Legendre's ODE III: Verifying/'Proving' Rodrigues' Formula
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- เผยแพร่เมื่อ 5 ก.พ. 2025
- In this video, I verify that the Rodrigues Formula is equivalent to the Legendre polynomial formula I derived in the previous video here: • Legendre's ODE II: Der...
Questions? Ask me in the comments!
Prerequisites: The first 3 videos of this playlist • Topics in Ordinary Dif...
Lecture Notes: drive.google.c...
Support my Patreon: www.patreon.co...
3:13 you say “notice the pattern here?” and i did. in fact in all your videos about ides that i have watched this far, i have always been able to spit the pattern. it’s not a sign of my intelligence (of which there is almost none lmao) but a sign of how good a teacher you are. you always bring me to the point where i can solve it myself. thanks and keep up the fine work
It has been a while from the first time i have watched this video and it still surprises me the simplicity you have to explain. Love it and use it a lot! thanks for sharing!
What can I say? It is always beyond words and tears. Thank you.
No problem!
you are a god to me. this is three blue one brown level - without funky animations but still excellent and with unbelievably deep insights
these are simply professional videos
thanks alot
That's much appreciated! Thank you for the kind words!
This was so much help. I am forever indebted.
Each and everything is very well explained thank you so much sir
Wow! I'll actually remember the formula's meaning now. Thank you!
That was the best proof out there!! Thanks, Sir.
No problem! Glad you liked it!
one piece
>On general relativity, tensor algebra and... lol.
Here we are, nearly two years later : ) .
I've had enough time to self study all this now but still, your videoes are top notch.
These videos make the idea of undergrad/phd seem appealing. Oh well lol, great video!
wow sir i really respect your work but i don't know why you have so little views i frankly didn't know how what world did the k'th derivative in Rodriguez formula come from until now gave me a really good insight will definitely recommend your channel to my friends and were you actually about the general relativity series because there are not many series out there on that anyways thanks for your troubles
Thank you! It would be a big favor if you could recommend my channel to your friends on social media (facebook, twitter etc). I would greatly appreciate it!
yes sure defintiely
bro this is one of the easiest method i have seen on the internet.. my college prof taught me this like this is some third world proof .. i was soo confused... im forever in debt
Thanks man, this video helped me a lot. Keep making these videos. And one more thing, it would be of more help if you could solve an example for each topic.
Thank you! And I'm going to try to add more examples for the topics I cover. Thanks for the suggestion!
Brilliant way of explaining....Really Brilliant
Thank You so much!! such pro videos are rare!
I wonder how Rodrigues come up with that formula in the first place... :q
My guess is that it's somehow related to the fact that the `-2·x` coefficient of the first derivative is itself a derivative of the `(1-x²)` coefficient of the second derivative ;J so the entire equation can be rewritten as:
d/dx[ (1-x²)·dy/dx ] + k·(k+1)·y = 0
so the `1-x²` part is important here, it's like a "kernel" that never changes, and has some potential for substitution.
Taylor series base on (x^2 +2)
Simple and Sufficient!
Thanks.
I finished all your Legendre courses and followed them while writing the derivations myself. Thank you for those very qualitative videos of yours. I just have one question: what are its applications ?
Can I ask what book should I have for self studying the subjects in these videos? I am so hyped after watching your videos.Your videos are in the best form to watch. But these subjects seem to be too difficult to learn without a book. I am sure it requires lots of practice. Can you recommend a book for me?
Funnily enough, I didn't really use a book for this video. I basically just did an exercise problem that showed up in an ODEs book (I think it was Applied Engineering Mathematics). And thank you!
Very Helpful Video! Which Programm do you use ?
Many thanks. Would you please consider doing a video about Associated Legendre Polynomial? Cheers.
Thank you, this helped a lot
Super helpful! Thanks
You said that Legendre polynomial get it's use in electromagnetic theory. Can you provide me with examples?
P.S. - Are they being used in defining magnetic field?
I don't think I mention the applications in this video. Perhaps you're referring to a previous video?
Regardless, Legendre polynomials are typically part of the solution to Laplace's equation (i.e. del^2 V = 0) in spherical coordinates. So when it comes to electromagnetic theory, you would encounter them when solving for the potential inside a sphere when the potential at the boundary (i.e. surface) of the sphere is already defined. You can refer to this document for more details:
www.me.rochester.edu/courses/ME201/webexamp/legendre.pdf
Hope that helps!
As for magnetic fields, here is an example from a book that mentions Legendre polynomials for magnetic fields:
www.physicspages.com/2013/04/11/magnetic-dipole/
Generally though, my experience is that Legendre polynomials are more popular when looking at electrostatic potentials.
I have read this is my Electromagnetic theory of Electrical Engineering course. But there was no explanation of how Legendre polynomial get in Electrostatic potential. But Legendre polynomial get it's use only at places where solution converges between 0 and 1.
That's the reason for asking it.
Well, in this case, the Legendre polynomial is defined as a function of cos(phi), which always lies between 0 and 1. Phi here is used to denote the angle relative to the +z axis. Anyway, if you have any more questions, let me know!
Helped a lot. Thanks.
you are awesome !!! thank you for good vdo !!!!
great work sir............
Thank you! Glad you liked it Zeeshan!
What about general approach to the Rodrigues formula
That is what i fount on wikipedia
Let's start from ODE
Q(x)f'' + L(x)f' + λf = 0
and assume that our orthogonal polynomial satisfies equation above
Let R'(x) = L(x)/Q(x)*R(x)
W(x) = R(x)/Q(x)
P_{n}(x) = 1/e_{n} *1/W(x)*d^n/dx^n(W(x)[Q(x)]^n)
In this approach R(x) seems to be the reciprocal of Wronskian
Thanks from foreign👍
Thanks so much!
Great
Thanx
👍
Thanks
That comment tho 1:52
lol I don't even have an undergrad education but I think anyone would look rather silly for emailing someone to ask for help about something as trivial as the nth derivative of a power function.
"Dat derivative is OP, pls nerf!" ;)
2:02 really cracked me up
Hoopla