What does the Laplace Transform really tell us? A visual explanation (plus applications)
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- Опубліковано 1 лип 2024
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This video goes through a visual explanation of the Laplace Transform as well as applications and its relationship to the Fourier Transform.
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I am a retired engineering professor and taught this material for years. Yet I still find it very useful to see it presented by someone else since there is always some new perspective that I acquire. Excellent presentation!
How did you become a professor in this area if you didn’t imagine this visualization in your head? By just memorizing the formulas?
@@joy2000cyber Thank you for your question. I am now 82 years of age and we didn't have the visualization tools available today way back when I was in college. As an undergraduate I probably memorized the formulas and learned to use them to solve differential equations. When I was in graduate school we had to "dig deeper" and actually derive many of the formulas and procedures. Since my teaching was in engineering, we had procedures that vary slightly from those of mathematicians. I have taught much of the concepts using different approaches for much of my career and have written a couple of books based on electrical circuit analysis using transforms. Yet it is always helpful to see someone else develop a procedure because we can always learn something from another person's point of view. My best wishes to you!
William Stanley Good for you having other approaches in understanding the topic. I was a student in feedback control and didn’t really understand FT and LT, which might be the reason I became a programmer. What portion of your students do you think had a good grasp of the topic by the end of the semester?
@@joy2000cyber I suspect that only a small percentage really understood it on the first exposure and I suspect many never really understood it. To be honest, when I was first exposed to some of the more analytical subjects as a student, I obtained only a superficial understanding. Indeed even at my advanced age, there are still lots of topics that I don't understand! For example, electromagnetic theory has always been an area that has challenged me! Maybe in my next life (??) I will try to master that area. Thank you for asking and my best to you!
William Stanley I suspect most my classmates didn’t understand FT, LT, control theory too. I thought we didn’t have enough practice with the symbols of jw, s, etc. Now I think we just need the visualizations and am a little upset that my teacher didn’t show it. Thank you for relieving me of the guilty thought of not being smart enough.
I'm seriously amazed that people from the 1700s figured this out while i could only understand it fully after seeing the 3D plots...
Imagination
Seriously those guys were awesome.,
Then think about Einstein who give theories which work at speed of 💡🔦...
THEY DIDNT HAVE PORN
@@alex_linhares Ah yes they had to imagine all the sex scenes since they were teens which improved their imagination power.
Imagine people like me who grew up without UA-cam and had to try to understand this stuff from textbooks and teachers drawing diagrams on a blackboard. I'm convinced my teachers didn't understand the intuition and only knew the formulas. UA-cam is the greatest thing to happen to education in human history.
When you procrastinate so much that you watch a video about what you are procrastinating
Lol so true xd, i should get my ass out of bed
I feel personally attacked
Yep that’s me. I should be actually studying this topic for a test
thats me high af right now
Yah sometimes when I’m lazy I’ll watch some UA-cam videos related to what I need to study while I’m bed hoping that I make myself feel better. It’s not nothing, but definitely far less productive than working problems out and attentively listening to what is being said.
Probably the most productive 20 minutes of my engineering life
Bad engineering life you've got
@@anders5611 LOL
Honestly same. Did what an entire semester long course couldn't do
@@AlexJoneses Even though it was a semester long, to me, it felt like a rush. That's because it was packed with so many topics, that you would not be able to cover any topic this deeply.
This was brutal. I've been sitting in Differential equation/signals and systems classes in university and just performed laplace trafos without ever having a clue about it. Thanks
Mee too man
Most of classical education student did the same, It is not just you .
@@certyfikowanyprzewracaczhu3390 True...but ouchh. What did engineers ever do to you man??
I always thought that s was real not complex. But that didn't really matter since the end goal was to determine Y(s) and then take the inverse. Differential Equations (1 semester) class was already packed with so many topics that you don't have the time to stop and analyze the form of Y(s) and what it meant.
As a retired engineer, I find the use of visual tools and youtube really enhances the intuitive understanding of this topic! E.E. Professors beware!
Universities and your dilettante-creating accreditation boards beware, you'll soon be peripheralized.
