The juxtaposition of the howls and the seriousness of the exposé is absolutely hilarious - you can't make that up. All in a sudden, I want to re-read Ginsberg poetry.
It's the z transform next, and then you will have the joy of discrete signal processing! I envy you, I just loved that so much. Just think for a minute, you have all of these new vistas opening up for you to explore. If it pains you, then you are on the wrong course.
Honestly.., been seeing commendable comments so far but as for me I rather feel ur not breaking this down enough and rather just jumping into solutions without even telling our it was brought about in the first place
I was thinking the same thing, then I realised our man here is being very specific about the topic he is discussing. One is expected to already have mastered primitive functions and integration. If you look at it from that angle, it makes perfect sense that the format of the video is what it is. It would make for a several hours long mammoth of a video if he had to explain this by starting from the law of identity. Besides, you only need to look at it and you should be able to tell it's mechanics, if you've done any meaningful integration in the past. After that, all you need is to cobble together a few lines of code and never have to touch this ever again.
I was just starting my journey on laplace today and i love that you uploaded this today. Honestly on of the best yt channels there is. Keep doing great things sir because you make a great impact
@@DrTrefor Thats great to hear, its sad honestly about this outbreak however it is really awe inspiring how all of us are coming together for this. Keep up the great work sir, you have helped me and many others and you will help more.
Laplace didn't come up with this method. He had a similar transform that is more like the modern Z-transform. It has properties in common with the Laplace transform, but isn't the form we know today. Heaviside and Gustav Doetsch are the ones who came up with what we call the Laplace transform today. The Fourier transform is rightly named in honor of Fourier.
Yeah its actually lap lace transform. Back in the days women wore lace skirts and they bunched up in their lap. So they invented a transform for solving the mechanical PDE of the skirt folds, hence the lap lace transform was born.
@@im_cpk a system you're designing or modelling in which parameters change over time. For instance, in chemical engineering you use laplace transforms to design reactors and model their reactions so you know how big the reactor should be, what the reaction conditions are etc.
How am I even supposed to understand something that's not fully explained (anywhere), like no one bothers to explain what even the purpose of laplace transforms is, you're just supposed to do it. Yet that's what I'm graded for and even if I get a good grade I would still have no clue what I'm actually doing. Kind of bizarre.
They are useful when predicting the performance of mechanical systems. ODE’s are needed to solve these types of problems and often involve initial conditions, and complicated systems have complicated ODE’s which require the Laplace method in order to effectively solve them because all the derivations get very complicated. That being said I have no idea how to solve these things.
Outstanding presentation! Incredible clarity. I never knew that the gamma function is the connection to the factorial, thank you so much for making this!
Thank you for your effort on this video. You should start with the Fourier Transform. Even better is to start at the Taylor/McLauren Series. Can’t expect newbies to relate to this in depth material.
Your cadence (the way you speak) is very helpful in retaining attention and making the material easier to stick with and follow. Thank you for the video!
It has been ages since i learnt and later forgot about this topic. I am now looking forward to re-learn it from you. Please speak slowly throughout so that it becomes easy to understand your words. Except for this, you are simply wonderful. Can you give examples of application of Laplace Transform in financial mathematics?
Thank you SO much for creating this playlist. Would be greatly appreicated if you could kindly create a PDE playlist. Your videos provide an initution approach which are incredible.
Where was this video 40 years ago during my undergraduate diff eq class? I recall it being much harder, including the gamma function giving me cold chills down the back of my spine
@@DrTrefor Yes sir. I feel like I have most of the intuition down, now I just need to amass a large amount of solving problems. Probably work my way through a few previous exams, that should do the trick
Believe it or not, when I started to watch this video, I was in the bad mood. But now I'm smiling and my feeling is changing ... Great job ..... Thank you so much 💞
It's coming actually! About 3-4 months away. Finishing Vector Calculus first then moving to differential equations and it will be part of that playlist.
