In case it is helpful, here are all my ODE videos in a single playlist ua-cam.com/play/PLxdnSsBqCrrHHvoFPxWq4l9D93jkCNIFN.html. Please let me know what you think in the comments. Thanks for watching!
AE 501: Another great video demystifying The Laplace transform. Your explanations help my understanding so much! I am looking forward to watching the next videos in this series!
Amazing lecture professor. I love the way you go in-depth in each and every single step and also the enthusiasm you transmit with your way of teaching.
Hiyori, great to hear from you again, thanks for the kind words. Please let me know what you think about any other videos you happen to view. Thanks for watching!
AE501. That was a long lecture but it flew by! I still can't really wrap my head around how Laplace came up with the Laplace transformation but I guess the man is a genius. Thanks to him, we can use this wonderful calculation method and thanks to Professor Lum, we now know how to use it!
AE501: Fantastic presentation Professor. I’ve always had issues with some of the other examples you have done, but your explanations were clear and concise! Thank you.
AE501 I was so thrilled that you mentioned a video game with respect to the Laplace Transform! When I first learned it, I likened the transform to Super Paper Mario, and how he flipped into 3D from 2D to make problems simpler! So cool to see the video game analogy being commonplace!
Super Paper Mario was a great game! Although the sad thing is that this was probably one of the last games where I actually had enough free time to play it through completely.
AE501: Definitely forgot about how useful Laplace transforms can be. Great refresher. Particularly love the examples in this one! It really helped my understanding
AE501: Ok... I think i'm with you so far.... hesitant for the next video haha. Great lecture. Included practical application of how laplace transform is done in real life... including a short tutorial on how to let Mathematica do the hard part for us. Appreciate that Professor Lum realizes that we as professionals have access to such tools in real world application.
AE501. This was a great lecture it really help me understand how to use the Laplace Transform. I didn't really get it during undergrad but your physical/engineering examples really helped. Thank you
[AE501] 9:21 It is nice to see the two methods (classical and laplace) side by side and why the laplace method is better before diving into how to actually do it.
#AE501 I think I finally understand the Laplace Transformation! During my undergraduate education, I've been using the "table of Laplace Transform" to solve problems and not really understand what's going on. I wish I knew about this video back in my old days.
AE501 - The LaplaceTransform function in Mathematica is very nice. Saves the trouble of looking up Laplace transforms in a table and will come in handy when checking my work on the homework problems
AE501 Looking forward to applying inverse Laplace transform In 38:10, you said the exponential is crushing the t so the value will be 0. How to know the exponential decrease faster than the t?
AE501: Thanks for the physics-based examples that helped explain why laplace transforms are so useful, I feel like much of my mathematics exposure to laplace transforms did not have that. I particularly enjoy how easy it is for Mathematica to do symbolic operations that would leave Matlab speechless.
AE501 @13:20 f piecewise continuous on (0,\infinity) and f has slower than exponential growth is sufficient for the existence of Lf. Is the 'f piecewise continuous' condition necessary for Lf? Needn't we only have f locally integrable? Are piecewise continuous functions dense in locally integrable functions?
AE 501: I really like the real life examples in this video as well. The spring mass damper system helps put abstract math concepts into real life problems
AE501 - Great video! Just to double check, at 57:40 when you're demonstrating how the 'time delay term' can be used with other functions, the alpha (a) in "e^(-as)" need not be the same value as the alpha in "A/(s+a)" right?
AE501 1:10:00 At my current job, we do a lot of work with Pyrotechnically Actuated Devices.. These systems utilize propellants in a confined setting which can lead to very short event times. I've seen events with certain propellants be as short as .0025s, and many events are less than .010s. They are very interesting and fun types of systems to design and test. And I think you see where I'm going with this, I could potentially see use cases for the impulse forcing function in my line of work, but could also potentially just use a very short pulse in this application. These types of events certainly create interesting design, analysis, and test challenges! -Nathan
Nathan, that sounds like a very interesting job you have :). You might be in one of the rare positions where impulses can actually be applied to a real system. Often the end effect of an impulse is simply a change in initial condition so this is an alternative way to think of them.
