will be developed for nature, and the ultimate result will be discoveries and inventions.
This was nice! Fluid dynamics is very similar to electric field theory we did in physics. The source is like a positive charge, sink being a negative charge and the velocity vectors are like electric field vectors. We do use potentials in electrostatics but I don't remember using complex potentials. In that way electrostatics might be a bit simpler. The mirror method is elegant indeed! Visualizing images of source in mirrors and doing the calculations. In electrostatics the wall is in fact a conducting surface. The infinite channel example was particularly enlightening.
Thoroughly enjoy these collaborations with Grant. I think the visuals with barriers and reflections would make a great 3Blue1Brown video (like a followup to the Maxwell's equations video).
When I saw the first example with the source and the wall - 12:41 - I thought about an additional step: to imagine every point of the wall as a kind of source itself, but a linear one in particular direction alpha, depending of its position relative to the source of the flow. But, wait a minute!... this is the definition of the mirror, and as we already know, we can imagine the second source behind the wall, placed at its special spot as it was shown in the video. (By the way, the method of mirror images is also used in the field of electrodynamics, which is my speciality - so, you see, I was taught to think in this manner.) How clever it is! What mathematical wonders are hidden in Fluid Dynamics... I can only guess! Ah, and the last example was also very elegant! What am I talking about - all they are! These things must be popularised and the host of this channel is doing great, I admire his efforts... as with the same favour for mathematics that 3Blue1Brown is doing with his magnificent visualisations... big fan!
"It's like you're looking at the mirror and then you give him a high five. Of course, it will stop there" That's a very good analogy for a method of images.
Thanks for the great video! Really interesting and well-explained 😊 I just have one question that's been bothering me since the beginning of the video: why do you take the potential to correspond to u *minus* iv, and not u+iv? Is there some physical or mathematical logic behing this choice?
We very briefly touched on this in the video, but the idea is so that when you calculate the derivative of the potential as dw/dz the velocities match up with the real and imaginary parts. If we instead define dw/dz as u + iv then the vertical velocity would be the negative of the imaginary part of the derivative.
You start of with the derivative being the sun from -inf to inf of 1/(z - 2nai) since the derivative of ln is 1/z. Then multiply top and bottom of each term of the summation by its complex conjugate. You get sum from -inf to inf of (z - 2nai)/(z^2 + 4n^2a^). Then extract the n = 0 term and group each of the rest with its corresponding negative term. You get 1/z + sum from n=1 to inf of (z - 2nai + z + 2nai)/(z^2 + 4a^2n^2). Cancelling out the 2nai terms and adding the z's and factoring them out of the sum you get the desired result. (Tom got it slightly wrong, the sum should start at 1 instead of 0. You could also include 0 in the sum but that would then negate the 1/z term we pulled out earlier).
The thing that strikes me - is the simplicity with which it all has to be approached. If nothing else this shows how to make maths accessible; or that even some of the best minds in math, still when introduced to a new topic, take it from the most basic forms and build on that. Sublime!
I needed a quick revision on this topic for my PhD Quals and there you are Grant. Awesome collab Tom xD. Also, the series should start from n=1,inf after taking the derivative. Sorry it had to be done :)
Won't we see multiple reflections even in the case of corners? When the boundaries are aligned at 90 degrees? Why did we consider only a single reflection there?
Which magically yields the same results as the method of conformal maps : there is nothing trivial , nor obvious about that coincidence , Which has to do with the existence of smooth complex maps …. An analytical discipline . Which has a non trivial relationship with the discipline of iteration.
A 2D paper. Can manipulate into a mobious strip by a 3D being. To double the distance a 2d character would experience with out mass increase. A 4D. Is time. A memory is 5D. Men in black 3. 5D being can see through time and space. =memory. What is 1D. Senses. The brain dychpers patterns. The movie about black women being the computer for NASA. Burned diseased bodies after the blubonic plague. Causes a chemical reaction to rise to the air. The disease transferred to the air. Ferngully. A pure body should be burned. And diseased body should be buried. Slower particles because not heat. And slower reaction to the environment. A town poisoned the aquifer by burying bodies in one location. Because rain in Spain goes to the drains. The water collected built up decomposing bodies and sunk to the town's drinking water. Someone poisoned the watering hole. And no one admits it. Who poisoned it and everyone has proof with videos. But have a town with a graveyard over an aquifer then thay is the source. Native americans almost died out from blankets. Warm water is used to build cultures. Cold water is best to prevent bacteria breeding. Only boiling water eliminates bacteria. No one should be washing their hands in boilingwater. .
