Professor Strang is awesome, but I found this video to be a bit rushed. He ignored many subtleties. For instance the series produces a periodic function, so the example at the end actually gives (... + δ(x+2pi) + δ(x) + δ(x-2pi) + ...) which is a periodic version of the Dirac delta spike. Another point is that (given f is periodic) we can integrate over any full cycle, it doesn’t have to be -pi to pi. It’s also worth mentioning that we could produce functions with any period T by replacing x with 2(pi)x/T, but this slightly changes the coefficient formula (1/pi in front becomes 2/T). Also it’s not hard to show the orthogonality (it comes down to a simple trig identity) instead of just assuming it. Still a good video just a little too brief. Much respect to professor Strang though.
not really you have professors with amazing skills pretty much everywhere in the world, but in the mit all the professors also some of the best researchers in the world, and the ones that reinvented so many fields
Good teachers make the materials easy to understand, so that students can learn. The problem is that many teachers teach poorly and that's why I'm here learning from YT videos. Thank MIT teachers!
LOved this! Im studing electrical engineering and really needed this! I got low second five times, and I am on my second retake of my second year. Im also on acedemic probation at oxford brooks! this single-handedly SAVED MY DEGREE!!! thanks again! :)
To me , fourier was marvlously msthematical genius of geniuses. With much awe as to how he conceived the idea of heat propagation that can be expressed in terms of sines and cosines. With reverence to his life and works, services. Thanks.
I've read a couple of explanations and read several videos, and I find something missing. I remember old Gilbert Strang and what he tought me about Calculus and Linear Algebra, get here, I see the board, and just by looking at it I get enlightened. Thank you for everything!
3 years ago when I first learnt Fourier series this had been the most confusing part in that semester. (my professor didn't spent much time on this because for some reason this was not going to be in the exams) I tried to work it out and with my own interpretations but failed. and since then I had been haunted by it, I come across Fourier from time to time in my study, I know how to apply the equations but never understand why these equations come to be like this, I never comprehended it. Thank you professor Strang for saving me again! Your 18.06 lectures also helped me a lot!
I'm still here MIT. Though i know i failed my jee journey, be it due to my lack of effort,laziness,other life things. I WAS and AM still here.I might've been intrigued in the past and stayed here for what, maybe a couple of seconds?, but i roughly know where am i headed.I promise i will be here again even if on and off but one day, one day i will gain all the possible knowledge.All the things i need to know to atleast try to understand this complx world.I will definately one day fix myself and offer my works if god bless im able to do. I might not have it today, not tomorrow or maybe the day after.But one day i will.I still have not lost hope. I think i have tired even god helping me.I may be skeptical of everyhing but i will be there.I know i still got this.
Hello bro just here to remind you after 8 months that you’re not alone, keep going no matter what and let’s go beyond the limit of our natural perception and understanding.
I ran it in Python to test the Fourier series from the delta function, and incredibly, the series just plotted the delta function like a charm. Unbelievable!
A LOT OF WORDS FOR SOMETHING SIMPLE. Simple because functions like f(x) are just vectors! Thus, the a's, b's and c's are components and the cosines, sines and e's are basisvectors. That's why mr Strang claims that this is true: 6:06. Of course, when you dot a basisvector with a vector f(x) you get a component. When V = x . i + y . j + z . k, then: y = j . V. Just compare: 8:29.
You nailed it elucidately , Prof. Strang. Now lam at peace with Fourier series.You have been precise , and hammered home the orthogonality point home, which is crucial to understanding of the Fourier series. REPLY
how exactly did he hammer the orthogonality point home? he never explained what the inner product represents graphically or logically as an integral and how that reflects on the functions we're looking at
I use to attend MIT, but not as a student. I was a janitor, but I had a penchant for non-linear equations and Fourier systems. One of the professors, a noble Fields recipient, would put equations for students to solve on a board outside the classroom.
I really hope at his age to be able at least to remember about ak and bk... I really love Fourier series but time will tell how all this will end up for me!
@7:25 when the professor said"this times this when i integrate gives zero"why is that i mean the orthogonality gives zero when talking this function how to relate between the two cases of vectors and functions? and from where the cos (kx) came ?& what is it or its nature ?
@@ekhliousful Here it is: Functions like f(x) are just vectors! Thus, the a's, b's and c's are components and the cosines, sines and e's are basisvectors. That's why mr Strang claims that this is true: 6:06. Of course, when you dot a basisvector with a vector f(x) you get a component. When V = x . i + y . j + z . k, then: y = j . V. Just compare: 8:29.
