i remember watching this in high school when i wanted to be a physics major just because i liked your content, at the time none of it made too much sense as i had zero education on linear algebra or differential equations. its funny now almost three years later that i am taking math methods as a physics major struggling with greens functions and i remembered you published these resources. thanks so much for the helpful video!
You gave me a better intuition about how Green's functions work than my Classical Electrodynamics professor did. He just jumped straight into the math. You're already a better grad student than I am, and I had a two-year head start. Keep up the good work!
Amazing intro to Green’s functions! I’ve had to use them on multiple occasions but I never felt comfortable with them, but your explanation of them helps so much!
At 6:08, the delta function picks out the value for X’ = X, not the other way around since X is assumed to be known inside the integral while X’ is the integration variable. This was very helpful! Thank you!
I have known that you had made videos about subjects that would be important to me later on as a physics major for quite some time, but now that I got there and am using these resources, I am very thankful. This is the most intuitive introduction to green's function I've seen so far and it really helps! Thank you Andrew!
Damn, Dirac's delta function is similar to the "unit" matrix but in a continuous case! You've made my day man :) Would also like to see some sort of a "smooth transition" between the discrete and the continuous case! Thanks a lot!
This was a really good explanation. I never understood the intrinsic meaning behind the delta function as the driving force, but now it makes sense. many thanks!!!
"...if you have the Green's function of the (linear differential) operator, you have solved the differential equation..." thank you so much for that insight
Thanks for the video! I’m currently studying greens function in my physics degree and my professor went through this topic rly quickly and I was confused asf. Very intuitive explanation!
Great explanation! Can you do a video related to the application of Green's function to solve the acoustic wave propagation problem i.e. to calculate the velocity potential?
Yes! I showed my AP Physics teacher your channel today. He told me isn’t this a little to advanced for a 9th grader. However I do feel like I’m ahead of my class.
You totally just saved me on my math methods midterm! I think I'm a year behind you in grad school and it's like you're the point goose in a flight formation of grad students
Hi! It’s excellent that you had such good teaching skills 5 years ago! One comment is that your constant parallels drawn between the indicial representation of a matrix multiplying a vector to the delta basis distracts from some main points. Making a comparison between the basis set in R^3 as a vector space for example along with their relation to vectors and the delta distribution along with its relation to distributions on a compact support or (uncountably) infinite dimensional vector spaces. Then moving on to linear operators like greens functions and their integral representation and comparing those to the M_ij*f_j (eins notation) term you had written, would’ve been perfect.
Hey Andrew, thank you for the content. I was wondering is there a chance that you could do a video about how you study textbooks and get the most out of them? Because I feel like either I am a really pedantic person since I can't move to next chapter without spending a lot of time on the previous one or I am doing something fundamentally wrong. Thanks again, wish you good luck in your studies.
what your doing is fundamentally correct, master the previous chapter before you go on, so you get a better basis, and an easier time understanding the newer material that will probably build upon the knowledge of the previous. You're on the right track, gl mate
@@Goku17yen honestly man thank you for the comment. I needed some sort of reassurance on this. Because I mean the emotional pressure is really high some days, feeling incompetent, not being able ever reach the heart of the subject and so on. Thank you for the support, good luck in your studies.
