I say it's mean because i want to juat study number theory and all this geometry is annoying me.

Hi mathemaniac, I have a question I hope you can address. I see you are passionate about finding the meaning behind the math, and not just focused on computation like most of the people. I am an engineering student graduate and have been struggling with understanding infinitesimal usage in engineering and physics. In math we deal with differentials, and they are treated as a miniscule change in one variable affecting a miniscule change in a dependent variable (think speed = distance / time = dx=dt), usually in the context of a slope of a non-linear function. But on the other hand, many physical and engineering formulas come from a different sort of application of derivatives, ie. infinitesimals - they aren't treated as changes in many cases, but as small amounts, unrelated to any non-linear function (If you google Capstan Equation Derivation and search up google images for it, you will see what I mean). There we have all sorts of infinitesimals, such as a dN (a small amount of force, rather than a change in said force), dθ (small amount or change in the angle), and dT (a small amount of tension). I have always been confused when such derivatives would present themselves: p = df/da, ρ = dm/dV etc. Those are supposed to be small amounts, rather than change/change. Why would you have a change in mass for a change in volume, if we are considering a static system where the volume doesn't change? We are using this derivative to calculate the mass of an object with uneven density distribution. I would like you to address these two different points of view (mathematical and physical/engineering), since there isn't a solid explanation for it, and most professors in university don't bother to address this either. I would be extremely thankful if you could clear my doubts, I have a pretty good intuitive understanding of it, but having it explained by you would make mine and many students' lives easier! I think this would be a decent idea to explore on your main channel, once you're done with uploading the Jacobian's trilogy vids, if you're willing to consider my proposal! Cheers lad, and keep it up. All the best with both channels!

Hey, this was great work as always, man!! Keep it up, I always love seeing your videos pop up in my subscription feed :) This was particularly insightful and accessible for an explanation of a concept that kicked my ass once upon a time, I can only hope that more people see it!!

I would've just considered it as the mean of the two eigenvalues of the matrix 🤷‍♂

The reason I asked for comments is, well, partly because of engagement, but partly because the detailed arguments interrupt the flow. The trace one at the end is actually quite nice, and thinking about it is probably useful for the next main channel video. Please LIKE, SUBSCRIBE, COMMENT to get this into KZfaq algorithm :)

so basically spherical integral

normally I integrate the surface on spherical coordinates

Equivalent statement, works for non-circle arcs! Imagine a car travelling along the boundary of a spherical triangle. It would have to rotate pi - [angle] at every point. So the rotation should be 3pi - [sum of angles], and the excess is 2pi - [rotation]. This also works on any spherical region, even ones defined by non-great circle curves. Just define rotation as deviation from the great circle, and it should be good

Best video on this topic

Gauss Bonnet for sure 👍🏻

how have i never seen this before

Immediately shared, very nice result and the proof is almost trivial!

so satisfying! great video!

Just a question. The area of this region inside a sphere would logically change if the sphere’s size changed. I don’t see how, only with the angles, you’re able to accurately determine the area for A.

Beatiful explanation... It's very clear and easy to understand!!

Me not knowing how to measure angles on the surface of a sphere

kzfaq.info/get/bejne/j56Gmtmgqt6beI0.htmlsi=7OoLdri32GNyyan2

For a hyperbolic plane just negate this result

I was going to say straight into the action! I don't mind that at all aha

Why is it a/2π and not 2π/a?

Would be cool if someone came up with an equation for mapping the coastlines of the world with curved vectors between continents that don’t touch so that it was vectorized and open sourced

Beautifully rendered. Appreciate the wisdom of the principles.

Good explanation very satisfying to watch👍

I was studying non Euclidean and was curious about this

The area depends on the angles only 😳

Oh my Euler! There is a second channel!!!

Geogebra 😉

wow you explained this really well, good job

hey, great video!!! i was just thinking, how about just extending 2 out of the 3 sides of the "quadrilateral", ig, then then computing the area? kinda like how we can compute the volume of a frustum of a cone, if that makes any sense?

This is purely for a future video on the main channel. Try to guess what it could be!

Wait at "2:35" can we directly substitute mu with mu1?

You should finish one case then the other

You are unbelievable !

I totally agree. I've thought this same thing in all my math classes. I'm glad somebody else agrees.

What software do you use to create these ?

Excellent Videos ..

Very very nice stuff

according to Curry-Howard and type theory, proof = computation😅, it literally is!

1:22 What happens to the X in the numerator/how is this term manipulated?

Ok I know I' just stupid but how the hell does a probability diverge (a.k.a. be greater than 1)

Thank you

Hi, first of all, I want to thank you because your videos almost make me cry for their beauty and simplicity. But I respectfully disagree here. It's not that math is compulsory, is that many teachers do not know how to teach it! We are all familiar with the formula for 1 + 2 + ... + n being discovered by Gauss because his teacher was lazy/incompetent. And to further forroborate this point, most people fall in love or start to hate Math at a very young age. I believe we, therefore, need more research and training into how to teach math. It's definitely true that too much emphasis is given to computations. One of the books that hooked me forever to the subject and luckily enough made me study it at one of the most prestigious uni (aka "the other place") is "The number devil" and I later realised it was the case because it was able to explain with utter simplicity very beautiful topics. Another book I recommend is "Math without numbers". What is needed, on top of different "avenues" to captivate more people that might otherwise even become poets or philosophers into math, is a different way to assess math capability, and I fully agree on this end. But alas, to find something that is standardized is not an easy task and I would say that perhaps the creativity of certain Math Olympiads questions is what could be used, together with perhaps small research projects but I am not sure how this would be feasible.

Very informative and well done!

Yes, I literally just need the mean and the variance of Poisson distribution as a prerequisite for the next video, but while introducing Poisson distribution sounds a little more related to the Poisson process, I have a hard time pushing the variance part into the script without making it very irrelevant to the narrative in the main video. So that’s why I made a second channel video for this. I have also considered making a second channel video explaining the solution to the differential equation, but upon further research, I found this to be not only tedious, but also conceptually difficult, requiring Laplace transforms, and knowledge about the Bessel functions. I genuinely think the method of arriving at the solution is quite ingenious, but I just don’t think it is accessible enough for me to cover it, even as a second channel video.

Reminds me of how Maximum Likelihood Estimator in statistics is a single word in norwegian, "sannsynlighetsmaksimeringsestimator"

This is really amazing but also extremely counterintuitive

haha i’m a first year math student but i managed to hang on this explanation somehow. great explanation

Could we get a video on logarithms... please and thank you 🤗

Don’t even mention the GRE … devolutionary

The truth is that no one knows, but if it is aplicable to Math, then it is to everything else. I believe one of the best systems is a small mandatory number of hours a year (I believe people need to know a bit of everything, and these classes would be the essential bits of each subjects), and the rest can be chosen by the student (up to fufilling the needed hours of class a year)