That's exactly what I was thinking.... everyone who already took applied math with Fourier Transforms thinks this is great. If you don't know anything about Fourier Transforms, you have no idea what the heck he is doing or why. That's why I always hated textbooks from Caltech and MIT. They were great if you wanted to go back and re-learn materials, but terrible at explaining basic concepts to someone who has no exposure. I mean who starts a lecture by saying I'm not sure where to start with Fourier Transforms, but what we are trying to get is a function with a coefficient for sine and a coefficient for a cosine value? My first question is WTF would you want that?
Anyone please explain the part from 3:40 - 4:00 a bit clearly .... Pls explain how the lower and higher values of k will change the frequency more visually??/
you can plot for example the graph of sin(2x), sin(5x), sin(9x) using for example:www.desmos.com/calculator you will see that the higher k the more it oscillates
at around 14 mins, why isnt a0 equal to 0 for delta function? we split the integral up from -pi to 0, 0 to 0, 0 to pi, so wont the integral evaluate to 0 + 0 + 0? ir doesnt matter that the function is one for only x=0
I was confused by this too. The en.wikipedia.org/wiki/Dirac_delta_function is "a function that is equal to zero everywhere except for zero and whose integral over the entire real line is equal to one".
it's because the delta function's definition is the derivative of the step function. thus the integral of the delta is the step, and since the step equals 1 at pi and 0 at -pi, the integral of the delta in that interval is 1-0=1
Thank you very much. But why we can find Fourier transform for delta function since delta function is not a periodic function. And why can we substitute delta(0) = 1. In the video, the prof say that delta(0) is infinite.
He doesn’t say the delta function at 0 is equal to 1, but its integral is. This is a property of the delta function, namely that its integral is equal to 1 whenever the extremes of integration include 0. Why this property holds is explained very nicely in ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/ (pdf: Delta Functions: Unit Impulse -- ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/MIT18_03SCF11_s24_3text.pdf) where delta is pictured as the limit of box functions of area 1 (11:08 picture of the delta function: ua-cam.com/video/vA9dfINW4Rg/v-deo.html)
Everything's clear to me except for one point. Coskx cancelling all the SinNx terms make sense. But how come coskx knocks out all other cosnx terms except the one with n=k? After all, its a dot product.
Did he teach the origin of Fourier series like why anybody want to represent any function in terms of sine and cosine functions.. If he did so, pls tell..
Good video! I think I discovered a little tiny mistake on the first board at the beginning of the video. Sir, you write Sum(...cos)+Sum(...sin)=sum(...e^i(...)). As the complex form gives complex numbers for each term to sum up, the left side only contains real numbers. I think you forgot the "i" in front for the sin-summation. :)
I've always been skeptical of this delta "function". How can we anchor this wild thing to a firm foundation? What value can we give delta(0) such that integrals of delta work the way we want them to??
You cannot give a value to *delta(0)* . Even worse, you cannot construct an integrable function *f* with the property you need for *delta(x)* : *\int_{-e} ^e f(x) dx = 1 for all e > 0* If you really want a rigorous foundation for *delta(x)* , you need to look into Schwartz' "Theory of Distributions" (sometimes called generalized functions). Just a warning: That is advanced math quite a bit beyond real analysis! But if you manage it, you will be able to prove amazing things like Shannon's _Sample Theorem_ rigorously, so please do not be discouraged.
Reason why MIT is at the top is because teachers can teach.
And students are willing to learn
@Mr. Gang Banger True, but, in addition, Strang is a terrific teacher. I am an engineering teacher myself and I want to be like him when I grow up ;)
Professor Strang is awesome, but I found this video to be a bit rushed. He ignored many subtleties. For instance the series produces a periodic function, so the example at the end actually gives (... + δ(x+2pi) + δ(x) + δ(x-2pi) + ...) which is a periodic version of the Dirac delta spike. Another point is that (given f is periodic) we can integrate over any full cycle, it doesn’t have to be -pi to pi. It’s also worth mentioning that we could produce functions with any period T by replacing x with 2(pi)x/T, but this slightly changes the coefficient formula (1/pi in front becomes 2/T). Also it’s not hard to show the orthogonality (it comes down to a simple trig identity) instead of just assuming it. Still a good video just a little too brief. Much respect to professor Strang though.
ua-cam.com/video/JF6skf4eaD4/v-deo.html
not really you have professors with amazing skills pretty much everywhere in the world, but in the mit all the professors also some of the best researchers in the world, and the ones that reinvented so many fields
"Ao has a little bit different formula. The π changes to 2π. I'm sorry about that."