Lies again? Lap dance
@@carnivalwrestler so its the same situation everywhere! Same here in India.
What UA-cam can't offer are credentials that employers will accept on a resume. Maybe if your employer is young he can relate with it if it comes up in an interview, but that is probably the only case, at least in the foreseeable future.
I have been an engineer for 12 years. I fully understood and loved the fourier transform but NEVER understood Laplace untill this moment. This was truly beautiful and comforting to watch. Thank you!
And I thought me not understanding it as a sophomore after a 90 minute lecture was bad...
I just finished my 3rd semester studying Electronics in IIT Kharagpur(the best college in India to get a degree in this field)....We were taught about Laplace and Fourrier transformations, pole zero diagrams, Transfer function, feedback, time shift, frequency shift, convolutions....yet I did not have this mathematical intuition behind Laplace transform...Zach please keep on doing this good work
100% truth.
You're to engineering what 3Blue1Brown is to linear algebra. Keep up the great work. Hey why don't you do some electromagnetics problems, those are fun.
Hay check out Applied Science.
To be fair, 3Blue1Brown has a lot more than just linear algebra. He has different playlists if you check his channel
You have an interesting definition of fun haha
@@rotatoe Lmao dont we all?
Bruh, you and 3b1b must do a collab.
Is it wanted to mix beer and wine?
@@DiamondSane Never drink and derive
I can Understand your Proposal but looks to me letting having Different Approaches, Different Sensibilities and Levels, ENRICHES; uniformity would be A RISK OF LOSS OF POTENTIAL RICHESSES, in my humble opinion. So I'd rather discourage Them to do so for the Very Benefit of All Of Us All ^_^
No,
I hope not, 3b1b has super annoying background noise (called "music"). Here, the background noise is a lot less audible, although it's still there. Honestly, I don't understand the point of adding distractive "music" to the speakers voice.
What was understood in class: "Laplace transform is that thing you do if some letters have more dots above them than usual"
HAHAHAHAHA Exact words of one of my professors when he got mad cause no one was paying attention and got tired of making and effort for students that don´t care. I had to go to his office for him to explain to me personally,
And in the textbook for my senior-year instrumentation class: "All you need to know is: Laplace means derivatives get an s and integrals get a 1/s"
That´s for mechanical engineers; we, the Electronics guys, don´t put dots on variables to represen derivatives. We use prime signs instead, and only for one semester, for all the next semesters we live in the Laplace realm, a bunch of "s´s" everyday.
@@dielaughing73Basically operational calculus.
This is one of the most enlightening videos on UA-cam. I have a hard time understanding things if I don't have an intuition about them, and this makes the concept super clear to me. This video should be required material for anyone learning this subject in school!
*Me:* _nods while not understanding anything_
Hi Mr. Obama! I didn't know you were into math like this. How was it like as a president?
Every time a new concept is introduced that adds on to, rather than explains, the previous concept.
"But of course..."
😆 almost no one gets this the 1st time. Gotta meditate on it, sleep on it, review it and try to explain it to someone else. Next thing you know, your an expert! Youse gots dis!
I’m just over here wondering what S is
@@Gideonrex1 yeah, majoring in Math or Physics is college on hard mode. They tried to teach us about Laplace transforms in sophomore year of a physics major. It made no intuitive sense to I always felt lost and confused.
In 20 minutes I learned much more than what I learned in 2 semesters of my sophomore year.
Don't worry. You'll forget about all of this faster than you might think.
@@lonestarr1490 please write those down in a word file and keep.
He even points out in this video that you likely wouldnt learn this in class because it’s not important. What’s most important in class is where the poles are?
@@lonestarr1490 true 🤣
So you didn't learn anything
You deserve a prize for this. It helps to "feel" what i have been doing just blindly for years in my engineering studies. Just perfect. Thank you.
ChemE here. When I took Process Dynamics and Controls (20 years ago) I never really understood Laplace transforms. I could do the math, but I didn't really understand it. And after watching this, I still don't really understand Laplace transforms and my 20 yr old PTSD has been triggered...
That being said, loved all the visuals in this and how you slowly stepped through the progression. If you had been my prof, there's a chance (although certainly no guarantee) that I would have done better in the class.