Thanks for the video. I need to understand how an exponent can be complex, s = σ + jω, and what it means. This is not explained. Also, as far as I know, Laplace transform is used to cenvert a continuous function in the time domain, into a function in the frequency domain. Normally, poles and zeros are presented in the complex s plane.
To understand what it means for an exponent to be complex, it all comes down to Euler's formula, to make sense of the imaginary part of the exponent. Essentially, it rotates the number in the complex plane, instead of scales it, like a real exponent does. Given a general complex exponent of a+b*i on Euler's number, we can split the exponent with properties of exponents. a and b are real, and combine as discussed to form a complex number. e^(a + b*i) = e^a * e^(b*i) e^a is a positive real number, so it's just a scaling factor. e^(b*i) is what we unpack with Euler's formula, which gives us cos(b) + i*sin(b) What's behind Euler's formula, is the Taylor series. Use the Taylor series of e^x, and plug in an imaginary value for i*theta for x. We can do this with first principles of complex numbers, because a Taylor series is just arithmetic and integer powers. You'll get an infinite series of real terms with even exponents, and an infinite series of imaginary terms with odd exponents. These two series, are Taylor series of cosine and sine respectively.
Is good, managed to easily understand everything. But, with all due respect, it lacks a lot in terms of explaining. I mean, is entirely theory, but nothing about how it comes to appear this Laplace Transform. I think you may agree with me that, when it comes to maths, there is ever a logic and somewhat simple explanation to the very reason because "a thing" is "created" (or, well, defined. You get the point). Integrals has all that Riemman's Sum behind, Taylor Series all that convergence thing behind, and so on so on so on, what I am trying to say is that there is a reason for "something to be like it is", and for newcomers or just people that doesn't fully understand this, the explanation (that it is almost always an "intuitive" explanation) could be of very very great help. Don't misunderstand me, the video is of course excellent
Hmm, interesting. Utilizing e^x's property to stay the same despite being integrated, such that you can integrate over and over again? Makes a lot of sense. Question to self: what other functions do that? The sine functions do something similar, which I guess allows us to display waves over and over again.
Sine functions are linear combinations of exponential functions, so no surprise there. If you have some polynomial of the derivative D, say p(D), and you have the equation p(D) = 0, then the solutions are going to be some linear combination of exponential functions. This is because the exponential functions are the eigenfunctions of the derivative operator.
I am so damn mad that no one ever explained the Gamma Function and n! like that! I had to learn that on my own when I was in college (My Calc II professor was horrible). It was a good thing I did because when I took Differential Equations (Last semester in college), I had this insight and things were not confusing for me. I appreciate that you explained the Gamma Function with rich substance because many students do not get the explanation to why it is equal to the factorial.
What he means more accurately, is that the real component of s has to be greater than a, for there to exist a Laplace transform of an exponential function, e^(a*t), in order for the improper integral to converge.
@@nathangmail-user8860 There's an extension which has all the historic dislikes from before December 2021 and any new dislikes after are estimated from the current users with the extension, I'd recommend it 👍
Well, it gives people an opportunity to engage in the discussion and that in turn enables the algorithm to realise what a great video this is. Otherwise you have to wonder at people even clicking on a maths video when they obviously don't like maths.
Sir, can you put a video for Gamma of half integers input and how really this gamma function was brought into this form .... you really explain very well
With all do respect.. you had to focus on just Laplace transform and stick to it giving more examples about it. The Gama transform is another subject that confused me much while I am trying to understand Laplace, also the u function is confusing. Anyway.. your explanation is great. The winds give more horrifying feeling of the complex stuff. You could record your voice separatly and add it later to the video.
What is the point of using the Laplace transform? Aren’t we supposed to transform back to what we started with? I thought the purpose of the transform is to make the integration of the initial problem doable.
Ya that's right. Basically you transform, then you do algebraic manipulations to clean stuff up, then you transform back. We do this a bit more further down the playlist.
