Hi @AndrewLappeman-w4d thanks so much for your comment, as always, and for watching! 😊 I am so happy you enjoyed the video and I hope you have an amazing day/evening/night! 😊
Hi @MridulGupta94 thank you so much for your comment, as always, and for watching! 😊 I am so happy to read it and I hope you have an amazing day/evening/night! 😊
Wonderful! The two dimensional version sounds very interesting too. I wonder if there's a way to visualize said function in three dimensional space, considering its domain consists of a particular subset of the xy plane. Thank you for the explanation! (This exact proof actually recently showed up in my real analysis homework :D)
Hi @Petrushka_Editor thank you so much for your very positive comment + feedback and for sharing your very interesting questions/thoughts! 😊 I am so happy to read it! I think in terms of visualization, it's a bit tricky since a function f:D^2 -> D^2 will have a graph that lives in 4-dimensional space D^2 x D^2 and in this case, we are trying to prove that its graph (a 2-dimensional surface in 4-dimensional space) intersects the graph of the identity function g:D^2 -> D^2 (g(x) = x) (another 2-dimensional surface in 4-dimensional space). The way to go about visualizing this is in terms of cross sections etc. (which are lower-dimensional). However, it still requires deeper machinery to truly understand in the realm of algebraic topology. I plan on doing another (accessible) video on this in the future though! 😊 Also, that's so cool it recently appeared in your real analysis homework! 😊 Yes, it is a very classical question in real analysis! What textbook are you using/what are you learning now in your class? I hope you have an amazing day/evening/night and Happy New Year!!! 🥳🎉🎊
Check out ua-cam.com/video/zz4v2KlJCgI/v-deo.htmlsi=Ozh8kSA5TV8q9u8C 🎁 if you want to see another fundamental proof with continuous functions that is central in calculus/real analysis! The proof characterizes which continuous functions f: R -> R satisfy f(x + y) = f(x)f(y)! 🥇 If you are enjoying my content, please don't forget to leave a like ✅, and subscribe 🥳 to always be notified of new videos - it's infinite free accessible math education across all topics and levels of math! 😊
That was beautiful
Hi @AndrewLappeman-w4d thanks so much for your comment, as always, and for watching! 😊 I am so happy you enjoyed the video and I hope you have an amazing day/evening/night! 😊
Very interesting! 😊
Hi @MridulGupta94 thank you so much for your comment, as always, and for watching! 😊 I am so happy to read it and I hope you have an amazing day/evening/night! 😊
Wonderful! The two dimensional version sounds very interesting too. I wonder if there's a way to visualize said function in three dimensional space, considering its domain consists of a particular subset of the xy plane. Thank you for the explanation! (This exact proof actually recently showed up in my real analysis homework :D)
Hi @Petrushka_Editor thank you so much for your very positive comment + feedback and for sharing your very interesting questions/thoughts! 😊 I am so happy to read it!
I think in terms of visualization, it's a bit tricky since a function f:D^2 -> D^2 will have a graph that lives in 4-dimensional space D^2 x D^2 and in this case, we are trying to prove that its graph (a 2-dimensional surface in 4-dimensional space) intersects the graph of the identity function g:D^2 -> D^2 (g(x) = x) (another 2-dimensional surface in 4-dimensional space). The way to go about visualizing this is in terms of cross sections etc. (which are lower-dimensional). However, it still requires deeper machinery to truly understand in the realm of algebraic topology. I plan on doing another (accessible) video on this in the future though! 😊
Also, that's so cool it recently appeared in your real analysis homework! 😊 Yes, it is a very classical question in real analysis! What textbook are you using/what are you learning now in your class? I hope you have an amazing day/evening/night and Happy New Year!!! 🥳🎉🎊
Check out ua-cam.com/video/zz4v2KlJCgI/v-deo.htmlsi=Ozh8kSA5TV8q9u8C 🎁 if you want to see another fundamental proof with continuous functions that is central in calculus/real analysis! The proof characterizes which continuous functions f: R -> R satisfy f(x + y) = f(x)f(y)! 🥇 If you are enjoying my content, please don't forget to leave a like ✅, and subscribe 🥳 to always be notified of new videos - it's infinite free accessible math education across all topics and levels of math! 😊
god
Hi @tsunningwah3471 thank you so much for commenting! 😊 I hope you have an amazing day/evening/night! 😊
Jed
Be more specific.