3D visualisation of both Laplace & fourier clear all my barriers between Lap & Fourier. Thank u.
I've been wondering what's the point of linking poles and zeros of control systems with transforms of signal processing. After years in dilemma, now I breath a good relief.
Simply phenomenal.
I've been struggling with Fourier and Laplace for a year and this simplified everything. Thank you from the bottom of my heart.
This is the video I was waiting for since years. Laplace transform and Fourier transform gave me a lot of uncertainties and I spent a lot of time trying to figure this out. You did an excellent job with this. Keep going man, this is the quality content that UA-cam needs and deserves.
I've just learnt whole control system engineering in 20 min. respect
so glad i can just chill out and learn this w/ no time restraint
I'm in my last semester of electrical engineering at university. I've dealt with Laplace transforms for over a year now but I gained a much deeper understanding of the Laplace domain in just the first 2 minutes of this video. I've known how useful they are for a long time, but for whatever reason your wording finally allowed me to understand why. Thanks for making this, and the visuals were beautiful.
How you doing now?
Engineering educator here. Love the visualization and the explanations! However, I find the arbitrary use of upper and lower case letters a little unfortunate (you could say it's all over LaPlace...). The common convention in engineering is to write the time domain function in lower case, the transform in upper case, and all variables in lower case (t as well as s). correct me if I'm wrong, but you mix and match for no apparent reason.
I agree
Fourier Information, I really enjoyed this comment. Sad laugh ensues.
You are right.
@@Methodwake noo, you can't just make me die laughing in l'hospital
@@proloycodes hopefully your health stays in acceptable limits that are determinate
The learning curve for advanced maths always seems so steep until I stumble across videos like these. Truly amazing. You explained half of my module in 20 mins.
I've never watched a 20 mins MATH video at one go, trust me this is GOD LEVEL !
THANK YOU !
Thank you so much. You really opened my eyes to the beauty hidden behind Laplace and Fourier transforms ! We need more people like you on the internet.
This is incredible. I've just spent nearly a week fumbling around the internet trying to understand the Laplace transform because my professor swears up and down that he can't explain it any better "because there's no visual explanation." This was the visual explanation I needed. Thank you so much.
Great take on this! I haven't revisited the Laplace transform since college. This is a very useful intuitive understanding of the concept that tends to be masked by the mathematical rigor of academia.
My man! I've been thinking about this for quite a while.
The way you explain and introduce the visualizations of the various transformations is just so boss!
You just Made me easily understand 2 years of University trying to understand Laplace transform, specially when everything came together in DSP this semester which I just failed. THANK YOU, please keep doing what you're doing, more people and institutions should teach the way you do
Really, really good work to create such accessible visuals. This is an excellent instructional resource!
Oh my fucking god! I had studied this almost like 8years ago... And I couldn't understand a bit of what I was doing! Matter of fact I didn't even realise then what's its importance... After watching this video everything is so much crystal clear. This video helps sooo much more! Infinite gratitude to the creator!!
This is absolute gold! As a freshman taking Signals and Systems, this provides much needed intuition!
I LOVE YOUUUUU, I looked everywhere for an intuitive explanation of laplace transforms. I undestand what the fourier was doing but struggled to grasp laplace until this video!
I wish I saw these videos when I was taking analog signals and systems, and digital signal processing. Great video.
Years of institute training wasted
Now these are brand new
My English sucks, i know. But im nơt Chinese
gachiHYPER
This is the most liked comment i have ever had, probably because of my broken English? :)
@@abcdefghijklmnopqrstuvwxyz1062 nah you are very right; lots of great education out there now, which does make old forms of schooling, feel wasted.
Quân Đặng some Chinese with poor English would not approach this kind of videos.
Due to my poor time management and the amount of material to study, I ended up blindly memorizing most of the stuff from control theory class, which comes to bite me every now and then. This is brilliant and much appreciated. Thanks a lot!
Zach, you are seriously incredible at explaining this and then there is also your very clear graphics. This would have made a real difference to me when I was studying this formally before the internet. Don't ever doubt you are contributing something very important to the world.
