Thanks Professor for this beautifully animated introduction to this theorem. I'm revising for my Year 11 exams here in Australia and wondered about the name of the underlying theorem used in a taught technique to identify a repeated root of a polynomial, given its multiplicity - differentiate the polynomial down to a quadratic trinomial, and then test each solution in the original polynomial. I'm sure I will come back to this video once I learn to work in the complex plane. But until then at least it's nice to know the theorem name even though I can't grasp it's geometric meaning.
Hi! Thank you for the proof! It is very exciting! I have a question: can I somehow estimate positions of roots of polynomial derivative given only its roots (I mean not involving direct differentiation of the polynomial)? It looks like there roots of derivatives are some kind of barycentre of original roots, but i'm not sure.
Distractive pronunciation of root. Last time I checked, "root" is spoken as in "rooting for someone". Not sure the dramatisation at points is necessary. Good animations though. I liked the phase plot, is there a previous video where you go into more detail regarding the phase plot? Somehow the steps where you start talking about wiggling f(w) and introducing the gradient 04:45 were confusing to me. Not sure I understood your logic. Have to watch again later.
I mispronounce many things -- root rhymes with foot rather than boot for me, which I guess is typical of Minnesota (though the rest of my accent isn't very Minnesotan, making it very confusing). Room and roof would also be annoying to hear me say, but mercifully those words rarely come up in my videos. I'm glad you like the animations. I'm not sold on the "drama" here either -- basically just trying out different formats for videos. In the queue there are some other videos which are all screencaps and no reaction shots so maybe those are better. I also should have explicated more on the gradient story -- perhaps there is a different picture (like drawing level sets) which would tell that story better, and might also cohere with the broader narrative around phase plots. ua-cam.com/video/HMH31MPzTHY/v-deo.html is a video introducing phase plots, and that points to kisonecat.github.io/phase-plot/ which you might like.
@@kisonecat I watched it again with a clear mind. Content-wise I was suspicious when you constructed p. I wasn't sure if a factorised polynomial exists for every polynomial but it seems like this is the case (and pretty fundamental which says something about myself lol). From there, the proof is pretty intuitive and I'm a sucker for geometric math. What I didn't realise before was that the whole angle talk at 6:35 was to make sure the gradients are not cancelling each other out (or at least that is what I think it was for). The step going from f' not 0 to p' not 0 is easy to understand at first. I had to first understand f which at that point in the video I had not fully grasped. I'd recommend breaking down such small steps more explicitly so the viewers can follow on their first watch. In general, I notice that I tend to enjoy math videos much more where the why and how of each step is explained even if it is a trivial step. The triviality of a fact is inversely proportional to the amount of time spent thinking about it which makes most new facts non-trivial and in need of explanation. Thanks for the links! I think your videos are great and definitely worthy of more views. I subscribed and I'll stick around for some more :)
@@aBigBadWolf Maybe a future video will discuss the fundamental theorem of algebra, though I think ua-cam.com/video/qdl1D8uci2I/v-deo.html does a nice job. (I would like to do a video about roots of quaternionic polynomials.) I very much agree with you that this video could be expanded and it would be better. There are some fun uses for Gauss-Lucas so maybe discussing some applications would be a good reason to revisit this. (And perhaps I'll get to teach complex analysis again next year and make some more videos.)
Hah, well, I hope you didn't spend too long mulling over that question. At Ross (the math camp), I am frequently mocked for it. In some of my older videos, I say 'innergral' and I now find that really grating. I'm glad you like the video!
Insightful yet simple explanation of a beautiful theorem!
Thanks! I only learned this argument recently, despite having taught a complex analysis course for the first time a decade ago.
You were my favorite professor!!!!!!!! You made linear algebra super interesting
Wow, thanks! That was years ago... I hope you're doing well!
@@kisonecat are you still teaching at OSU?
@@andy_hay Yes, I'm still at Ohio State. I haven't actually been to campus since last summer... My teaching this term is online.
Thanks Professor for this beautifully animated introduction to this theorem. I'm revising for my Year 11 exams here in Australia and wondered about the name of the underlying theorem used in a taught technique to identify a repeated root of a polynomial, given its multiplicity - differentiate the polynomial down to a quadratic trinomial, and then test each solution in the original polynomial.