It's not clear why Tom says there is no divergence in the field with a source. There would obviously be a divergence, wouldn't there? Isn't that what a source is? I also don't understand the complex factor z that is added to the expression for w. It appears to already have a magnitude and a direction everywhere, without z. And it just appears out of nowhere. Why is there a z in there?
The series at the end actually doesn't converge. But since a potential is only defined up to a constant, we can subtract from each term an appropriate constant just so that it converges. This way we can use the Weierstrass product formula for sin(z) and get the same result.
Hold on.... doesnt this assume that the source is positionally static. as in, not migrating in space due to its changes upon the flow field... What occurs for a migratory source?
Im a little confused, why are we able to just add another potential flow when using the method of images? won't that equation be different than the 1 source case (This is for the infinite plane case)
adding in the other sources acts to create the 'wall'. the flow pattern is in fact the same for two symmetrical sources and one with a wall - and you can verify this with experiments
I studied physics and then went on with a not related degree. This video reminded me of when I used this mirror method for potentials in electrodynamics where there is e.g. some point charge (Punktladung in German) in a plane. In such moments I dont know if I feel sad to have "abondend" the world of physics/maths and their methods.
Mathematically, the point perpendicular to the mirror (15:00) is fine, but physically what would happen to the atoms and building up of the energy around that point?
Mathematically, the point perpendicular to the mirror (15:00) is fine, but physically what would happen to the atoms and building up of the energy around that point?
Grant also taught me something - check out the 'Power Tower' video here: ua-cam.com/video/dnZ3xlif9VA/v-deo.html
What does z represent at 6:15..you didnt say
@@leif1075 z is a complex-valued number - z = x+iy. Think of this as the "coordinate" of the complex plane.
:v
hello papa flemmy
Ah, yes, two sexiest mathematicians in one video
Grant is so attractive he is the only man who could seduce me.
I'm glad this is the first comment, cause I wanted them to kiss from the moment I looked at the thumbnail
BONK
you must be unaware of Ed Frenkel :D
@@davidgjam7600 👀
im taking a break from learning maths with a maths Video, weird isn´t it ^^
somehow the college experience is just about getting distracted from doing maths by another different kind of maths
yeah i m watching this video instead of practising for my circuit theory test.
I have a class in 4 minutes
Same!
Just in time for my fluid dynamics exam! Haven't seen the video yet but man, this collab is epic
Hehe ikr!!
Good luck :D
@@aiyopasta lots of maths, not so easy but super interesting!
I had my exam like 4 weeks ago... We had a question on method of images and it was AWFUL, i just think this video was suggested to mock me
@@aiyopasta depends on your lecturer
In general it should be easy
Oh goodness.... Math bursting at the seems, all we need now is an onlyfans. Hahahahaha.... Kidding not kidding.
Fluid flow, for sure.
Tom is a good example why there isn't a nobel for math.
I hope you got the joke.
Is the word "seems" intentionally misspelled?
@@lloydgush not me explain pls lol
@@20031bibi The joke is that nobel didn't made a nobel for math because a mathematician was fucking his wife.
Tom is heavily flirting with a married man.
Therefore, a joke.
But he flirts with everyone, after this christimas season I'd say he had an only fans, but who am I kidding, this is youtube, everyone has an onlyfans.
@@lloydgush LMAOOOOOO
Maths, fluid dynamics, Tom and Grant. Simply amazing.
Is Grant really huge or is Tom really small? Or am I just bad at understanding camera angles?
Grant is large.
Why, man, he doth bestride the mathematical world. Like a Colossus.
i heard "sauce" when Tom says "source" and honestly that didn't asked myself any question before finding out it wasn't sauce
These equations can describe sauce flow too... For the right choice of sauce (incompressible, inviscid, irrotational sauce)
Maybe you guys haven't seen it or it has been a while, but I would check out Feynman's Lectures on Physics. In volume 2 there is a neat section on the method of images for electrostatic potentials!
I was looking for a video on method of images of electrostatics but ended up watching this amazing fluid video ❤️
Awesome! Thank you!
i see tom i see 3b1b i click
Next: Laminar flows with Dustin, the epic fluid collab, and it doesnt get any better than that
I'm down
@@TomRocksMaths Are those walls supposed to be solid walls or walls of stationary fluid or something?