Amazing! Really helped understand this concept. Keep up the great content!
Might get too specific but I would love a follow up video about poles, zeroes, stability and root locus (all that good controls stuff) reinforced with a real world application.
Damn, This channel just keeps getting better and better
Couldn't grasp it in a full semester.. you explained it in 20 mins. Awesome work bro.
This video is just too good. You (and some other UA-camrs) have revolutionized the teaching of some very difficult subjects in Math and Engineering.
This video would have been perfect last term when I was studying control engineering!
This has to be the best math video I’ve ever seen. Well explained, dense in information, and understandable by experts and those only with basic math knowledge.
I like the visualization of graphs with changing the values of parameters in expressions that imparts a good understanding. a beautiful explanation of the relation between LT and FT.
I think this is the best, and most clear explanation of the Laplace transform i´ve ever seen, thank you for having done this amazing work
This bloke just explained almost half my degree in 20 mins. What a champ
05:42 for people checking the math, there's a t missing in the laplace equation after substituting s = alpha + i•omega
also, the next equation would lead to a t^2 in the exponent
Excellent video. Excellent refresher overview with excellent graphics.
I cannot thank you enough for this video. I've taken my calc classes almost half a decade ago, and having recently been able to return to studying, this was the explanation that finally completed this huge gap that this specific subject had left in my education as an E.E undergrad.
Great Tutorial. As a BSEE, I learned more in a few minutes than I did in years at the university.
it's crazy because visualization is ignored in schools and colleges
Thank you so much! You can't imagine how much time I waited for this explanation in class (and it never came) and also in UA-cam. I always had this doubt about what did you win changing from Fourier to Laplace. This graphical way of describing it helped me a lot.
Btw I am a graduated engineer and physics student, but with control systems and differential equations is clear that you never stop learning new things and new interpretations to things you think you understand
what are you talking about lol
I don't even know rn hahah
A slice of Fourier. So well explained! Needed this now. Thanks.
The visualization, the explanation, that all was so magnific
thank you
Sweet! I've been secretly waiting for this video from you!
This was really good, it provides intuition of the idea of the Riemann surfaces (graphs) of analytic functions. Splitting into trig and exponential sums is the polar equivalent of rotations and dilations/stretches. It’s Abelian so it doesn’t matter the order, I can rotate the plane by 30 degrees and then expand by two, or expand the whole plane by two and then rotate 30 degrees. You end in the same place. That’s analytic transformations, thought of geometrically. This is them thought of in terms of the underlying calculus and complex differential equations. It turns out to be the case in even higher dimensional complex geometry. The geometry is one side of the coin, and the other side is a bunch of complex PDEs. As a geometer I prefer and work in the geometric side because it’s easier, it’s mostly fiber bundles over semisimple Lie groups, but the analysts doing the PDE side do amazing and technical work I don’t know if I have the patience for lol. And I’m mainly talking about complex analytic geometry, where only noncompact and even dimensional geometries are allowed/ only even dimensional make sense. There’s of course the other dimensions, and that’s contact geometry and CR geometry, basic example like real analytic hypersurfaces inside complex spaces like spheres, and the boundaries of domains in C^n.
Amazing Presentation! I was looking for this kind of explanation for a long time and thanks to you who made such good content. This video explanation killed all my doubts about Laplace's transformation. Lots of love and respect!
Best explanation I have ever seen. Thank you so much, I now understand Laplace way better than I did from my university.
Now I understand why camels have humps: camel drinks periodically, and the amount of water decays in its organism with time. The L-transform of that results in appearance of 2 humps or sometimes 1 hump, depending on alpha - related to genetics?
Sorry bruh, but i laughed hard at this one 😂
Maybe this is deeply related with DNA in human body
😂😂😂
God bless you. I wish that from the bottom of my heart. I'm so elated after watching this video. In my college years I used to have discussions on control system with my friend Aditya and used to imagine what you so brilliantly have explained here. Every child out there deserves a teacher like you.
What an explanation?
Mind blowing...!! Amazing lecture explaining about Fourier and Laplace
This video is such an amazing job!