AE 501 Hi Professor Lum, at time 1:27:05 we assume that S is positive. I am confused on why we can make this assumption and what would happen if it were not positive, would the laplace transform then not be possible since it no longer goes to 0 as t goes to infinity? Also, what does it mean for s to be positive? does this mean that both real and imaginary components must be positive? Thank you
The short answer for what s represents, is complex frequency. It is a mixture of a real component that corresponds to exponential decay, and complex component that corresponds to sinusoidal oscillation. The Laplace transform scans the original function for a spectrum of amplitudes as a function of all values of s with a positive real component, and any imaginary component. Since s being negative would correspond to exponential growth, rather than exponential decay, negative values of s generally aren't applicable to reality. For exponential terms with positive poles, you get the explosion of exponential growth, which is considered unstable. There is the bilateral Laplace transform, that considers s being negative. I'll leave it to you to search for more about it.
AE501: Professor Lum, thank you for demonstrating the pulse vs impulse example. Math is much more fun when there is an example attached to it. Now got me thinking, where could we use impulse mathematical expressions in real-world issues?
That is exactly the right question. Impulses can be used to simulate initial conditions. For example, a system with an initial velocity is similar to having an impulse applied at t = 0.
AE501 At 14:21 when you're discussing requirements to do the Laplace transform i"m a little confused of use of the term "exponential order". f(t) must be of exponential order means that the exponent is a number and not a function, correct? Does it have to be a whole number?
AE501: Hello Professor, thank you for the refresher on Laplace Transforms. This was one topic I struggled with in the undergraduate program. For the homework problem 1, would it be sufficient to use the Euler's formula or do we have to proof it from scratch? -Natalia Ermolaeva
AE 501 - A Link to the Past is one of my favorite games! That analogy will be helpful to remember how the Laplace transform is a different representation of a function I'm more familiar with.
Isaac, yes you are correct, in situations where you need to instantaneously impart an initial velocity to an object, the impulse can be used. As a side note, thanks for being a Patreon supporter. If you have questions or would like to chat in the future, I'd recommend using Patreon messaging as I'll be sure to respond using that platform. Due to the volume of comments I get on UA-cam videos it is easy for your messages to get lost in the noise. Thanks for watching! -Chris
AE501 So for f(t) that are not of exponential order, the classical method is the only way to solve the ODE? Also, is there a reliable way of determining if f(t) is of exponential order without computer tools?
AE 501: Is there any reason why you had us skip the laplace transform of a time derivative when you stressed how important it was? Just deciding if I can skip or should watch.
Wait a sec, perhaps I messed up the timestamps but I believe you should be watching the part on the time derivative because as you mention, it is important. Can you tell me why you thought you should skip this section?
AE 501: I enjoyed the learning about Laplace transforms now I'm curious how this can apply to the aerospace realm. I also enjoyed the Legend of Zelda shoutout :)
AE 501: Professor Lum, I didn't understand when you added the unit step at t = alpha (min 50). If t = alpha then 1*(alpha - alpha) and that would result in g(t) = 0 ?
You did it well, but with showing practical example it would be 3ven better because, Inustrial application may slightly varies with theoretical approch, it may not be possible this theoretical approach into industrial applications.
@@ChristopherLum I can access the pdfs, but I feel like that results in me simply skimming your notes and work instead of having any chance of understanding these concepts myself to be able to resonate. Then eventually pass a test, or be able to recall later. Reading the notes I don't believe is enough.
@@ChristopherLum I do appreciate you walking through each of these functions in the detail you did to provide additional levels of understanding than just jumping to a Laplace Transform reference table after covering the basic method.
In case it is helpful, here are all my ODE videos in a single playlist ua-cam.com/play/PLxdnSsBqCrrHHvoFPxWq4l9D93jkCNIFN.html. Please let me know what you think in the comments. Thanks for watching!
AE 501: Another great video demystifying The Laplace transform. Your explanations help my understanding so much! I am looking forward to watching the next videos in this series!
AE501 - Thank you for this explainer on the Laplace Transform. These videos really help me as I do not find these topics to be the most intuitive
Amazing lecture professor. I love the way you go in-depth in each and every single step and also the enthusiasm you transmit with your way of teaching.