@@leif1075 those would be work with any walls along which the potential flow is 0 so perfectly stationary fluid walls with infinite mass would definitely work and solid walls would as well. Not sure about other stationary fluids with finite mass
Definitely one of the coolest ideas in fluids! One of my favorites is if you have a source at (1, 0) and 2 walls leaving from the origin at slopes of +30 degrees and -30 degrees, then you can replace it with 6 sources at the 6 roots of unity (hexagonal symmetry).
Your comment made me start thinking. Have you realized you can generalize this method? If you have a similar setup but with an angle of 2pi/2k between upper wall and x axis you can construct a solution with k charges on the vertixes of a poligon with k edges.
And if k goes to infinity? I think in some sense it converges to the infinite channel example
reading this comment felt really good @@marcocecchi9853
I didn't know Tom had his own channel! I just saw the drag equation video on the Numberphile channel and the recommended video was this one. Thank you algorithm!
It's good to know it does its job sometimes :)
This is what happened when mathematicians go with the flow!
I laughed.
Next video: "Grant and I are a couple now! #MathLove"
Are you kidding me? I just found your channel and you have a collab with Grant? Christmas came really early this year :)
Welcome :)
Also a question how do people even think of this abstract idea it feels magical
Kutta, blasius, zhoukovsy etc they were all extremely gifted
Its today's computer,, internet and technology, which paralyzed our mind and creativity. we are so much dependent on computers that we even don't try to imagine things, we search youtube for animations etc, which feels super easy to grab the things but in long term our brain gets lazy.
the time when there were no computer machines, all computations were supposed to be done in the brain, as a matter of fact, the more you use the brain the more it gets trained and powerful. and then curiosity will be developed for nature, and the ultimate result will be discoveries and inventions.
Greatest math video ever created. I am in awe at how amazing this journey how perfect the format is.
Thanks Connor :)
The method of images is so amazing that it deserves a background music of its own: "Mirror on the wall, here we are again...."
Early gang
This was nice! Fluid dynamics is very similar to electric field theory we did in physics. The source is like a positive charge, sink being a negative charge and the velocity vectors are like electric field vectors. We do use potentials in electrostatics but I don't remember using complex potentials. In that way electrostatics might be a bit simpler.
The mirror method is elegant indeed! Visualizing images of source in mirrors and doing the calculations. In electrostatics the wall is in fact a conducting surface. The infinite channel example was particularly enlightening.
As Richard Feynman put it - the flow of "dry water"...
Epic colab!
Yay! More Grant and Tom collabs!
Where was this video when I was doing fluid mechanics last year 😭
HENCE PROVED TOM ROCKED
Now I'm remembering fluid dynamics... Oh no... the screams. The terrible screams.
What about a wall with two holes in it and does it generalize to n dimensions.
Ahhh, that's what PotentialFOAM does.
Youre a really good teacher wow
I was looking for a video on method of images of electrostatics but ended up watching this amazing fluid video ❤️
Thoroughly enjoy these collaborations with Grant. I think the visuals with barriers and reflections would make a great 3Blue1Brown video (like a followup to the Maxwell's equations video).
i never saw this aproach in fluids, since i just knew it from EM topic ... epic greetings from colombia
Hello in Colombia!
When I saw the first example with the source and the wall - 12:41 - I thought about an additional step: to imagine every point of the wall as a kind of source itself, but a linear one in particular direction alpha, depending of its position relative to the source of the flow. But, wait a minute!... this is the definition of the mirror, and as we already know, we can imagine the second source behind the wall, placed at its special spot as it was shown in the video. (By the way, the method of mirror images is also used in the field of electrodynamics, which is my speciality - so, you see, I was taught to think in this manner.)
How clever it is! What mathematical wonders are hidden in Fluid Dynamics... I can only guess!
Ah, and the last example was also very elegant! What am I talking about - all they are!
These things must be popularised and the host of this channel is doing great, I admire his efforts... as with the same favour for mathematics that 3Blue1Brown is doing with his magnificent visualisations... big fan!
Not just maths but also chemistry!! ❤️
I needed this for electrodynamics
why complex values? why not x y?
Great video! The method of images is very useful when dealing with phenomena that can be treated as linear, e.g. in (linear) acoustics.
"It's like you're looking at the mirror and then you give him a high five. Of course, it will stop there"
That's a very good analogy for a method of images.
agreed :)
Eye candy, brain candy. Also happpy 2021 to the both of you!
Happy New Year John!
