I say it's mean because i want to juat study number theory and all this geometry is annoying me.
@djt6fan5 күн бұрын
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!
@lexinwonderland574114 күн бұрын
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!!
@darkshoxx17 күн бұрын
I would've just considered it as the mean of the two eigenvalues of the matrix 🤷♂
@2maniac17 күн бұрын
Which was what I thought it meant! However, I think it makes more sense that it is the mean of the curvature of the cross sections!
@2maniac17 күн бұрын
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 :)
@sclearDevelopment4 күн бұрын
Can you write the explanation? I've just finished linear algebra and I don't know how to approach the proof.
@TauGeneration2 ай бұрын
so basically spherical integral
@ralvarezb782 ай бұрын
normally I integrate the surface on spherical coordinates
@nanamacapagal83422 ай бұрын
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
@strikerstone2 ай бұрын
Best video on this topic
@user-mc7bi3mk8l2 ай бұрын
Gauss Bonnet for sure 👍🏻
@takyc78832 ай бұрын
how have i never seen this before
@berkeunal57732 ай бұрын
Immediately shared, very nice result and the proof is almost trivial!
@eduardoeller1832 ай бұрын
so satisfying! great video!
@timefuzzball80972 ай бұрын
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.
@2maniac2 ай бұрын
I would acknowledge that I didn't make it clear that it should be at a unit sphere (I only very briefly mentioned at 0:40 when I mentioned the sphere has surface area 4*pi).
@timefuzzball80972 ай бұрын
@@2maniac I see, but what if I wanted to measure the are of A in a bigger sphere? Sorry, my math understanding is not very advanced yet.
@pettanshrimpnazunasapostle19922 ай бұрын
@timefuzzball8097 general formula for surface area of a sphere is 4pi*r^2. So replacing all 4pi with it should work for spheres with larger or smaller radius
@S1GMATHS2 ай бұрын
Beatiful explanation... It's very clear and easy to understand!!
@aymanmansoori89362 ай бұрын
Me not knowing how to measure angles on the surface of a sphere
@gdtargetvn24182 ай бұрын
The angle between 2 great circles is defined to be the angle between 2 planes that respectively consist of those circles. Equivalently, the angle is also the angle between 2 tangents of those circles
I was going to say straight into the action! I don't mind that at all aha
@janbendrixmalagayo4902 ай бұрын
Why is it a/2π and not 2π/a?
@2maniac2 ай бұрын
Out of the full 2 pi possible angle, we take a slice of alpha, so the proportion is alpha/2pi.
@ttmfndng2012 ай бұрын
a is smaller than 2pi, so if it was 2pi/a you would get a number greater than 1. When multiplying by 4pi, you would then get a number greater than 4pi, and find the section of the sphere you're calculating the area of has a greater surface area than the sphere it's on
@ValidatingUsername2 ай бұрын
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
@anandarunakumar68192 ай бұрын
Beautifully rendered. Appreciate the wisdom of the principles.
@languafranter34502 ай бұрын
Good explanation very satisfying to watch👍
@duckymomo79352 ай бұрын
I was studying non Euclidean and was curious about this
@txikitofandango2 ай бұрын
The area depends on the angles only 😳
@fangjiunnewe36342 ай бұрын
Well, technically all the angles are multiplied by r^2, but it is typical to use r=1. And yes being independent of the arc length is nice
@samueldeandrade85352 ай бұрын
Oh my Euler! There is a second channel!!!
@mzg1472 ай бұрын
Geogebra 😉
@FlwxX2 ай бұрын
wow you explained this really well, good job
@dhruvarai18952 ай бұрын
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?
@2maniac2 ай бұрын
Not sure what you mean by 3 sides and then a quadrilateral? Do you mean that if we have a quadrilateral, then we can compute the area by "extending" some sides to form a larger triangle, computing the area of the bigger triangle, minus the area of a smaller one? If yes, I think that's possible IF the quadrilateral is made of "great circle" sides, so you can't use it on the quadrilateral-like region in the video for example.
@dhruvarai18952 ай бұрын
@@2maniac my bad. excuse the typo. And yeah, that's what I meant by extending the sides. Thanks for the clarification 🙏
@2maniac2 ай бұрын
This is purely for a future video on the main channel. Try to guess what it could be!
@leif10752 ай бұрын
Thanks fpr sharing. Can you share hiw ypu don't get bored and fed up and frustrated with math? Hope to hear from you!
@PMA_ReginaldBoscoG2 ай бұрын
Gaussian curvature?
@dirtydanthephysicsman17502 ай бұрын
No way if you do a video on Gauss Bonnet I’d go crazy! It’s such a beautiful theorem that I’ve thought about how a video on it would go but if I trust anyone on math KZfaq to do it it’d be you!
@jasdeepsinghgrover24703 ай бұрын
Wait at "2:35" can we directly substitute mu with mu1?
@wenwensi95975 ай бұрын
You should finish one case then the other
@Dr.HarshTruth7 ай бұрын
You are unbelievable !
@retiredharvest8 ай бұрын
I totally agree. I've thought this same thing in all my math classes. I'm glad somebody else agrees.
@user-ef2nc8ic7r8 ай бұрын
What software do you use to create these ?
@user-ef2nc8ic7r8 ай бұрын
Excellent Videos ..
@StratosFair11 ай бұрын
Very very nice stuff
@kaa1el960 Жыл бұрын
according to Curry-Howard and type theory, proof = computation😅, it literally is!
@chsgs9009 Жыл бұрын
1:22 What happens to the X in the numerator/how is this term manipulated?
@iagojacob3785 Жыл бұрын
Ok I know I' just stupid but how the hell does a probability diverge (a.k.a. be greater than 1)
@Dubai_life_ Жыл бұрын
Thank you
@Forzalnter1996 Жыл бұрын
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.
@Nylspider Жыл бұрын
I genuinely think a lot of teachers would actually be perfectly capable of teaching math well. The issue is foundational: these teachers are being forced to teach curriculums that rely on computation. It isn’t the teacher’s fault, it’s the system as a whole, which is what the message of the video was
@ShenghuiYang Жыл бұрын
Very informative and well done!
@2maniac Жыл бұрын
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.
@jacekstankiewicz1594 Жыл бұрын
Reminds me of how Maximum Likelihood Estimator in statistics is a single word in norwegian, "sannsynlighetsmaksimeringsestimator"
@jasdeepsinghgrover2470 Жыл бұрын
This is really amazing but also extremely counterintuitive
@andrewdupper973 Жыл бұрын
haha i’m a first year math student but i managed to hang on this explanation somehow. great explanation
@princehickmon2170 Жыл бұрын
Could we get a video on logarithms... please and thank you 🤗
@HubertMarekD Жыл бұрын
Don’t even mention the GRE … devolutionary
@muriloporfirio7853 Жыл бұрын
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)