6:31 I don't understand where the "1" comes from. This is the part where I'm supposed to input "e^{-st} * f(t) dt" where f(t) = u(t-a), am I correct? How does f(t) become 1?
The unit step function, u(t), is defined as an abrupt jump from 0 to 1, at the value of t=0. The general unit step function, u(t - a), has the abrupt jump happening at t=a.
Great explenation but i have got a question: if s is a variabel how can we then integrate with redpect to x? You can't integreat a function with two variabals with tespect to only one of them.
You can! Basically what you do is hold s as a constant and integrate with respect to x where you treat anything with s identically to how you would if it was a constant.
@@DrTrefor sorry that i have to ask u again but if we can treat s as a constant when integrating with respect to t coudn't we solve any differential equation like that (at least 1st order odes) . What i mean is coudn't we just multiply both sides by dx and then integrate the one side with respect to x and the other with respect to y even if the x and y terms are not sapetated?
You can use a Fourier transform (special case of Laplace transform) to filter out the wind noise in the video.
This deserves an award! LOL
It's a varying frequency, I don't think so
lmao:D I didn't think of this so I just got a new office and a new mic instead:D
@@DrTrefor Ha. Nonetheless, thanks for the great videos!
@@DrTrefor lol, we should be able to practically implement what we have learnt
The wind is the soul leaving my body as i learn Laplace Transformations
😂😂
I was thinking the same thing 😂😂😂
The juxtaposition of the howls and the seriousness of the exposé is absolutely hilarious - you can't make that up. All in a sudden, I want to re-read Ginsberg poetry.
هههههههههه
It's the z transform next, and then you will have the joy of discrete signal processing! I envy you, I just loved that so much. Just think for a minute, you have all of these new vistas opening up for you to explore.
If it pains you, then you are on the wrong course.
some say you can even hear the screams of the horrified students...
I really heard some sound oooooooooooooooooooooohhhhhhhhhhhh
0:42. Bruh😂😂😂
Waahhhhhhgggg
Lol😂
Once it hit me - this prof looks and sounds just like my barber - the subject got a lot easier.
You are one of the few that made a proper series of the Laplace transform. Much appreciated. Keep up the good work!
Honestly.., been seeing commendable comments so far but as for me I rather feel ur not breaking this down enough and rather just jumping into solutions without even telling our it was brought about in the first place
I was thinking the same thing, then I realised our man here is being very specific about the topic he is discussing. One is expected to already have mastered primitive functions and integration. If you look at it from that angle, it makes perfect sense that the format of the video is what it is. It would make for a several hours long mammoth of a video if he had to explain this by starting from the law of identity. Besides, you only need to look at it and you should be able to tell it's mechanics, if you've done any meaningful integration in the past. After that, all you need is to cobble together a few lines of code and never have to touch this ever again.
The Wikipedia article on this topic freaked me out. It is so outstandingly presented and I like his style.
I was just starting my journey on laplace today and i love that you uploaded this today. Honestly on of the best yt channels there is. Keep doing great things sir because you make a great impact
@@DrTrefor Thats great to hear, its sad honestly about this outbreak however it is really awe inspiring how all of us are coming together for this. Keep up the great work sir, you have helped me and many others and you will help more.
How did it go? Where are you now in terms of math?
@@brandonmohammed9092 I'd like to know too.
The video becomes more exciting because he is happy to explain the topic.
Shout out to Pierre-Simon Laplace for this life hack
Laplace didn't come up with this method. He had a similar transform that is more like the modern Z-transform. It has properties in common with the Laplace transform, but isn't the form we know today. Heaviside and Gustav Doetsch are the ones who came up with what we call the Laplace transform today. The Fourier transform is rightly named in honor of Fourier.
Yeah its actually lap lace transform. Back in the days women wore lace skirts and they bunched up in their lap. So they invented a transform for solving the mechanical PDE of the skirt folds, hence the lap lace transform was born.
Your enthusiasm makes your video much more interesting.
