The teaching techniques used in this video are vastly superior to classical math education. The animations are a massive improvement. And the open why and how approach is refreshing.
@gagadaddy8713 Жыл бұрын
This is the fantastic visualization of how Jacobian works. It may not be a very secrets concept for those who attended the higher level math course. But for those student new to advance calculus, this is definitely a precious gem --- Quickly guide them to the Multivariable mapping world.
@ibgarrido Жыл бұрын
If only math professors learn you like this. 3D sketch are fundamental
@krishyket Жыл бұрын
@@gagadaddy8713 The Jacobian is the key to the total derivative, which is one of the most important concepts in diff geo and advanced calc.
@gagadaddy8713 Жыл бұрын
@@krishyket Can't agree more!
@AdrianBoyko Жыл бұрын
I keep telling people that local educators are almost never going to be able to teach as well as the best KZbin video.
@mikexbox12 жыл бұрын
I remember my cal prof at the time saying "this is the formula for the jacobian, no need to know what it is just compute it" and i never even realized the areas of the shapes don't match when going from cartesian to polar ... you just blew my mind. Thank you
@ull893 Жыл бұрын
He himself didn't know. What do you expect 😂😂😂😂He is just glad, he has a well paid job that pay his bills🙏
@frieden62989 ай бұрын
I don't expect to learn everything from university. The teachers don't give me the rigor of detail that makes me satisfied. I learn more from self-study and these external sources. I just think of university as being there to give me an outline of what to learn and to hand me my diploma. To be fair, the laboratory facilities and libraries also facilitate my learning, but never the professors.
@kristopherwilson5067 ай бұрын
@@frieden6298I’m sorry you and many others have had this experience. This is a common/standard undergraduate experience. During my degree obtained at my university, I was very fortunate to dive into the “why”, and that motivated me to pursue graduate studies. My standard undergraduate multivariable calculus course went over the conversions from Cartesian to polar-and spherical-and discussed the Jacobian and its importance at length
@artyatbk2 ай бұрын
Serbian? I’m in 53 rn
@BangkokBubonaglia2 жыл бұрын
A great refresher on stuff I've forgotten over 20 years. After you've been out of school for a while, you just robotically insert "r dr dθ" for "dx dy" without even considering why you are doing it. Jacobians are just things you vaguely remember when trying to code up some algorithm you found in a paper and cursing how much linear algebra you have forgotten.
@blakegray3427 Жыл бұрын
What do you do integrals for
@StratosFair2 жыл бұрын
Easily the best explanation I've seen on the Jacobian determinant. You're an absolute legend
@paultaylor43589 ай бұрын
As a disclaimer I am a graduate of A&M college of engineering (graduated 20 years ago and never looked back). I came across this video skeptically, and to my surprise, I'm very impressed.
@Saujas24 күн бұрын
wow dude if i was not a broke student i would pay for this so you can keep producing such high quality videos for everyone to watch for free truly what a world to live in
@Saujas24 күн бұрын
but i wish you could have explained why we neeed to take that specific determinant
@necaro2 жыл бұрын
If Mathematics were taught in schools and universities more like this...imagine the world that would be possible. Thank you!
@horacioguillermobrizuela42958 ай бұрын
The best of all videos explaining the Jacobian I´ve seen.
@josephjacobus3022 жыл бұрын
There is a simpler way to think of this concept without having to understand why the jacobian is neccessary. dy and dx are graphically the deriviatives of lengths. The corresponding derivatives of lengths in polar coordinates is dS and dr, not d(theta) and dr, where S is arc length. dS = rd(theta). dA then = dy dx = dr dS = r dr d(theta). It works better with visuals, but I always found this a helpful way to understand this concept.
@ultimaxkom87282 жыл бұрын
A simple, clear and concise explanation. Thanks. That was very helpful.
@viniciuslourenzettiferreira2 жыл бұрын
Understanding the Jacobian helps to generalize whatever transformation you'd like to do, regardless how much you can "visualize" the new integral. However, the geometric example using dS, as r dθ, helps to visualize "what's going on" when beginning in integral change of variables.
@bherdzrob1557 Жыл бұрын
Jacobian is general
@tayebtchikou1646 Жыл бұрын
thank you very much Joseph, this is waht I am looking for.I can't tell you what you did for me
@JoeMama-hj1zk Жыл бұрын
If your confused A is area and switch the “then” and the “dA”
@james007900 Жыл бұрын
This was the most fabulous explanation on change of variable, which I have ever come across!
@comoenclase2 жыл бұрын
Magnificent way to visualize the Jacobian. Congratulations!!
@plantnt4895 ай бұрын
For someone who just started learning integral techniques, this video was surprisingly well explained
@moneymo64177 ай бұрын
Best explanation ever on the Jacobian. Amazing. I graduated almost a decade ago, but this was fun to revisit.