This is cool. I've yet to be exposed to G functions outside of solving the Poisson, so this is a cool exposition to what I assume to be the "standard" way to introduce G functions. I actually came up with my own way of solving differential equations exploiting linearity, and it turned out that my way is the G function all along. So let me explain my method as an alternative take on G functions. Let L be a linear operator. Suppose we want to solve the equation Ly = f. Now we know that since L is linear, for constants a and b and vectors y_1 and y_2 L(ay_1 + b_y2) = aLy_1 + bL y_2. This is where it gets a little hand wavy, because we shall assume that there exists a function G such that LG = 𝛅, where 𝛅 is the delta distribution. There is of course, no motivation given as to why G should exist, and indeed it seems rather implausible, which is why the way the video does it is more rigorous. But anyway, I'm sure there is a way to justify the existence of G from distribution theory or something. Not that physicists care either way. What is the motivation behind finding G? Because of the linearity of L. We can write f down as a linear combo of the delta function. Just as the differential operator was defined as a distribution in the video, we can write f down "as a distribution", or intuitively as a linear combination of vectors. If we had a finite basis of dimension n we could write down basis vectors as e_1, e_2... e_n and a vector f = ∑a_i e_i where a_i are scalars. If we have an infinite basis, this turns into an integral. Now we know that the basis vectors are defined by them following a Kronecker delta relation (e_i • e_j = 1 if i = j, 0 otherwise) and that the continuous version of the Kronecker is the delta distribution. So our basis vectors are now delta distributions 𝛅(x' - x). Our scalars obey f(x) = ∫a(x') 𝛅(x') dx', so a(x') = f(x'). Therefore f(x) = ∫f(x') 𝛅(x' - x) dx'. My interpretation is that we are basically making every point in the range of f a vector and then we need to weigh each point by the value of f at that point as a scalar. Then f is a linear combination of all these points. Physically, we can think of f as the total charge. The total charge is the sum of each point charge 𝛅(x' - x), weighted by their charge f(x'). This is very similar to how we solve the Poisson equation because of course, this is exactly how we solve it. Anyway, why is this useful? Because LG(x, x') = 𝛅(x - x'). Therefore, f(x) = ∫f(x') LG(x, x') dx', but an integral is a linear combination , so it commutes with L and we can pull the L out from under the integral f(x) = L(∫f(x') G(x,x') dx'). Now ∫f(x')G(x,x') dx' is itself a linear combination. Since Ly = f(x) as well, we conclude that y = ∫f(x')G(x,x') dx'
Very good, although difficult to follow. 50 years ago I got a D in electrodynamics, partly because I had no idea how to use Green's function. Hopefully, I'll understand it one day.
Hey Andrew u should make a video on Green’s Functions in QFT. Literally all of QFT in both Particle and Condensed Matter is built on Green’s Functions as they are the fundamental objects which contain all the information about the theory analogous to the Wavefunction in QM.
@@AndrewDotsonvideos Btw since u know QFT and use QCD for research are u familiar with Nonperturbative methods like the Bethe Salpeter Equation or Schwinger Dyson Equations.
I believe that you missed an small additinal opportunity to continue this prominent analogy to matrices in the following part: you say that the inverse should be an integral because integrals are kind of inverses of derivative; however, the intuition might be completely different: the function f(x) might be depicted as a "continually dimensional" vector whose components are indexed not by numbers 1,...,n (as in the usual finite dimensional vectors) but by real numbers. When one multiplies a finite-dimenstional vector v by a matrix M, one computes the sum \sum_{i=1..n} v(i)M(i,j) for each j from 1 to n. Now the "continually dimenstional" analogy for the sums is the intergration (most people know that the intergal is the limit of specific sums): we compute the "sum" \int_{x in R} f(x) M(x,y) dx for each y from R. This analogy is truely very far fetching and helps to explain many theorems of the functional analysis and differential equations.
Hello Andrew I don't understand one little thing: From what differential equation the propagator is the green function? From Schrödinger equation? And how exactly is related with time evolution operator? Sorry for my bad english
I have a little question, if I calculate [ -y"=cos( πx)] equations by using normal methods then I fınd the reasult as a y(x)=(1/ π^2)(cos( πx) but if ı use green functions then ı find y(x)=(1/ π^2)cos( πx) -1/ π^2 +2x/ π^2, that result also satisfies the equation but why do extra terms exist ??
Ch. 8 on Poisson's Equation of "Modern Electrodynamics" by Andrew Zangwill www.cambridge.org/core/books/modern-electrodynamics/E5448C70CBF3651B2056F28EBF859AE9 has a detailed presentation on the construction of Green Functions. It's on the level of Jackson without the sharp edges.
lol it's so funny to be back here at this video when I'm actually working on my grad school homework instead of being a wee baby physics major tryna shove math into my brain and hoping something sticks
The number of people that would say no to any video on the whiteboard ever is equal to the flux of a conservative vector field on a closed boundary. EDIT: *work done on a particle subject to a force field moving on a closed path
Ehm. You are thinking about line integrals. But flux is the surface integral. In a conservative field a closed line integral is 0, but a surface integral is not.
OOF you're right I had the picture of all the little normal vectors to a curve being summed over but forgot that you can do line integrals on more than scalar fields :)
i remember watching this in high school when i wanted to be a physics major just because i liked your content, at the time none of it made too much sense as i had zero education on linear algebra or differential equations. its funny now almost three years later that i am taking math methods as a physics major struggling with greens functions and i remembered you published these resources. thanks so much for the helpful video!
literally me right now
Colour me surprised, when I searched for Green's functions and Andrew was legit one of the top results that popped up
You gave me a better intuition about how Green's functions work than my Classical Electrodynamics professor did. He just jumped straight into the math. You're already a better grad student than I am, and I had a two-year head start. Keep up the good work!
