Wow! this was incredibly well animated and presented! Thankyou so much for making this video. I really hope you continue making videos like these and that your channel gets the recognition it deserves 😁
I think these videos are a great tool for people who struggle to understand the differences between the definitions of pointwise and uniform convergence. I had to make my own little drawings to help me understand what it meant, and that's really what helped differentiate them for me.
THANK YOU VERY MUCH!! I'm now studying uniform convergence in calc 3 and the formal definition is quite a mouthful (especially that uniform convergence is established when somehow the choice of big N only depends on ε and not on x), but I tried picturing the entire thing and had a sense that uniform convergence meant the ability to bound the entire function within the ε-neighborhood of f_n (so to speak) for n ≥ N, so that it is not specifically pointwise and therefore no longer depends on x, I knew it would be nice to visualize, and voilà, I found your video which explains it nicer than I thought possible! Again thank you very much for this video!! Why does it only have 35 likes and 7 comments?? I'm sharing this to my friends, it's too good! :D very much underrated! EDIT: Forgot to say, about timestamp 5:46, I think "uniform" here is supposed to be a synonym for "simultaneous" after all (we have points uniformly converging together), so yeah ;)
you really shouldn't worry about these formal definitions in calc 3, 2, or 1 (although epsilon arguments can be intuitive). They are defined this way for advanced mathematics in which they utilize the formal definitions to prove other theorems.
@@simonribas4625 Understandable. It just so happens that I took Real Analysis 1* alongside Calculus 3 (as my faculty recommends), so there was some temptation to view it rigorously. *Here, the teaching of introductory real analysis (using Bartle's textbook) is split into two consecutive classes. I have passed Calc 3 and I am now taking Real Analysis 2, discussing continuity, derivatives, integrals, and eventually uniform convergence. Can't wait!
Let (M, d) and (N, rho) be metric spaces. Let f_k be a sequence of functions. Assume \forall k, f_k is a mapping from A \subset M \to \N, and assume f is a mapping from A \subset M \to \N. Pointwise convergence: \forall \varepsilon > 0, \forall x \in A, \exists K such that if k \geq K, then rho(f_k(x), f(x)) < \varepsilon. Uniform convergence: \forall \varepsilon > 0, \exists K such that if k \geq K, then rho(f_k(x), f(x)) < \varepsilon \forall x \in A. For uniform convergence the K must not depend on x. That is, there must be a universal K that works for any x. This is similar to the idea of uniform continuity compared to continuity (delta cannot depend on x for UC, but delta can depend on x for continuity). Theorem: if f_k is continuous for all k and f_k converges uniformly to f, then f is continuous. Contrapositive: if f is not continuous, then there exists a function f_n such that f_n is not continuous or f_k does not converge uniformly to f. Hence if f_k is continuous for any k but f is not continuous (on A), then f_k does not converge uniformly to f.
bro, you are awesome!!!!!!!! please continue to do what you do, its amazing. if you could also make a visualize video for uniform convergence using supremum theorem. Also in the future it would be pretty good to explain the series convergense and weierstrass M-test. good luck for you!!
visualizing math feels like eating a delicious cake
just changed the trajectory of my life - a math major with two midterms tomorrow morning
THE best video on pointwise and uniform convergence!!
Wow! this was incredibly well animated and presented! Thankyou so much for making this video. I really hope you continue making videos like these and that your channel gets the recognition it deserves 😁
Phenomenal video. The visual "contrapositive proof" at the end really drives the point home.
I think these videos are a great tool for people who struggle to understand the differences between the definitions of pointwise and uniform convergence. I had to make my own little drawings to help me understand what it meant, and that's really what helped differentiate them for me.
Pleasing visuals and straight to the point explanation!
I really enjoyed composing the Intro.
THANK YOU VERY MUCH!! I'm now studying uniform convergence in calc 3 and the formal definition is quite a mouthful (especially that uniform convergence is established when somehow the choice of big N only depends on ε and not on x), but I tried picturing the entire thing and had a sense that uniform convergence meant the ability to bound the entire function within the ε-neighborhood of f_n (so to speak) for n ≥ N, so that it is not specifically pointwise and therefore no longer depends on x, I knew it would be nice to visualize, and voilà, I found your video which explains it nicer than I thought possible! Again thank you very much for this video!! Why does it only have 35 likes and 7 comments?? I'm sharing this to my friends, it's too good! :D very much underrated!
EDIT:
Forgot to say, about timestamp 5:46, I think "uniform" here is supposed to be a synonym for "simultaneous" after all (we have points uniformly converging together), so yeah ;)
Hey man I'm very glad I could help you :) Thanks alot for the comment!
you really shouldn't worry about these formal definitions in calc 3, 2, or 1 (although epsilon arguments can be intuitive). They are defined this way for advanced mathematics in which they utilize the formal definitions to prove other theorems.
@@simonribas4625 Understandable. It just so happens that I took Real Analysis 1* alongside Calculus 3 (as my faculty recommends), so there was some temptation to view it rigorously.
*Here, the teaching of introductory real analysis (using Bartle's textbook) is split into two consecutive classes. I have passed Calc 3 and I am now taking Real Analysis 2, discussing continuity, derivatives, integrals, and eventually uniform convergence. Can't wait!
this is probably the best visualisation I have ever seen before.
It's fascinating being intuitive and mathematically rigorous in same video! Very well explained.