@rudypieplenbosch67522 жыл бұрын
This is the best explanation I found on YT even much more popular channels just come up with the transformation matrix, but fail to come up with a good example.
@lenoxbaloyi79822 жыл бұрын
The animation did the trick, I didn't understand anything before with just formulas but now after watching the video everything is clear ☺
@donkeyknight1453 Жыл бұрын
By far the most concise and clear explanation on change of variable I have watched.
@sshrek199610 ай бұрын
Wowwww! I could never understand the Jacobian when I was studying in college. This is by far the best video I have come acrossed
@ishraqshams101 Жыл бұрын
ive been trying to understand this for the longer part of an hour, your visuals helped me understand it within minutes
@mohan153doshi Жыл бұрын
Superb explanation of the Jacobian as a scale factor. And fantastic animation. Makes the concept of the Jacobian so clear. Great video.
@billcipher37372 жыл бұрын
Amazing video! Very well explained. Its unusual how a channel of such quality doesnt have at least a couple of thousand subscribers.
@phd_anime_cats2 жыл бұрын
The best explanation for the Jacobian I found out there! Thanks!
@GamblerRuined Жыл бұрын
If I had to rate the best videos on change of variable integration that I've ever seen, this is 1 through 7. maybe even 1 through 8.
@bismajoyosumarto12372 жыл бұрын
This is too good!! Why didn't I come across this video back when I was taking Calc 2 class in early 2022?!?! Thank you sooo much for this videoo! I might need it for a future encounter with the Jacobian
@pianodan160810 ай бұрын
U did stuff like this in calc 2? I’m only familiar with it being a calc 3 concept
@bismajoyosumarto123710 ай бұрын
@@pianodan1608 I think every math department classifies it differently. With the current curriculum, my uni has four calculus classes: - Calc 1 (single variable calculus): limits, derivatives, integrals, integration techniques, FToC, IVT, MVT, trigonometric and hyperbolic functions, surfaces of revolution - Calc 2 (multivariable calculus): partial derivatives, iterated/double/triple integrals, polar/cylindrical/spherical coordinates - Calc 3 (sequences and series): improper integrals, Taylor series, sequence/series convergence (Dirichlet's test etc), orthogonality conditions, Fourier series - Vector calculus: divergence, curl, Gauss' divergence theorem, Stokes' theorem
@pianodan160810 ай бұрын
@@bismajoyosumarto1237 I see!
@JhonatanNeneh2 жыл бұрын
Brazilian book of calculus from the mathematician Hamilton Guidorizzi is the best book of calculus! He explained every detail of the Jacobian with high class!
@georgesadler78302 жыл бұрын
Thank you for an exceptional video/lecture on the Change of Variables and the Jacobian in Calculus Three.
@physicsdoneright2 жыл бұрын
Excellent. This is good explanation for canonical coordinate transformation from one set of coordinate to another. I am happy to see this.
@wilisonilagi11432 жыл бұрын
I always learn well with animation.. thank you sir for this perfect teaching
@blemishingbohemian20752 жыл бұрын
This is a great video! Your explanation is very calm and clear, which makes it easy enough to follow what is happening in what can sometimes be a little overwhelmingly confusing math. The video really helped me understand the topic better, thank you!
@kavin43889 ай бұрын
Oh my god this video is amazing, such an intuitive explanation of it! It clicked to me right as you said the areas change and the jacobian is the determinant of the matrix (which represents the area of the thing, and since things are in derivatives, I’m guessing it translates to the change in area). Definitely subscribed
@keithkeller78954 ай бұрын
Excellent video and explanation of the Jacobian - thank you for creating !!!
@subtlethingsinlife10 ай бұрын
You helped me understand jacobian in one small lecture ... God , engineering was easy, if I had teachers like you ... gosh .. But thank you so so much
@willyh.r.12162 жыл бұрын
Excellently explained. If all math concept is explained with a visual, and concise comment, all math learners will love math. 100% guranteed.
@nkugwamarkwilliam8878 Жыл бұрын
Best explanation of jacobian i have ever heard thank you so much
@jamesbroughton21992 жыл бұрын
These illustrations are literally so nice
@aashutosh.mishra Жыл бұрын
The most satisfying video on change of variables. Thanks❤
@Parirash1232 жыл бұрын
Best explanation of Jacobian on KZbin
@nicoleworoniuk1544 Жыл бұрын
this is the best video i've seen on Jacobian explanation !!
@plwn6468 Жыл бұрын
Great, easy to understand reminder. Allowing me to pass it on to my son.
@carletpierre18952 жыл бұрын
This video got me excited for calc 3 next semester, this was so fun
@dcterr15 ай бұрын
Excellent explanation of the Jacobian! If I'd been able to watch this 40 years ago when I first learned this, I could have learned in 15 minutes what took me a month or so to learn!