I'm sure I will come back to this video once I learn to work in the complex plane. But until then at least it's nice to know the theorem name even though I can't grasp it's geometric meaning.
Professor I saw your vedio first time it awesome for me.Your explanation is so simple and easy.
Truly beautiful explanation and proof.
Better than the one in our class notes any day
Thank you for the kind words! I very much appreciate your taking the time to comment.
What a beautiful theorem. Thank you for the video!
Glad you liked it!
Hi! Thank you for the proof! It is very exciting! I have a question: can I somehow estimate positions of roots of polynomial derivative given only its roots (I mean not involving direct differentiation of the polynomial)? It looks like there roots of derivatives are some kind of barycentre of original roots, but i'm not sure.
Distractive pronunciation of root. Last time I checked, "root" is spoken as in "rooting for someone". Not sure the dramatisation at points is necessary. Good animations though. I liked the phase plot, is there a previous video where you go into more detail regarding the phase plot? Somehow the steps where you start talking about wiggling f(w) and introducing the gradient 04:45 were confusing to me. Not sure I understood your logic. Have to watch again later.
I mispronounce many things -- root rhymes with foot rather than boot for me, which I guess is typical of Minnesota (though the rest of my accent isn't very Minnesotan, making it very confusing). Room and roof would also be annoying to hear me say, but mercifully those words rarely come up in my videos.
I'm glad you like the animations. I'm not sold on the "drama" here either -- basically just trying out different formats for videos. In the queue there are some other videos which are all screencaps and no reaction shots so maybe those are better.
I also should have explicated more on the gradient story -- perhaps there is a different picture (like drawing level sets) which would tell that story better, and might also cohere with the broader narrative around phase plots.
ua-cam.com/video/HMH31MPzTHY/v-deo.html is a video introducing phase plots, and that points to kisonecat.github.io/phase-plot/ which you might like.
@@kisonecat I watched it again with a clear mind. Content-wise I was suspicious when you constructed p. I wasn't sure if a factorised polynomial exists for every polynomial but it seems like this is the case (and pretty fundamental which says something about myself lol). From there, the proof is pretty intuitive and I'm a sucker for geometric math. What I didn't realise before was that the whole angle talk at 6:35 was to make sure the gradients are not cancelling each other out (or at least that is what I think it was for). The step going from f' not 0 to p' not 0 is easy to understand at first. I had to first understand f which at that point in the video I had not fully grasped. I'd recommend breaking down such small steps more explicitly so the viewers can follow on their first watch. In general, I notice that I tend to enjoy math videos much more where the why and how of each step is explained even if it is a trivial step. The triviality of a fact is inversely proportional to the amount of time spent thinking about it which makes most new facts non-trivial and in need of explanation.
Thanks for the links! I think your videos are great and definitely worthy of more views. I subscribed and I'll stick around for some more :)
@@aBigBadWolf Maybe a future video will discuss the fundamental theorem of algebra, though I think ua-cam.com/video/qdl1D8uci2I/v-deo.html does a nice job. (I would like to do a video about roots of quaternionic polynomials.)
I very much agree with you that this video could be expanded and it would be better. There are some fun uses for Gauss-Lucas so maybe discussing some applications would be a good reason to revisit this. (And perhaps I'll get to teach complex analysis again next year and make some more videos.)
Simply mind-blowing!
Thanks!
Love your videos as always, professor!
I am glad you like them!
Beautiful!
Thanks! I'm really pleased with the animated phase plot pictures.
Excellent!
Glad you liked it!
yes!!!
It's definitely a very satisfying argument!
I can't tell if I love or hate the way you say roots
Update: I hate it. Great video though!
Hah, well, I hope you didn't spend too long mulling over that question. At Ross (the math camp), I am frequently mocked for it. In some of my older videos, I say 'innergral' and I now find that really grating.
I'm glad you like the video!
@@kisonecat Got over it pretty quick haha! Hopefully you're only mocked in kindhearted jest
@@Jop_pop Oh yes, it is all kind-hearted!