Lol, legend. I love Prof Strang.
Good teachers make the materials easy to understand, so that students can learn. The problem is that many teachers teach poorly and that's why I'm here learning from YT videos. Thank MIT teachers!
Dr. Gilbert Strang is legendary -- absolutely love his lectures!
the man explained both linear algebra and Fourier in 15 minutes, while school failed to teach me in three classes. Astonishing mastery!
LOved this! Im studing electrical engineering and really needed this! I got low second five times, and I am on my second retake of my second year. Im also on acedemic probation at oxford brooks! this single-handedly SAVED MY DEGREE!!! thanks again! :)
I’m your professor and I hate you
After searching for countless articles on fourier series , this one really helps , many thanks professor !
ua-cam.com/video/JF6skf4eaD4/v-deo.html
My professor literally was like “yeah I’m not a great lecturer, MIT puts all their stuff online though you should check it out” 😐
This is an amazing opportunity to go back to the roots. Thank you for making this possible
Yes...... "Roots"
ua-cam.com/video/JF6skf4eaD4/v-deo.html
I don't think he could have presented this introduction to Fourier Transforms any better! Spectacular job, professor!
I feel more relieved for my midterm tomorrow now. Thank you loads, Professor. You're super awesome!
Thank you for the open courseware so we can learn from MIT around the world. Cheers. :)
after one year of searching finally i found a good stuff about Fourier series
wiche helped me to get evry thing
Thanks
Absolute mad lad. Cheers Professor Gil from down under! Loved your book on Linear Algebra.
By far the best explanation on UA-cam. Thank you!
Prof Gilbert Strang .. got me through Lin Alg all the way to graduating as a math major with honors. Wish I could take a real class at MIT
It blows my mind how any function can be represented as harmonics, truly something to know :)
This is gold
Pure gold
Diamond
What all else couldn't do in hours he did in minutes. But he is Gilbert Strang then.
God bless any institution that sets out to teach for the betterment of humanity, not selling sealed papers.
This is the best teacher I have seen in my entire life😮
I tried to look for other lectures about this subject, but nobody's better than Prof. Strang.
Really a awesome and comprehensible lecture on the basic concept of Fourier series.
To me , fourier was marvlously msthematical genius of geniuses. With much awe as to how he conceived the idea of heat propagation that can be expressed in terms of sines and cosines. With reverence to his life and works, services. Thanks.
God tier course, Gilbert Strang
is the best teacher I have seen.
I've read a couple of explanations and read several videos, and I find something missing. I remember old Gilbert Strang and what he tought me about Calculus and Linear Algebra, get here, I see the board, and just by looking at it I get enlightened. Thank you for everything!
3 years ago when I first learnt Fourier series this had been the most confusing part in that semester.
(my professor didn't spent much time on this because for some reason this was not going to be in the exams)
I tried to work it out and with my own interpretations but failed.
and since then I had been haunted by it, I come across Fourier from time to time in my study, I know how to apply the equations but never understand why these equations come to be like this, I never comprehended it.
Thank you professor Strang for saving me again! Your 18.06 lectures also helped me a lot!
Thanks for making this possible, MIT.
My professor "teached" us all the fourie and basic signals in 5 lessons... a true legend
I have achieved enlightenment watching this video.
Watch most other any video on Fourier Transforms and you'll see what a gem the teaching of Prof. Strang is.
This lecture helps me understand Fourier Series from start to finish.
I'm still here MIT. Though i know i failed my jee journey, be it due to my lack of effort,laziness,other life things. I WAS and AM still here.I might've been intrigued in the past and stayed here for what, maybe a couple of seconds?, but i roughly know where am i headed.I promise i will be here again even if on and off but one day, one day i will gain all the possible knowledge.All the things i need to know to atleast try to understand this complx world.I will definately one day fix myself and offer my works if god bless im able to do. I might not have it today, not tomorrow or maybe the day after.But one day i will.I still have not lost hope. I think i have tired even god helping me.I may be skeptical of everyhing but i will be there.I know i still got this.
Chup lazy lodu
Hello bro just here to remind you after 8 months that you’re not alone, keep going no matter what and let’s go beyond the limit of our natural perception and understanding.
I bow my head and salute to your teaching Sir. :)
Awesome Professor.