Amazing intro to Green’s functions! I’ve had to use them on multiple occasions but I never felt comfortable with them, but your explanation of them helps so much!
Hey we have the same name lol
woah
That's impossible
At 6:08, the delta function picks out the value for X’ = X, not the other way around since X is assumed to be known inside the integral while X’ is the integration variable. This was very helpful! Thank you!
1 week suffering about Green's function. Thanks for sharing me a good explanation !!!
OMG, I get greens functions now. They are just the result of an infinite dimensional inverse problem. haha
Thanks Andrew
I have known that you had made videos about subjects that would be important to me later on as a physics major for quite some time, but now that I got there and am using these resources, I am very thankful. This is the most intuitive introduction to green's function I've seen so far and it really helps! Thank you Andrew!
Love to hear it, thank you!
Thank you so much for that explanation. Sometimes a hand-wavy explanation is the first kick I need to really start understanding.
Damn, Dirac's delta function is similar to the "unit" matrix but in a continuous case! You've made my day man :) Would also like to see some sort of a "smooth transition" between the discrete and the continuous case! Thanks a lot!
This is a great introduction and compliments chapter 1 of Zee's QFT nicely because he also goes the matrix->continuous route.
just looked this up and found your channel. fantastic. thank you
This was a really good explanation. I never understood the intrinsic meaning behind the delta function as the driving force, but now it makes sense. many thanks!!!
This is more intuitive... Thanks for your valuable service to math admirers
I don't know why I found this video so late, thanks a lot to this simple but great video!
Great！ I am writing a book and this viewpoint really inspire me.
wow so that's what green's function is all about.... good job this one man.... really appreciate it
Sweet mother of god thank you for this video. I knew how to derive greens functions, but I had no idea why they worked.
"...if you have the Green's function of the (linear differential) operator, you have solved the differential equation..." thank you so much for that insight
Yesterday I said good beard and you shaved off your beard today lmao.
Quahntasy - Animating Universe because I hate your approval. Jkluvu
Great one man. Your selfless efforts are extremely respectful. Good work man and goodluck!
Nice explanation man. Examples are always helpful if you could.
Thank you for a very understandable explaination for us normies. This was really helpful.
this video is a piece of art
the analogy to linear algebra really helped, thanks!
Please solve an example and mention what it can be used for in theoretical maths or physics
Thanks for the video! I’m currently studying greens function in my physics degree and my professor went through this topic rly quickly and I was confused asf. Very intuitive explanation!
This is a brilliant intro! Thanks man!
Great explanation, very different to what I was told in class.
That helped A LOT! The first part really helped clarify things up for me! Thanks!
Nice! I am being introduces to Green functions in a mathematical methods right now and this video has explained it pretty good. Ty man
Simply fabulous explanation! Thank you!!!
Best video of this topic! Thanks for letting me understand this thematic.
Great explanation! Can you do a video related to the application of Green's function to solve the acoustic wave propagation problem i.e. to calculate the velocity potential?
Yes! I showed my AP Physics teacher your channel today. He told me isn’t this a little to advanced for a 9th grader. However I do feel like I’m ahead of my class.
Jon r/imverysmart
C R I N G E
Jealous
Jealous'
Thank you so much, im studying Physics too, and this is so useful for me.
You totally just saved me on my math methods midterm! I think I'm a year behind you in grad school and it's like you're the point goose in a flight formation of grad students
I did not know this. Wonderful. Thank you.
Hi! It’s excellent that you had such good teaching skills 5 years ago! One comment is that your constant parallels drawn between the indicial representation of a matrix multiplying a vector to the delta basis distracts from some main points. Making a comparison between the basis set in R^3 as a vector space for example along with their relation to vectors and the delta distribution along with its relation to distributions on a compact support or (uncountably) infinite dimensional vector spaces. Then moving on to linear operators like greens functions and their integral representation and comparing those to the M_ij*f_j (eins notation) term you had written, would’ve been perfect.
this explanation was amazing, tysm
I just took a break from studying Math methods and this pops up in my feed. TH-cam sure has been spying on me.
Nice presentation. Well done.