I could understand so much in a few minutes than all my teachers wanted me to know by 2 years.
Thank you ao much!!!
I noticed a minor mistake at 5:21. After rewriting s as 'a+iw', the t disappears from the second line. Not important but can cause confusion for those following closely.
I have no clue how I passed this course back in University. No clue.
Truth. I took this class as a senior in high school at a local university. I loved every minute of this material, and I have no idea now how I understood it then when 3D graphics were not nearly as available. (I'm 40 now)
Bless me too
Great work! I think your visualization and explanation of the Laplace transform using a simple to understand spring-mass-damper system really helps!
I like how everything just comes together at the end. Great stuff!
Could you kindly do samething on Wavelet transform ?
This video is great,which tell insight of of laplace transform root locus
Great video majorprep !! Really love this content and appreciate the efforts you put in for this video... could you make video on real world application of bessel function ? I heard that in Japan somebody use this to make letters on water surface ..not sure how they made it
This is the best explanation of Laplace and Fourier transform I've ever seen. Great video!!!
Amazingly done. Kudos!
Great video, but I needed this 10 years ago. ^^"
Well, now it's a great reminder of what I used to know in college. And it makes me a little sad considering how much I've forgotten since then.
It's impressive how much I understood from this even though I haven't learned calculus. There were occasional things I didn't get at all, but I learned a lot just from conceptualization.
if you can't visualize it and manipulate it, you can't understand it.
This is wonderful. I had the basic concepts down but never really fully visualised the s-plane in 3d. We were pretty much taught to treat transforms like a machine where you crank the handle and get results. I've always wanted to understand the underlying formulae and what they actually do. This visualisation is on another level and really clears it up. Really well done! Thanks.
This video has literally changed my entire experience on my degree course. Thank you!
Studied this 45 years ago.
Could do the sums and get the correct result
Finally, after watching your vid, I have an idea of what was happening, thanks
that's how I see it too.
This video is a must for all ee
We all took controls engineering, passed the exam but I think very few people actually understood why a pole on the left was stable. I sure didn’t.
Totally agree, the Profs I had (with Big egos) were more interested in imparting their math skills rather then teaching us visual control theory which we needed to gain a livelihood.
@@dol3980 I've noticed there seem to be to kinds of engineering professors, those who like the math, and those who like its applications.
The ones who like the math are terrible. Sure, everyone can pass their class, but nobody actually learns a thing while they do it.
The moment I saw the two poles in the video(keyframe) with the word Laplace transform written in title and views approaching a million... I knew it. I have finally got to something I eagerly awaited. Laplace transforms were difficult to understand in college. Just unbearable is the pain of teachers with a 2D black board to draw and explain what they perceive in their minds. Looking at the comments here I agree with most... some very funny. 7% humans who were ever born on earth are alive today. May be millions and billions more will take birth and watch this video. Out of 7% humanity alive now may be 0.07% of humanity has got an opportunity to learn engineering ever. Forget those who invented these things or knew about them but are no more alive...I bet every engineering professor who hasn't watched these kinds of videos would salivate and thirst for more.. Let alone the students. Very much appreciate the greatness of all minds which are able to comprehend these things. Just wondering where all are Laplace transforms getting applied.
Explained this better than my control systems univeristy lecturer, very concise and easy to follow.
Damn throwing shade on the engineers. 16:14 "Like good engineers will say if the mass is 1 then the force is 10N" oof
Is it really throwing shade though? Setting the mass as 1 and gravity as 10m/s^2 allows for really quick calculations as shown there. Allows for an understanding of the system with very little effort at all.
@@TheVivi13 engineering = lazy maths
@@Blox117
Engineering = Making things work, Even when your theory is wrong ... !!
@@Blox117 It's crazy how lazy maths leads to literally everything in the world around you working huh. If you REALLY think that when it actually comes to developing and building stuff, Engineers still use estimations in their designs, then you probably should go to school more.
so many butthurt engineers lmao
Hi major prep love yo vids
Great presetation with graphs and 3D representation. Really well explained. A huge thanks.
