Recent content by caffeinemachine

Thank you for the suggestion. I tried Manim for a bit but found that the time it takes to make one video using manim is way more than what I can do in the handwritten style. Maybe when my coding skills improve Manim would be an option.

Hello Everyone! I created a YouTube channel (here's the link) a few years ago in which I post detailed lectures in mathematics. I just started a series on General Topology. Following is a snapshot from a video. I mean to deliver a comprehensive course with a lot of pictures and intuition and...

Hey thanks so much man!

Thank you for the encouragement. This is only one video in a series of videos. Ordered fields were discussed in detail in a previous one. The beginning was meant as a quick recap. I agree that 'subtraction' and 'division' are derived operations. I meant to only quickly capture the main idea of...

Some time back I posted about my videos on Group Theory on YouTube and got valuable feedback from the PF community. With the response in mind, I made substantial changes to my presentation. One of the main complaints was that I was speaking too fast. Here is my recent video on Real Analysis...

Thank yo for the feedback. I will try to fix my speed and reupload the videos eventually. I cannot, of course, change my accent.

Thank you for the valuable feedback. I agree that my earlier videos had an audio-clarity problem. Do you face the same problem even with the video I linked (if not, can you please link the video that is unclear?) Also, the subtitles are accurate. Though I understand that using subs may...

As of now I am only focusing on math majors. In the future I would like to include topics from theoretical physics and theoretical computer science.

I created a YouTube channel (here's the link) a few months ago in which I post detailed lectures in higher mathematics. I just finished my Group Theory Course. Here is a sample video. Apart from that, so far I have uploaded A first course on Linear Algebra (which I am currently renovating). A...

Problem: Evaluate $$ \int_{0}^\infty \frac{e^{3x} - e^x}{x(e^x + 1)(e^{3x} + 1)}\ dx $$ Attempt. I substituted $y=e^x$, thus $dx = dy/y$, which turns the above integral to $$ \int_{1}^\infty \frac{y^2 - 1}{(\log y)(y+1)(y^3+1)}\ dy = \int_{1}^\infty \frac{y-1}{(\log y) (y^3+1)} \ dy $$ I am...

Just a typographical comment. Instead of writing $m*(m-1)$ etc, one should simply write $m(m-1)$. Aslo, one should typeset all the math using TeX. You have left some math in plain text.

The two statements that you have mentioned are equivalent provided you assume that $U_1$ is not the trivial subsspace and the containments $U_i\subseteq U_{i+1}$ are strict. With this, you, in your argument, need to mention $U_1$ is necessarily one dimensional (do you see why) and that $v$ can...

If you do not assume connectedness then the simplest example where $\alpha(G)$ is more than $|U|$ is when there are no edges. But one can constrcut such exampels even with connectedness.

Assuming $f$ is an odd function, and then replacing $x$ by $-x$ and $y$ by $-y$ and adding we get $$f(x+y+2xy) + f(-x-y+2xy) = 4f(xy) = f(4xy)$$ where the last equality is by your observation. This looks like $f(a) + f(b) = f(a+b)$ since $(x+y+2xy) + (-x-y + 2xy) = 4xy$. So one can make some...

Let $f:\mathbb R\to \mathbb R$ be a function satisfying $f(x+y+2xy) = f(x)+f(y) + 2f(xy)$ for all $x, y\in\mathbb R$. Then I need to show that $f(2017 x) = 2017 f(x)$ for all $x\in \mathbb R$. I am not sure where to start. All I could note is that $f(0)=0$ which one obtains by susbtituing...