0 dislikes thats awesome. Thats the power of a great video. Keep up the good work Sir.
Sohams Day
There's 2 now :(
Now there are 12!
20 now
27!
These assholes!
A brilliant gem of a lecture. Thanks Prof.
He started this lecture where he left in laplace equation video, amazing series of lectures to vizualize each and every steps.
Whoah came from Mattuck's lecture on it and this is much clearer. So quick and easy to understand
I ran it in Python to test the Fourier series from the delta function, and incredibly, the series just plotted the delta function like a charm. Unbelievable!
I would like to see your code. hh not that I can't find it or write it on my own. but I would like to see it
A LOT OF WORDS FOR SOMETHING SIMPLE. Simple because functions like f(x) are just vectors! Thus, the a's, b's and c's are components and the cosines, sines and e's are basisvectors. That's why mr Strang claims that this is true: 6:06. Of course, when you dot a basisvector with a vector f(x) you get a component. When V = x . i + y . j + z . k, then: y = j . V. Just compare: 8:29.
His body might seems like old but his spirit and knowledge is high 👍
Old people are the ones with knowledge...
You nailed it elucidately , Prof. Strang. Now lam at peace with Fourier series.You have been precise , and hammered home the orthogonality point home, which is crucial to understanding of the Fourier series.
REPLY
how exactly did he hammer the orthogonality point home? he never explained what the inner product represents graphically or logically as an integral and how that reflects on the functions we're looking at
Strang is an awesome an professor makes the difficult subjects comprehensible
We all just witnessed MASTER at work!
You made this so much easier than my professor did today.....
Wooo! Prof. Strang is great! Even a dumbass like me finally understood the Fourier series!
Thank you
Wow. My whole semester in 10 minutes. Genius
This video literally made my jaw drop
one of the best lectures I have ever seen
This shows why MIT is good one!
This professor is just AMAZING .... hats off.
I use to attend MIT, but not as a student. I was a janitor, but I had a penchant for non-linear equations and Fourier systems. One of the professors, a noble Fields recipient, would put equations for students to solve on a board outside the classroom.
this man is more of a god i realized this when i listened to his lectures on linear algebra
Professor strang, you freaking legend.
Thank you so much. I’ll be eternally grateful.
like a boss. That was a very useful lecture. I got more out of that than other bits on the topic.
What if you don’t want the domain of ‘x’ to be limited to -pi < x < pi?
wished I had this professor when I was in school
I'm crying. It's so beautiful.
I really hope at his age to be able at least to remember about ak and bk... I really love Fourier series but time will tell how all this will end up for me!
Really good and great opportunity for the students
Thank you Mr. Strang, very well explained.
best platform to learn and concept clearence thanks
i love this guy and his explanation
@7:25 when the professor said"this times this when i integrate gives zero"why is that i mean the orthogonality gives zero when talking this function how to relate between the two cases of vectors and functions? and from where the cos (kx) came ?& what is it or its nature ?
Just read my reaction.
@@jacobvandijk6525 where is that?
@@ekhliousful Here it is:
Functions like f(x) are just vectors! Thus, the a's, b's and c's are components and the cosines, sines and e's are basisvectors. That's why mr Strang claims that this is true: 6:06. Of course, when you dot a basisvector with a vector f(x) you get a component. When V = x . i + y . j + z . k, then: y = j . V. Just compare: 8:29.
@@jacobvandijk6525 thanks for your time i really appreciate it
@@ekhliousful You're welcome, Ahmed.
This is quite informative IF you already have pretty good knowledge of fourier series
That's exactly what I was thinking.... everyone who already took applied math with Fourier Transforms thinks this is great. If you don't know anything about Fourier Transforms, you have no idea what the heck he is doing or why. That's why I always hated textbooks from Caltech and MIT. They were great if you wanted to go back and re-learn materials, but terrible at explaining basic concepts to someone who has no exposure. I mean who starts a lecture by saying I'm not sure where to start with Fourier Transforms, but what we are trying to get is a function with a coefficient for sine and a coefficient for a cosine value? My first question is WTF would you want that?
@@FergusScotchmani mean thats what youre lookin for when doing harmonic analysis
But i agree this isnt much of an introduction to the concept
Wow...Best video on Fourier series..
This is the probably the best class I ever watched(I already know the topic, I am just refreshing my memory)
But damn, I wanna take class from him.
Thank you Prof Strang for the wonderful explanations.