Glad to hear that!
Cheers for these vids im currently doing Laplace Transforms for Maths Undergrad so this came at a perfect time.
This is an excellent, little lecture. Thank you Sir, for this and other fine series in the field of mathematics!
You're very welcome!
Laplace transform is very important when you try to design a dynamic system.
But , what is Dynamic System?
@@im_cpk a system you're designing or modelling in which parameters change over time. For instance, in chemical engineering you use laplace transforms to design reactors and model their reactions so you know how big the reactor should be, what the reaction conditions are etc.
you are a good man, thank you
Kudos Trefor - great contribution to subject - much appreciated
My pleasure!
I got 90 on my math systems exam. Hurrayy!!!
I remember solving these problems in undergrad!! Well explained
Happy Teaching!! ✌️✌️✅
How am I even supposed to understand something that's not fully explained (anywhere), like no one bothers to explain what even the purpose of laplace transforms is, you're just supposed to do it. Yet that's what I'm graded for and even if I get a good grade I would still have no clue what I'm actually doing. Kind of bizarre.
i can relate to this so much lol
They are useful when predicting the performance of mechanical systems. ODE’s are needed to solve these types of problems and often involve initial conditions, and complicated systems have complicated ODE’s which require the Laplace method in order to effectively solve them because all the derivations get very complicated. That being said I have no idea how to solve these things.
Its super useful for mathematically modeling systems for simulation. Eg control simulation in simulink
The hum in the background adds a vast loneliness atmosphere. I've got different emotions while listening this lecture and lost in deep thoughts.
Outstanding presentation! Incredible clarity. I never knew that the gamma function is the connection to the factorial, thank you so much for making this!
You're the best Sir. The explanation is very clear, much appreciated
Thank you for your effort on this video. You should start with the Fourier Transform. Even better is to start at the Taylor/McLauren Series. Can’t expect newbies to relate to this in depth material.
All I can say is thank you very much, I love the way you explain.
What are you doing step function?!
It helps illustrate concepts since its values are 1 and 0 (it's also causal).
@@paschikshehu7988 bruh
it's helping you out since you're stuck
@@sowickk Hey step function, you must really... like math, huh....
I just say ....outstanding❤❤
your presentation is awesome
Your cadence (the way you speak) is very helpful in retaining attention and making the material easier to stick with and follow. Thank you for the video!
Love your content and I am doing my dissertation on the theory and applications of Laplace, this is a great help!
Glad it was helpful!
It has been ages since i learnt and later forgot about this topic. I am now looking forward to re-learn it from you. Please speak slowly throughout so that it becomes easy to understand your words. Except for this, you are simply wonderful. Can you give examples of application of Laplace Transform in financial mathematics?
Bro just play the video on 0:75x speed ....that's good to understand us.
You Are The Best....I Can't Explain In Words...
Very useful for me thanks u so much dear sir 🙏🙏🙏 Namaste because i am an Indian.❤❤❤
MASTER CLASS!
I was doing a video on this topic. I referred to this just for additional knowledge 😊
Thank you for teaching!
Thank you SO much for creating this playlist. Would be greatly appreicated if you could kindly create a PDE playlist. Your videos provide an initution approach which are incredible.
I do plan to do more pde/Fourier stuff in the future:)
Professor, Your Affection with us greatful !
That's really helpful and will be to everyone watching this pls continue posting vid like thse
amazing explanation of the formula @ 2mins
Where was this video 40 years ago during my undergraduate diff eq class? I recall it being much harder, including the gamma function giving me cold chills down the back of my spine
my left ear really enjoying this
Thank you for this great explanation!
I like the explanation..will re listen this on repeat 🔁
Excellent video sir.
Ah, yes, beginning yet another one of your series. Amen.
haha you are crushing these, did you make it all the way through vector calc?