This was super intuitive!!! Thanks.
I don't say much to others but u r very good teacher..
Hey Andrew, thank you for the content. I was wondering is there a chance that you could do a video about how you study textbooks and get the most out of them? Because I feel like either I am a really pedantic person since I can't move to next chapter without spending a lot of time on the previous one or I am doing something fundamentally wrong. Thanks again, wish you good luck in your studies.
what your doing is fundamentally correct, master the previous chapter before you go on, so you get a better basis, and an easier time understanding the newer material that will probably build upon the knowledge of the previous. You're on the right track, gl mate
@@Goku17yen honestly man thank you for the comment. I needed some sort of reassurance on this. Because I mean the emotional pressure is really high some days, feeling incompetent, not being able ever reach the heart of the subject and so on. Thank you for the support, good luck in your studies.
No problem mate, we’ve all been there, and most still are ;D
This is superb! I'm taking math methods 3 this term and we gonna get into Green's fns, seeing this was pleasant :)
Alp Bartu good luck!
im not sure how to apply it though.
thank you for the background though as it makes more sense on why we use them now
The first integral you wrote should have a second derivative in the integrand!
nope he just rewrote a famous identity of the dirac delta distribution:
f(a) = int δ(x-a)f(x) dx from a - ε to a + ε for all ε greater than zero
Really appreciated. Thank you 🙏🏼
Helpful video, thanks !
this was sooooooo helpful. Thank you!!
This is cool. I've yet to be exposed to G functions outside of solving the Poisson, so this is a cool exposition to what I assume to be the "standard" way to introduce G functions. I actually came up with my own way of solving differential equations exploiting linearity, and it turned out that my way is the G function all along. So let me explain my method as an alternative take on G functions.
Let L be a linear operator. Suppose we want to solve the equation Ly = f. Now we know that since L is linear, for constants a and b and vectors y_1 and y_2 L(ay_1 + b_y2) = aLy_1 + bL y_2. This is where it gets a little hand wavy, because we shall assume that there exists a function G such that LG = 𝛅, where 𝛅 is the delta distribution. There is of course, no motivation given as to why G should exist, and indeed it seems rather implausible, which is why the way the video does it is more rigorous. But anyway, I'm sure there is a way to justify the existence of G from distribution theory or something. Not that physicists care either way.
What is the motivation behind finding G? Because of the linearity of L. We can write f down as a linear combo of the delta function. Just as the differential operator was defined as a distribution in the video, we can write f down "as a distribution", or intuitively as a linear combination of vectors. If we had a finite basis of dimension n we could write down basis vectors as e_1, e_2... e_n and a vector f = ∑a_i e_i where a_i are scalars. If we have an infinite basis, this turns into an integral. Now we know that the basis vectors are defined by them following a Kronecker delta relation (e_i • e_j = 1 if i = j, 0 otherwise) and that the continuous version of the Kronecker is the delta distribution. So our basis vectors are now delta distributions 𝛅(x' - x). Our scalars obey f(x) = ∫a(x') 𝛅(x') dx', so a(x') = f(x'). Therefore f(x) = ∫f(x') 𝛅(x' - x) dx'. My interpretation is that we are basically making every point in the range of f a vector and then we need to weigh each point by the value of f at that point as a scalar. Then f is a linear combination of all these points. Physically, we can think of f as the total charge. The total charge is the sum of each point charge 𝛅(x' - x), weighted by their charge f(x'). This is very similar to how we solve the Poisson equation because of course, this is exactly how we solve it.
Anyway, why is this useful? Because LG(x, x') = 𝛅(x - x'). Therefore, f(x) = ∫f(x') LG(x, x') dx', but an integral is a linear combination , so it commutes with L and we can pull the L out from under the integral f(x) = L(∫f(x') G(x,x') dx'). Now ∫f(x')G(x,x') dx' is itself a linear combination. Since Ly = f(x) as well, we conclude that y = ∫f(x')G(x,x') dx'
Brilliant explaination
Thanks for the awesome explanation! I have a question about the way you multiplied M^-1 and f(x). We should get two components, not ( )+( ). No?
Very good, although difficult to follow. 50 years ago I got a D in electrodynamics, partly because I had no idea how to use Green's function. Hopefully, I'll understand it one day.
thank you.. your explanation is great..!!!!
My boi andrew spitting fire again. Real shit tho why my professors be like that. I can't understand anything sometimes
Another mind blown.