@@TomRocksMaths Thanks Tom 😊
Please makes videos on streamline, streakline, pathline and stream functions etc 🙏 please 🙏.
Added to the video idea list - thanks!
Thanks for the great video! Really interesting and well-explained 😊
I just have one question that's been bothering me since the beginning of the video: why do you take the potential to correspond to u *minus* iv, and not u+iv?
Is there some physical or mathematical logic behing this choice?
We very briefly touched on this in the video, but the idea is so that when you calculate the derivative of the potential as dw/dz the velocities match up with the real and imaginary parts. If we instead define dw/dz as u + iv then the vertical velocity would be the negative of the imaginary part of the derivative.
could you explain how did you differentiate and apply the limits at 21:08
ln(f(x)) differentiates to f’(x) / f(x) and then I rationalise the denominator by multiplying the top and bottom by the complex conjugate
I love how Grant just knows the answer was some cotangent thing. You prove it with Parseval's identity, correct? Or is there a more fun way?
On second thought, I realized that you can do it with a contour integral, which is a bit less tedious.
I was thinking contour integrals...
Queue some amazing fluids visualizations @ 3Blue1Brown in 3... 2...
Could someone explain the differentiation and simplificatiom step? I don't even really know what he has written down there ..
You start of with the derivative being the sun from -inf to inf of 1/(z - 2nai) since the derivative of ln is 1/z. Then multiply top and bottom of each term of the summation by its complex conjugate. You get sum from -inf to inf of (z - 2nai)/(z^2 + 4n^2a^). Then extract the n = 0 term and group each of the rest with its corresponding negative term. You get 1/z + sum from n=1 to inf of (z - 2nai + z + 2nai)/(z^2 + 4a^2n^2). Cancelling out the 2nai terms and adding the z's and factoring them out of the sum you get the desired result. (Tom got it slightly wrong, the sum should start at 1 instead of 0. You could also include 0 in the sum but that would then negate the 1/z term we pulled out earlier).
The thing that strikes me - is the simplicity with which it all has to be approached. If nothing else this shows how to make maths accessible; or that even some of the best minds in math, still when introduced to a new topic, take it from the most basic forms and build on that. Sublime!
I needed a quick revision on this topic for my PhD Quals and there you are Grant. Awesome collab Tom xD.
Also, the series should start from n=1,inf after taking the derivative. Sorry it had to be done :)
Glad you enjoyed it!
I understand the concept, but the math leaves me in the dust !
This is 2nd year maths undergraduate level so don't feel bad!
Could not asked for a better new year gift. I am working with these for past 2 years and it still amazes me how beautiful the math is.
Won't we see multiple reflections even in the case of corners? When the boundaries are aligned at 90 degrees? Why did we consider only a single reflection there?
Thank you Tom and Grant! Just in time for my Fluids exam
Which magically yields the same results as the method of conformal maps : there is nothing trivial , nor obvious about that coincidence ,
Which has to do with the existence of smooth complex maps ….
An analytical discipline . Which has a non trivial relationship with the discipline of iteration.
A 2D paper. Can manipulate into a mobious strip by a 3D being. To double the distance a 2d character would experience with out mass increase. A 4D. Is time. A memory is 5D. Men in black 3. 5D being can see through time and space. =memory. What is 1D. Senses. The brain dychpers patterns. The movie about black women being the computer for NASA.
Burned diseased bodies after the blubonic plague. Causes a chemical reaction to rise to the air. The disease transferred to the air. Ferngully.
A pure body should be burned. And diseased body should be buried. Slower particles because not heat. And slower reaction to the environment. A town poisoned the aquifer by burying bodies in one location. Because rain in Spain goes to the drains. The water collected built up decomposing bodies and sunk to the town's drinking water.
Someone poisoned the watering hole. And no one admits it. Who poisoned it and everyone has proof with videos. But have a town with a graveyard over an aquifer then thay is the source.
Native americans almost died out from blankets. Warm water is used to build cultures. Cold water is best to prevent bacteria breeding.
Only boiling water eliminates bacteria. No one should be washing their hands in boilingwater. .
It's not clear why Tom says there is no divergence in the field with a source. There would obviously be a divergence, wouldn't there? Isn't that what a source is?
I also don't understand the complex factor z that is added to the expression for w. It appears to already have a magnitude and a direction everywhere, without z. And it just appears out of nowhere. Why is there a z in there?