@@DrTrefor Yes sir. I feel like I have most of the intuition down, now I just need to amass a large amount of solving problems. Probably work my way through a few previous exams, that should do the trick
Believe it or not, when I started to watch this video, I was in the bad mood. But now I'm smiling and my feeling is changing ...
Great job .....
Thank you so much 💞
Knew I'd love it before I even watched.
your way to explain this topic is so good.
Thanks a lot 😊
Danke you! Exellente explanation!
not know whcih language this hehe
Nice Explanation Thank you
thank you for this amazing explanation. very well presented 😌.
Glad you enjoyed it!
This was amazing
Thank you for making this series. I was waiting for for from a long time. Thanks alot ❤
It's a GREAT HELP. Thank you again.
Fantastic presentation! Outstanding explanation with excellent examples. 💯💯💯💯💯💯💯
Thank you so much!
You're acually goated. Thnx alot
Great Video. Thank you.
love the explanation. what a cute and happy teacher
your videos helped me a lot! thank you so much
Thank you for this!!!
You're so welcome!
Please make a similar playlist on the Fourier series and Transform.
It's coming actually! About 3-4 months away. Finishing Vector Calculus first then moving to differential equations and it will be part of that playlist.
That comment halfway through about the howling wind made me laugh out loud. Thought it was just me going mad 😂😂
God Bless You Great Video . BUT Where Did The N Come From
Thanks for the video. I need to understand how an exponent can be complex, s = σ + jω, and what it means. This is not explained. Also, as far as I know, Laplace transform is used to cenvert a continuous function in the time domain, into a function in the frequency domain. Normally, poles and zeros are presented in the complex s plane.
To understand what it means for an exponent to be complex, it all comes down to Euler's formula, to make sense of the imaginary part of the exponent. Essentially, it rotates the number in the complex plane, instead of scales it, like a real exponent does.
Given a general complex exponent of a+b*i on Euler's number, we can split the exponent with properties of exponents. a and b are real, and combine as discussed to form a complex number.
e^(a + b*i) = e^a * e^(b*i)
e^a is a positive real number, so it's just a scaling factor.
e^(b*i) is what we unpack with Euler's formula, which gives us cos(b) + i*sin(b)
What's behind Euler's formula, is the Taylor series. Use the Taylor series of e^x, and plug in an imaginary value for i*theta for x. We can do this with first principles of complex numbers, because a Taylor series is just arithmetic and integer powers. You'll get an infinite series of real terms with even exponents, and an infinite series of imaginary terms with odd exponents. These two series, are Taylor series of cosine and sine respectively.
Is good, managed to easily understand everything. But, with all due respect, it lacks a lot in terms of explaining. I mean, is entirely theory, but nothing about how it comes to appear this Laplace Transform.
I think you may agree with me that, when it comes to maths, there is ever a logic and somewhat simple explanation to the very reason because "a thing" is "created" (or, well, defined. You get the point).
Integrals has all that Riemman's Sum behind, Taylor Series all that convergence thing behind, and so on so on so on, what I am trying to say is that there is a reason for "something to be like it is", and for newcomers or just people that doesn't fully understand this, the explanation (that it is almost always an "intuitive" explanation) could be of very very great help.
Don't misunderstand me, the video is of course excellent
clear explanation, thanks
thank you so much!
fun fact: gamma of a integer is that integer factorial-1 ! that's how people define (1/2)! even that recursion is true for non integers how cool
using the same gamma function you can even do it for complex numbers!
mind_blown.png
thank you so much
Hmm, interesting. Utilizing e^x's property to stay the same despite being integrated, such that you can integrate over and over again? Makes a lot of sense. Question to self: what other functions do that? The sine functions do something similar, which I guess allows us to display waves over and over again.
Sine functions are linear combinations of exponential functions, so no surprise there. If you have some polynomial of the derivative D, say p(D), and you have the equation p(D) = 0, then the solutions are going to be some linear combination of exponential functions. This is because the exponential functions are the eigenfunctions of the derivative operator.