Real good video, making a lot of sense, thx!
Thanks for this a lot.
Why would someone wanna solve differential using this method as opposed to laplace or regular method? Is there any other app to this?
Really nice vid, though perhaps a bit handwavy like you said. Would love an example! 😊
Hey Andrew u should make a video on Green’s Functions in QFT. Literally all of QFT in both Particle and Condensed Matter is built on Green’s Functions as they are the fundamental objects which contain all the information about the theory analogous to the Wavefunction in QM.
Good idea!
@@AndrewDotsonvideos
Btw since u know QFT and use QCD for research are u familiar with Nonperturbative methods like the Bethe Salpeter Equation or Schwinger Dyson Equations.
Please solve an example. I'm struggling through Jackson.
Interesting approach
9-year-old Andrew is back.
Rip the beard. Also, good video man!
Bruh thank you so much for the video, really helped me a lot
This helped so damn much.
Broo we just covered Greens function in today's lectures and its use in oscillations.
But does it mean to be the kernel of an integral? All functions that make the integral zero?
You can have both scalar and vector Green’s function representations
Thank you!
Ah, now I get the swinging delta function reference for the pendulum lol.
Thanks a lot for the vedio Andrew . Can we say that greens function depend on both X and X prime. ? Is it a two variable function?
ye, i believ
I believe that you missed an small additinal opportunity to continue this prominent analogy to matrices in the following part: you say that the inverse should be an integral because integrals are kind of inverses of derivative; however, the intuition might be completely different: the function f(x) might be depicted as a "continually dimensional" vector whose components are indexed not by numbers 1,...,n (as in the usual finite dimensional vectors) but by real numbers. When one multiplies a finite-dimenstional vector v by a matrix M, one computes the sum \sum_{i=1..n} v(i)M(i,j) for each j from 1 to n. Now the "continually dimenstional" analogy for the sums is the intergration (most people know that the intergal is the limit of specific sums): we compute the "sum" \int_{x in R} f(x) M(x,y) dx for each y from R. This analogy is truely very far fetching and helps to explain many theorems of the functional analysis and differential equations.
Exactly the topic i need to understand better :D
That is so weird. I love it.
good explanation
Thanks for exiting ❤❤make drivtive videos...
I love you thank you
I really need to learn this for my research thesis I hope this helps lol
Good video!
Hello, what book did you use to study this? and what chapter?
Hello Andrew I don't understand one little thing: From what differential equation the propagator is the green function? From Schrödinger equation? And how exactly is related with time evolution operator? Sorry for my bad english
I have a little question, if I calculate [ -y"=cos( πx)] equations by using normal methods then I fınd the reasult as a y(x)=(1/ π^2)(cos( πx) but if ı use green functions then ı find y(x)=(1/ π^2)cos( πx) -1/ π^2 +2x/ π^2, that result also satisfies the equation but why do extra terms exist ??
Holy sheetu, thank you for this
Mind Blow!
Ch. 8 on Poisson's Equation of
"Modern Electrodynamics" by Andrew Zangwill
www.cambridge.org/core/books/modern-electrodynamics/E5448C70CBF3651B2056F28EBF859AE9
has a detailed presentation on the construction of Green Functions. It's on the level of Jackson without the sharp edges.
Rest in pepperoni the beard. press f to pay respects.
Can you tell me some reference to start reading the green function
lol it's so funny to be back here at this video when I'm actually working on my grad school homework instead of being a wee baby physics major tryna shove math into my brain and hoping something sticks
Interesting!
Godunov would be proud
I like to think this comment is from him and he's speaking in 3rd person.
I cameded Mr. Godunov
MOAR
The number of people that would say no to any video on the whiteboard ever is equal to the flux of a conservative vector field on a closed boundary.
EDIT: *work done on a particle subject to a force field moving on a closed path
Ehm. You are thinking about line integrals. But flux is the surface integral. In a conservative field a closed line integral is 0, but a surface integral is not.
yes! I thought "wait then the amount of charge inside doesn't like the videos ;c"
OOF you're right I had the picture of all the little normal vectors to a curve being summed over but forgot that you can do line integrals on more than scalar fields :)
Scary I forgot my multivariable so quickly.
Also thanks btw I had major conceptual errors in visualizing line and surface integrals. That kinda saved me from going into E&M very confused.
Woah! 10k finally
When did you learn this? In undergrad?
Really cool