Hiyori, great to hear from you again, thanks for the kind words. Please let me know what you think about any other videos you happen to view. Thanks for watching!
AE501: Great explanation of the Differentiation Theorem. Thanks Professor.
AE501: Thank you for the refresher. I liked the way you had it set up at the end and went step by step with the Laplace method, very helpful.
AE501 - such a thorough explanation of Laplace transforms, thank you so much!
AE501. That was a long lecture but it flew by! I still can't really wrap my head around how Laplace came up with the Laplace transformation but I guess the man is a genius. Thanks to him, we can use this wonderful calculation method and thanks to Professor Lum, we now know how to use it!
AE501: Fantastic presentation Professor. I’ve always had issues with some of the other examples you have done, but your explanations were clear and concise! Thank you.
Glad it was helpful!
AE501. Great lecture. I love the way you go in-depth in each and every single step.
AE501
I was so thrilled that you mentioned a video game with respect to the Laplace Transform! When I first learned it, I likened the transform to Super Paper Mario, and how he flipped into 3D from 2D to make problems simpler! So cool to see the video game analogy being commonplace!
Super Paper Mario was a great game! Although the sad thing is that this was probably one of the last games where I actually had enough free time to play it through completely.
AE 501 - Thank you for explaining this in such great detail. I really enjoyed the Darth Vader metaphor you used!
Glad you liked it, there are more nerdy analogies coming in the future!
AE501: Definitely forgot about how useful Laplace transforms can be. Great refresher. Particularly love the examples in this one! It really helped my understanding
AE 501: Thank you Prof. Lum for the amazing Laplace refresher. I really enjoyed the impulse vs pulse explanation with the car and bat.
Peter, I'm glad it was entertaining, please let me know how things go as the quarter progresses.
AE501: I appreciate your use of props and analogies to help drive previously abstract (to me) mathematical concepts home (29:53)
Awesome lecture, love your style of teaching. Thanks!
I'm glad it was helpful, thanks for watching!
AE501 Thank you for the explanation of laplace transformations. It has been a very long time since I have learned the basics of these.
AE501: Ok... I think i'm with you so far.... hesitant for the next video haha. Great lecture. Included practical application of how laplace transform is done in real life... including a short tutorial on how to let Mathematica do the hard part for us. Appreciate that Professor Lum realizes that we as professionals have access to such tools in real world application.
AE501: I remember struggling with Differentiation Theorem in undergrad, glad to see it makes more sense to me now after watching this.
The step function was especially helpful. Thank you for posting this video.
[AE501 Student] Good understanding and applications of LaPlace transforms. The examples of the Mathematica code are helpful as well.
AE501. This was a great lecture it really help me understand how to use the Laplace Transform. I didn't really get it during undergrad but your physical/engineering examples really helped. Thank you
AE501 - Really good lecture on the laplace transform, thank you for the refresher
I'm glad it was helpful!
[AE501] 9:21
It is nice to see the two methods (classical and laplace) side by side and why the laplace method is better before diving into how to actually do it.
Great lecture! Easily understood
AE501: Thanks for the review on the Laplace transform!
#AE501 I think I finally understand the Laplace Transformation! During my undergraduate education, I've been using the "table of Laplace Transform" to solve problems and not really understand what's going on. I wish I knew about this video back in my old days.
Never actually understood Laplace during undergrad. Thanks!
AE501: Graphically representing the pulse function was very helpful!
AE501 - The LaplaceTransform function in Mathematica is very nice. Saves the trouble of looking up Laplace transforms in a table and will come in handy when checking my work on the homework problems
It sure does make life easier!
AE501 Looking forward to applying inverse Laplace transform
In 38:10, you said the exponential is crushing the t so the value will be 0. How to know the exponential decrease faster than the t?
AE501: Thanks for the physics-based examples that helped explain why laplace transforms are so useful, I feel like much of my mathematics exposure to laplace transforms did not have that. I particularly enjoy how easy it is for Mathematica to do symbolic operations that would leave Matlab speechless.