Anyone please explain the part from 3:40 - 4:00 a bit clearly ....
Pls explain how the lower and higher values of k will change the frequency more visually??/
you can plot for example the graph of sin(2x), sin(5x), sin(9x) using for example:www.desmos.com/calculator
you will see that the higher k the more it oscillates
Sinwt
How is this man so easy to understand?
Ok I got Fourier series. Now on to Fiveier
this one kool professor. thanks for the fourier stuff.
This guy's version of hell is 30 hands raised up, all asking if he can write in + C
Correct me if I'm wrong but isn't the integral from 1/2pi *( integral -pi to pi of 1 dx ) = 1/2*pi *(pi-(-pi)) = 1?? Maybe I'm having a brain fart...
at around 14 mins, why isnt a0 equal to 0 for delta function? we split the integral up from -pi to 0, 0 to 0, 0 to pi, so wont the integral evaluate to 0 + 0 + 0? ir doesnt matter that the function is one for only x=0
I was confused by this too. The en.wikipedia.org/wiki/Dirac_delta_function is "a function that is equal to zero everywhere except for zero and whose integral over the entire real line is equal to one".
it's because the delta function's definition is the derivative of the step function. thus the integral of the delta is the step, and since the step equals 1 at pi and 0 at -pi, the integral of the delta in that interval is 1-0=1
Very clear explanations! Thank you!
goodness does it make a difference when the professor's actually SPEAK English.
Muito obrigado pela belíssima explicação.👏👏🇦🇴🇦🇴🇦🇴🇦🇴
Thank you very much. But why we can find Fourier transform for delta function since delta function is not a periodic function. And why can we substitute delta(0) = 1. In the video, the prof say that delta(0) is infinite.
He doesn’t say the delta function at 0 is equal to 1, but its integral is. This is a property of the delta function, namely that its integral is equal to 1 whenever the extremes of integration include 0. Why this property holds is explained very nicely in ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/ (pdf: Delta Functions: Unit Impulse -- ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/step-and-delta-functions-integrals-and-generalized-derivatives/MIT18_03SCF11_s24_3text.pdf) where delta is pictured as the limit of box functions of area 1
(11:08 picture of the delta function: ua-cam.com/video/vA9dfINW4Rg/v-deo.html)
Nicely explained and in a very simple way
I wish we had professors half as good over at ASU.
Just brilliant tuition thanks!
teaching was so clear .
thank you professor
11:00-11:25
As written in description for this video Fourier series is used for periodic functions.
Is the Delta-function periodic function?
Any function can be made periodic if you allow it to be defined on some interval [a,b]
Thank you from Algeria
is the Dirac delta function satisfies Dirichlet's condition? I think this function does not show periodicity,
X,2×+5=8[n3]
I want to take a class like this!! JUsT WOWW!
That was awesome!!
Everything's clear to me except for one point. Coskx cancelling all the SinNx terms make sense. But how come coskx knocks out all other cosnx terms except the one with n=k? After all, its a dot product.
His spirit and methodology
Did he teach the origin of Fourier series like why anybody want to represent any function in terms of sine and cosine functions..
If he did so, pls tell..
This guy is incredible
Good video! I think I discovered a little tiny mistake on the first board at the beginning of the video.
Sir, you write Sum(...cos)+Sum(...sin)=sum(...e^i(...)). As the complex form gives complex numbers for each term to sum up, the left side only contains real numbers. I think you forgot the "i" in front for the sin-summation. :)
This is mistake in original theory itself, some how imaginary number "i" was introduced in the derivation, so as to meet the equality.
can't b_n contain i as well
Excellent lecture
Professor says, it is going to take 2 sessions to explain Fourier series but video itself is 16 minutes.
I've always been skeptical of this delta "function". How can we anchor this wild thing to a firm foundation? What value can we give delta(0) such that integrals of delta work the way we want them to??
You cannot give a value to *delta(0)* . Even worse, you cannot construct an integrable function *f* with the property you need for *delta(x)* :
*\int_{-e} ^e f(x) dx = 1 for all e > 0*
If you really want a rigorous foundation for *delta(x)* , you need to look into Schwartz' "Theory of Distributions" (sometimes called generalized functions). Just a warning: That is advanced math quite a bit beyond real analysis!
But if you manage it, you will be able to prove amazing things like Shannon's _Sample Theorem_ rigorously, so please do not be discouraged.
Sir which book you have preferred for this
Adamsınız profesörümm
this has been very helpful. thank you.