Eureka !
Have never seen an epsilon area neither better visualized nor better explained.
Crisp animations.
Really amazing ! Please dont stop this channel have great potential
looks like he stopped
Wow. I was struggling to understand the intuition between pointwise and uniform convergence but now it all makes sense. thank you so much!
The production value is simply amazing!
This visualisations help a lot to understand these different concepts of convergence .
I never comment on videos, but this was amazing!!! Really beautifull
Finally, a god-damn easy to understand video
these are some stunning visualizations. made very well and explained very well, , thank you for that
Great video!
Wow🤯... that was just SOOOO WELL-EXPLAINEDDDD. Thank uuuu!
Absolutely brilliant demonstration and explanation. Thank you!
Firstly i thought that your channel has 278 000 subs but now im shocked that it has only 278( So helpfully video
This was amazing honestly. I finally understand the uniform convergence part . Thankss !!
Great. Please continue uploading.
WOW!! Thank you so much!!
Congratulations broo
Your explanation is way better than other.
You deserves more subscribers
Now that is a great video on pointwise and uniform convergence
visual explanation makes the concepts clear. thanks you
This is cool! Please upload more videos on the concept
Funny that this was the one topic I couldn’t find a video for and it’s a really well animated video on your channel, and it’s the only one.z
Lucky me!
Thank you so much! These videos are super useful to digest the material :D
Very neat and intuitive! I wish you'd post more videos.
What a amazing way of making content
What an amazing and easy to understand video!
This is an excelent video. Great voice and a very intuitive mix of defiiniton and visuals.
Clear explanation thanks lot
This video is absolutely amazing! Thank you!
Good job! You helped me a lot to understand calculus, you should continue presenting these types of concepts!
gracias amigoooo sos gigante saludos desde la argentina, FAMAF.
WOW! You're video helped me finally understand the different types of convergence for sequences of functions! Really great video, thanks!
Thanks 👍
From #INDIA 🇮🇳🇮🇳🇮🇳
Thank you for tthis video. I wish you would make more videos!
Wow, please the world needs more videos like this! How is it the only video on this channel?
Wow great video. It helped me out so much. The visualisation is exactly what I needed to grasp this concept. Thank you and keep up the amazing work!
you gained a subscriber for life. keep posting more
Amazing! Thanks. Keep going
Amazing video keep it up
i very much enjoyed this video
Thank you for this video!
i finally get this now , OMG thanks you so much for the explaination
great video man,thanks
Whaaaaaaaaat this is amazing!!!!!!!!!!
Amazing video, thank you so much!
Thanks mate, it really helped me out a lot! Nice music, you got the ID?
:'D IDs in the description mois
U deserve an Oscan man. Don't stop, please make more videos like this 😃
I want to see a much clearer video about the visual difference between pointwise and uniform convergence
Really great explanation!
Outstanding👍👍👍
We want more video like this
Beautiful 😌
Great Video! Keep up the good work
Bro has one video and it's perfect.
Outstanding animation. Thank you sir
Great video Man highly appreciated।
Great video man! Learned a lot from this😊
Thank you so much, outstanding work!!
Let (M, d) and (N, rho) be metric spaces.
Let f_k be a sequence of functions.
Assume \forall k, f_k is a mapping from A \subset M \to \N, and assume f is a mapping from A \subset M \to \N.
Pointwise convergence: \forall \varepsilon > 0, \forall x \in A, \exists K such that if k \geq K, then rho(f_k(x), f(x)) < \varepsilon.
Uniform convergence: \forall \varepsilon > 0, \exists K such that if k \geq K, then rho(f_k(x), f(x)) < \varepsilon \forall x \in A.
For uniform convergence the K must not depend on x. That is, there must be a universal K that works for any x.
This is similar to the idea of uniform continuity compared to continuity (delta cannot depend on x for UC, but delta can depend on x for continuity).
Theorem: if f_k is continuous for all k and f_k converges uniformly to f, then f is continuous.
Contrapositive: if f is not continuous, then there exists a function f_n such that f_n is not continuous or f_k does not converge uniformly to f.
Hence if f_k is continuous for any k but f is not continuous (on A), then f_k does not converge uniformly to f.
amazing! beautiful explanation and visuals
insane, good job!
Great visual explanation.
Nice, bro.
Thanks for the video.
The music is lit!!!
Super Video und Animation!
Your are very good at understanding the concepts of mathematics please make more videos
Thank You
great explanation !
thank you so much
Excellent work!
It's Just.... excellent ✨❤️
Excellent explanation!
Thanks a lot
thanks a lot.....
amazing video
Thank you!
great!
Thanks man!
Great video
why is this your only upload bro
Great content. ❤️
thank you very much, this was very helpful
Please continue the concept of maths through visualisation
Thank you, this helped a lot.
Woah, I will def teach my friends this.
great video
Beautiful explanation, thank you🙂🙂🙂
Nice explanation🙏🙏🙏🙏🙏🙏 sir
Great video bro :) and by the way may I ask if your German ;)
Thanks, you are very right. :)
bro, you are awesome!!!!!!!! please continue to do what you do, its amazing. if you could also make a visualize video for uniform convergence using supremum theorem. Also in the future it would be pretty good to explain the series convergense and weierstrass M-test. good luck for you!!
Hey, thanks for your feedback and your suggestions. I'm currently making a video on total differentiability/Jacobian matrix :)