Yeah, Mathematica is very powerful for symbolic math applications. I like Matlab for numerics and Mathematica for analytical.
AE 501 Good Video on Laplace Transformation. Helped me a lot to solve tricky ODEs
AE 501. Thank you Professor Lum!
AE501
@13:20
f piecewise continuous on (0,\infinity) and f has slower than exponential growth is sufficient for the existence of Lf.
Is the 'f piecewise continuous' condition necessary for Lf? Needn't we only have f locally integrable? Are piecewise continuous functions dense in locally integrable functions?
You are a beast, many thanks!
AE501: Great review of Laplace transforms!
AE501 Thank you for this explanation /review of the Laplace Transform and Differentiation Theorem from differential equations.
AE 501: I really like the real life examples in this video as well. The spring mass damper system helps put abstract math concepts into real life problems
Thank you for the refresher
Sam
AE501 - Great video! Just to double check, at 57:40 when you're demonstrating how the 'time delay term' can be used with other functions, the alpha (a) in "e^(-as)" need not be the same value as the alpha in "A/(s+a)" right?
Hi David, good question and you are correct, these can be different values,
I love Laplace, nice video
Great refresher
AE501 1:10:00 At my current job, we do a lot of work with Pyrotechnically Actuated Devices.. These systems utilize propellants in a confined setting which can lead to very short event times. I've seen events with certain propellants be as short as .0025s, and many events are less than .010s. They are very interesting and fun types of systems to design and test. And I think you see where I'm going with this, I could potentially see use cases for the impulse forcing function in my line of work, but could also potentially just use a very short pulse in this application. These types of events certainly create interesting design, analysis, and test challenges!
-Nathan
Nathan, that sounds like a very interesting job you have :). You might be in one of the rare positions where impulses can actually be applied to a real system. Often the end effect of an impulse is simply a change in initial condition so this is an alternative way to think of them.
AE 501 Hi Professor Lum, at time 1:27:05 we assume that S is positive. I am confused on why we can make this assumption and what would happen if it were not positive, would the laplace transform then not be possible since it no longer goes to 0 as t goes to infinity?
Also, what does it mean for s to be positive? does this mean that both real and imaginary components must be positive?
Thank you
The short answer for what s represents, is complex frequency. It is a mixture of a real component that corresponds to exponential decay, and complex component that corresponds to sinusoidal oscillation. The Laplace transform scans the original function for a spectrum of amplitudes as a function of all values of s with a positive real component, and any imaginary component.
Since s being negative would correspond to exponential growth, rather than exponential decay, negative values of s generally aren't applicable to reality. For exponential terms with positive poles, you get the explosion of exponential growth, which is considered unstable.
There is the bilateral Laplace transform, that considers s being negative. I'll leave it to you to search for more about it.
Thank you for the refresher
AE501: Professor Lum, thank you for demonstrating the pulse vs impulse example. Math is much more fun when there is an example attached to it. Now got me thinking, where could we use impulse mathematical expressions in real-world issues?
That is exactly the right question. Impulses can be used to simulate initial conditions. For example, a system with an initial velocity is similar to having an impulse applied at t = 0.
AE501
At 14:21 when you're discussing requirements to do the Laplace transform i"m a little confused of use of the term "exponential order". f(t) must be of exponential order means that the exponent is a number and not a function, correct? Does it have to be a whole number?
Thanks for the refresher!
AE501 - thank you for the review!
MAE 501 great video thanks for the info!!!!
AE 501 - Jacob Givens, I liked the explanation of Laplace transforms!
[AE501]
Haha that legend of Zelda reference was sick!
Good review on Laplace transforms
AE501: Hello Professor, thank you for the refresher on Laplace Transforms. This was one topic I struggled with in the undergraduate program. For the homework problem 1, would it be sufficient to use the Euler's formula or do we have to proof it from scratch? -Natalia Ermolaeva
Feel free to go ahead and use Euler's formula. Keep me posted on how things are going.
AE 501 - A Link to the Past is one of my favorite games! That analogy will be helpful to remember how the Laplace transform is a different representation of a function I'm more familiar with.