This is a truly BRILLIANT explanation! Thanks! Now I do understand, in particular, why poles in the left half of an s-plane indicate a stable system. I guess this topic in general will make much more sense to me from now on.
"Like all good engineers we'll assume that the mass is one, so the gravity force is 10N".
I too enjoy trolling physicists :P
Why stop there. π=3 e=3 sin(x)=x
@@poisonpotato1 π=e=3sinx/x..... The Engineer's Identity
Or we could gather up about a thousand miles of dirt from elsewhere in the solar system, and cover our planet. Then we can have g equal exactly 10 N/kg.
Differential equations was easily my favorite math class
The idea of rates being linked to states is super satisfying.
The best explanation in the entire youtube! Thank you!
Outstanding work Zach.
Masterfully done. Thanks. Subscribed!
This is a really nice explanation of the Laplace Transform. Btw, at 5:22 you are missing t in the exponent in Laplace transform.
a moment later t was back. Appearance from nowhere, like the virtual particles in quantum physics.
now I've seen the video 5 times, but I have to keep interrupting the video, the beautiful pictures have to sneak into my brain and anchor there.
0:28
he is going to assume the damping force as a 'multiple' of velocity.
for those who did not understand this, it simply means F = k v.
Thank you so much! I never knew how amazing the laplace transform was! It converts calculus to algebra and gives us information about the type of function at the same time!
watched lots of material before this video, this dude unlocked meaning of this transform to me in first three minutes good job, subscribed.
i just failed my laplace test today and here i am watching this lol
In order to succeed, you must learn from your failures and improve. Don’t worry, I believe in you. Just practice, practice, practice😃
@@navjotsingh2251 Practice does indeed help! Pretty comfortable with laplace now. Now I need to prepare myself for higher level circuits classes next semester...
Fuxk man why isn't there teachers like you people will become genius just after learning from you
What on earth makes you think there aren't teachers who know and teach stuff like this? If you didn't experience them, that doesn't mean they don't exist. I absolutely hate it when people disparage teachers like this. It's beyond rude.
@@michaelmann8800 how about replacing "isn't there" with "aren't there more"?
*aren't
@@siddarth_vader To answer that, let me ask you a few questions...Do you think the learning environment is created solely by the teacher or the quality thereof, or is there more that goes into it? Do students themselves have a role to play? How much time do you think is required to not only understand such topics to the degree that these people creating these videos do, and even more the amount of time it would take to create visuaiizations as these videos use to effectively impart that information to learners? Given just how much information teachers have to expose students to in order to provide as complete a picture as they possibly can over the breadth of the course, do you think they would have the time to do such stuff for every lesson? And let's say by some magic of physics they were able to create a time bubble where they would have all the time they needed, just for the sake of argument...are they compensated for that effort to the degree that they should be? I mean, if the idea is that teachers are supposed to be so good that they can "turn people into geniuses" just by teaching them (note, this apparently requires no effort on the part of the learner), then shouldn't they be getting paid like rock stars and CEOs?
My point is that education is not a trivial exercise. Too many people act as if it is a trivial exercise. There aren't more teachers able to do this to this level because there is limited time and much to cover, it is simply impractical to try to teach like this, and teachers are not paid anything even close to what they should be for the required level of effort.
I know people are going to dislike this because it seems to be a worldwide pasttime to hate on teachers.
@@siddarth_vader How about replacing "how" with "How"?
Nicely done, Thanks for giving the 'feeling' of these explanations.
I have been trying to understand laplace / fourier for so long and you did it in 20 minutes congrats
I took Controls 15 years ago and I didn't actually understand the root locus until right now.
I saw it 3 years ago and im happy I saw this video now and not 15 years later
@@YamilSG I asked my prof the real life application of root locus and he said "in designing microchips" thats all......its like i could have googled that, fucking show me how it works dude...
5:20 a t is missing in the exponent of e under the integral, but a very nice video nonetheless.
This video was far more useful and well thought that my entire Superior Mathematics and Control Theory classes. I got out of those knowing nothing. Now I have a sound comprehension on the material. Thanks!
I came in knowing Fourier transforms, but left knowing Laplace transforms as well.