I am so damn mad that no one ever explained the Gamma Function and n! like that! I had to learn that on my own when I was in college (My Calc II professor was horrible). It was a good thing I did because when I took Differential Equations (Last semester in college), I had this insight and things were not confusing for me.
I appreciate that you explained the Gamma Function with rich substance because many students do not get the explanation to why it is equal to the factorial.
Hi dr.Trevor , s is a complex number in general. And the complex numbers are not ordered set. Threrfore we can't say sa 4:03
What he means more accurately, is that the real component of s has to be greater than a, for there to exist a Laplace transform of an exponential function, e^(a*t), in order for the improper integral to converge.
these Videos are so great helping me for masters# student of University of Windsor ontario
Thank you
بسیار عالی بود...احسنت...
I read the comment and was wondering, what wind? And while going through the video, I laughed out loud! Haha good laugh!
Great explanation, this all makes so much more sense now.
Great job
i was like, "okay interesting choice to play owl noises in the background of a math video" XD
Awesome video, thank you
Why does this video has 85 dislikes? It's so helpful
ah yes, back in the day when we could all see the number of dislikes
@@nathangmail-user8860 There's an extension which has all the historic dislikes from before December 2021 and any new dislikes after are estimated from the current users with the extension, I'd recommend it 👍
Well, it gives people an opportunity to engage in the discussion and that in turn enables the algorithm to realise what a great video this is.
Otherwise you have to wonder at people even clicking on a maths video when they obviously don't like maths.
What is bigger, n! or infinite?🤓 Thanks for this great video.
At 7:15 how did negative s turn positive
Sir, can you put a video for Gamma of half integers input and how really this gamma function was brought into this form .... you really explain very well
With all do respect.. you had to focus on just Laplace transform and stick to it giving more examples about it. The Gama transform is another subject that confused me much while I am trying to understand Laplace, also the u function is confusing. Anyway.. your explanation is great. The winds give more horrifying feeling of the complex stuff. You could record your voice separatly and add it later to the video.
great video 👍
Love from INDIA ❤️❤️❤️
I'm in 10th grade like it... India
What is the point of using the Laplace transform? Aren’t we supposed to transform back to what we started with? I thought the purpose of the transform is to make the integration of the initial problem doable.
Ya that's right. Basically you transform, then you do algebraic manipulations to clean stuff up, then you transform back. We do this a bit more further down the playlist.
You are great sir
Sank you so much!
nice lecture
Great video, easy explanation ❤
Glad you think so!
i didn't even notice the wind noises until you pointed it out
My two year old brain is loving 2:30 with the English(auto-generated) Subtitles on
"Whaa.... whaat are you doing, Step Function!"
Excellent
6:31 I don't understand where the "1" comes from.
This is the part where I'm supposed to input "e^{-st} * f(t) dt" where f(t) = u(t-a), am I correct?
How does f(t) become 1?
The unit step function, u(t), is defined as an abrupt jump from 0 to 1, at the value of t=0. The general unit step function, u(t - a), has the abrupt jump happening at t=a.
can anyone say r these videos better than khan academy videos?
The unit step function is actually undefined at t=0
pesky wind, thanks for mentioning it i thought i was going crazy lol
Great explenation but i have got a question: if s is a variabel how can we then integrate with redpect to x? You can't integreat a function with two variabals with tespect to only one of them.
You can! Basically what you do is hold s as a constant and integrate with respect to x where you treat anything with s identically to how you would if it was a constant.
Thanks! I thought this was only possible with partial derivatives.(btw. sorry for the bad spelling I am from germany and I am only 14.
@@DrTrefor sorry that i have to ask u again but if we can treat s as a constant when integrating with respect to t coudn't we solve any differential equation like that (at least 1st order odes) . What i mean is coudn't we just multiply both sides by dx and then integrate the one side with respect to x and the other with respect to y even if the x and y terms are not sapetated?
@@erikawimmer7908hallo. Wie gut sind sie in Maths? Und was studiert sie?