Yes, this was a great game. Did you play any others? My kids just finished Zelda - The Minish Cap which has some similarities to Link to the Past.
@@ChristopherLum I've played a bunch of others! Probably one of my favorite video game series!
Hi Chris, actually the impulse probably have physical applications in destructive systems e.g a bomb and explosive for instance. Would you think so?
Isaac, yes you are correct, in situations where you need to instantaneously impart an initial velocity to an object, the impulse can be used. As a side note, thanks for being a Patreon supporter. If you have questions or would like to chat in the future, I'd recommend using Patreon messaging as I'll be sure to respond using that platform. Due to the volume of comments I get on UA-cam videos it is easy for your messages to get lost in the noise. Thanks for watching!
-Chris
Great material
AE501:
Thanks for the lecture professor, but I think at time stamp 27:15 you mention at t=0 but I think you meant t=inf.
You are right, I misspoke. The board is correct but I said the word "0" instead of '"infinity"
AE501: I enjoyed this lecture maybe because I love the Laplace transform so much!
AE501 So for f(t) that are not of exponential order, the classical method is the only way to solve the ODE? Also, is there a reliable way of determining if f(t) is of exponential order without computer tools?
AE501: Good discussion on the Laplace transform and great Zelda reference
Glad you liked it, there are more nerdy analogies coming in the future!
AE 501: Is there any reason why you had us skip the laplace transform of a time derivative when you stressed how important it was? Just deciding if I can skip or should watch.
Wait a sec, perhaps I messed up the timestamps but I believe you should be watching the part on the time derivative because as you mention, it is important. Can you tell me why you thought you should skip this section?
AE 501: I enjoyed the learning about Laplace transforms now I'm curious how this can apply to the aerospace realm. I also enjoyed the Legend of Zelda shoutout :)
I'm glad the reference is still valid, this is a bit of an older game :)
AE 501: Professor Lum, I didn't understand when you added the unit step at t = alpha (min 50). If t = alpha then 1*(alpha - alpha) and that would result in g(t) = 0 ?
Hi Marcos, does it make sense if you cross reference with the lecture notes? Let me know if this is still odd and I'll take a closer look, thanks.
[AE501] Mass-spring damper system brings me back to physics and helped me better grasp the concept
[AE501] 17:17
Why do you do "t_" rather than just "t"
Kickass Zelda ref, prof!
Jake, I'm glad someone understands the reference :)
Thanks for the video!
Great video
greatly appreciate!!!
Great job
great video, even better jokes! thanks professor
I'm glad it was entertaining. Thanks for watching!
Chris I am your biggest fan
Thanks, I appreciate the support! Please stay tuned as I've got a bunch more discussions on ODEs coming in the next few hours.
You did it well, but with showing practical example it would be 3ven better because,
Inustrial application may slightly varies with theoretical approch, it may not be possible this theoretical approach into industrial applications.
great vid prof lum
As t goes to infinity not zero. 27:16
Great catch, thanks for pointing this out. Please let me know of you see anything else in the future.
A bus full of nuns - I almost died laughing!
I'm glad it was entertaining, thanks for watching!
I love the lego car! (mostly, when you hit it with a bat)
AE501 Finally someone moves quickly to the easy solution methods. Embrace the dark world. Save the nuns
the impluse killed me lol
A 2hr lecture, set at 1.75speed, still took 4.5hrs to take notes and absorb. My hand hurts.
Danielle, are your able to access the pdf notes? Hopefully this will save you some time. I hope this was helpful!
@@ChristopherLum I can access the pdfs, but I feel like that results in me simply skimming your notes and work instead of having any chance of understanding these concepts myself to be able to resonate. Then eventually pass a test, or be able to recall later. Reading the notes I don't believe is enough.
@@ChristopherLum I do appreciate you walking through each of these functions in the detail you did to provide additional levels of understanding than just jumping to a Laplace Transform reference table after covering the basic method.
@@daniellerogers5959 thanks for the feedback, please let me know what you think about any future material as well
AE501 - Malachi Morris
AE501: Arda Cetken
[A E 501 student] watched - CW
AE501, Cody Smith
29:55 Darth Vader
AE 501: Bryce Foland