The series at the end actually doesn't converge. But since a potential is only defined up to a constant, we can subtract from each term an appropriate constant just so that it converges. This way we can use the Weierstrass product formula for sin(z) and get the same result.
My IQ just jumped 3 points.
THANKS OBAMA
Finally I got the images method! 👏🏼👏🏼👏🏼
Happy to help :)
That was so beautiful ❤️
I miss fluid dynamics
it's awesome isn't it?
Hold on.... doesnt this assume that the source is positionally static. as in, not migrating in space due to its changes upon the flow field...
What occurs for a migratory source?
i am doing a fluid mechanics master degree and this really brainstorming, thanks so much for sharing.
Glad it was helpful!
18:14 Grant is thinking: "did you see that throw? nailed it!" but, unluckily, Tom is focused on the graph :(
Haha I think I said a quiet 'nice' before immediately getting back to the maths...
@@TomRocksMaths Nice! I had missed that
I'm getting flashbacks from electrostatics class.
Both of you are assisting me with my physics maths degree, final year student ❤️
idk but change of variables always throw's me off for some reason
Cool just like electrostatics.
Yes, so I've been told!
Physics students learn this in Griffith em.
They still using Griffith? That's great. That was my book 35 yrs ago!
@@zornslemmon2463 yes.
Still sad to find out Grant is Married, Happy for him but still sad.
14:39 plot twist 🔥😂
Well, tom shows us the reason why we don't have a nobel for math...
lol!
brilliant content as usual, thank you
I dont understand a word but eh... fun :)
That's all that matters :)
Never seen the source in a channel before. Thanks Tom great video!
I now see why you chose potential flow and the method of images to show to Grant, you had a brilliant example :D
MAAAAN CAN YOU SOLVE THE VORTEX IN THE WAKE OF A CILINDER PLEAAASE
see also: stagflation glow 🤷🚬
thanks:)
Just in time for my EM exam lol thank u
Wow, amazing. That's one of those concepts you never forget once you've seen it.
I did mathematics well until no alphabets came .
Channel is growing my man, awesome!
Cool nerdy stuff
It's what we do :)
Why are complex numbers involved in potential flow? You could of used cartesian coordinates?
It's a neater way to write out the potential and allows us to use lots of nice results from complex analysis such as conformal mapping.
Amazing video. How would you calculate the flow with a curved surface instead of a flat plane?
Ah, now that requires a completely different theory... this only works for 2D flows.
for me every day
my color is pink
when I remember
This is the best video on the internet
A bold claim, but I'm not complaining - thank you
Are the conditions of incompressibility and irrotation equivalent to a meromorphic potential? - it looks like they should be.
U speaking the gods language hooman..
There is a lot of crossover with Complex Analysis which is where the 'complex potentials' idea comes from
Im a little confused, why are we able to just add another potential flow when using the method of images? won't that equation be different than the 1 source case (This is for the infinite plane case)
adding in the other sources acts to create the 'wall'. the flow pattern is in fact the same for two symmetrical sources and one with a wall - and you can verify this with experiments
then conformal mapping appears and renders everything more useful
For more difficult regions yes.
Electron energy levels for the hydrogen atom? Is that Tom's Tattoo on his right arm? If so, that's an awesome concept
great idea - but my tattoo represents the Basel problem
There's a cut at 0:38 because Grant snapped and said "I am not scientifically illiterate Tom!"
... this is some kind of niche superpower
Great :)
Thanks!
Is there a book that explains the computation steps of the last problem a bit more in depth? Tried to do the computations on my own but failed :D
I recommend 'Elementary Fluid Dynamics' by David Acheson
Love eeeeeeeeeeeeeetttttttttt!!!!
Lol, we do this in materials kinetics all the time. Same exact concept really. Momentum or particles.
Yep!
I studied physics and then went on with a not related degree. This video reminded me of when I used this mirror method for potentials in electrodynamics where there is e.g. some point charge (Punktladung in German) in a plane.
In such moments I dont know if I feel sad to have "abondend" the world of physics/maths and their methods.
Reminded me of the same thing
The same method is indeed used in electrostatics - well remembered :)
Mathematically, the point perpendicular to the mirror (15:00) is fine, but physically what would happen to the atoms and building up of the energy around that point?
Mathematically, the point perpendicular to the mirror (15:00) is fine, but physically what would happen to the atoms and building up of the energy around that point?
Sounds like Tom is... super StOkEd to do this
Ha nice.
Yes
every day is the Boys