@MajewskiMarcin10 ай бұрын
Super great and interesting video. You put so much effort in it. The outcome is impressive. Thank you so much.
@afroohar2 жыл бұрын
This doesn't explain why the Jacobian (as a determinant of partial derivatives) is the scale factor for the differential area of the transformation.
@Zxv9752 жыл бұрын
Good question. The tl;dr is that the typical 2D integral is wrong and is missing a wedge product. If you include the wedge product, then deriving the Jacobian is simple algebra and nothing mystical at all. A (very long) but more complete explanation is as follows. Honestly you could make an entire video on properly explaining this this idea... First, note that the double integral we're familiar with is missing something crucial. We know from 1D integration that integrals are signed; they carry information about orientation. This is still present in 2D with swapping bounds, but it isn't present when swapping the order of integration. For example, calculate the volume of a cube, given by f(x, y) = 1 from 0 to 1 ∫₀¹∫₀¹ f(x, y)dxdy = ∫₀¹∫₀¹dxdy= 1 This is positively oriented: you're tracing out the 2D square in the x-y plane in the anti-clockwise direction (we arbitrarily call this "positive") However, if we swap our order of integration we should get something negatively oriented, but we don't ∫₀¹∫₀¹dydx = 1 This *should* be negative because it traces out the square in the opposite orientation (clockwise), but it's not. What gives? The answer is that dxdy was the incorrect area form, and should instead be dx∧dy. This wedge operation behaves like the cross product in 2D and 3D (but it also works in arbitrary dimension), so it has the rule dy∧dx = -dx∧dy. This alone fixes the problem and recovers orientation. You could also arrive at the same conclusion by trying to integrate along axes that aren't perpendicular (i.e. using infinitesimal parallelograms as the fundamental unit instead of infinitesimal squares, as both can tile the plane both should be valid in principle). The standard dxdy has no nice way of handling anything other than perpendicular axes natively, but with the wedge product dx∧dy = dx×dy = sin(ϕ)dxdy we can handle it easily. This is the same answer you'd get through the Jacobian, but presupposing that would somewhat be circular logic. Now, with that established, deriving the Jacobian is a straightforward exercise in algebra. The entire reason that the Jacobian feels like a leap is because this entire backstory I presented here is swept under the rug and only implicit in the Jacobian. This means that you are indeed taking a leap as none of this is ever explained. We know that we can swap from Cartesian to polar, therefore x = x(r, θ) and therefore dx = ∂x/∂rdr + ∂x/∂θdθ is the total derivative, and same for y. For the integral, we already directly substitute for x and y, so why don't we also just substitute for dx and dy directly too? ∫f(x, y)dx∧dy= ∫g(r, θ)(∂x/∂rdr + ∂x/∂θdθ)∧(∂y/∂rdr + ∂y/∂θdθ) Where g(r, θ) = f(x(r, θ), y(r, θ)) is just what you get from plugging in polar coordinates into your original function. This looks like a mess, but it's honestly not that bad. The wedge product distributes like an ordinary product would, so expanding using the distributive law gives something which has the form Adr∧dr + Bdr∧dθ + Cdθ∧dr + Ddθ∧dθ for the stuff from the brackets. Each of the terms A, B, C and D are just products of partial derivatives. Now a straightforward consequence of the wedge product is that dx∧dx = -dx∧dx = 0. This means that the A term and D term disappear from the expansion, and we are left with only B and C. We also note that since dr∧dθ = -dθ∧dr, we are overall left with the term (B-C)dr∧dθ, so the integral becomes ∫g(r, θ) (B-C)dr∧dθ and if we evaluate B we get ∂x/∂r ∂y/∂θ and C we get ∂x/∂θ ∂y/∂r which is exactly what you get from the Jacobian method. This is all perfectly explained extremely directly if you use the proper wedge product in your integrals. The wedge product makes the absent terms disappear and inserts the correct sign in front of C. The last thing missing is the absolute value. This is again needed because of the missing wedge product. Since drdθ has no orientation information, we must have ∂(x, y)/∂(r, θ) = ∂(x, y)/∂(θ, r). But algebraically this clearly isn't the case, so to sidestep this issue we throw out all orientation information and focus solely on the magnitude, which is what the absolute value does. With the wedge product, we instead would need to have ∂(x, y)/∂(r, θ)dr∧dθ = ∂(x, y)/∂(θ, r)dθ∧dr, which is automatically true because both minus signs on the RHS cancel out. So there's no need for an arbitrary absolute value. For the life of me I don't know why this isn't taught like this. It's not like any of this is any more difficult than standard multivariable calculus coursework.
@afroohar2 жыл бұрын
@Bryte H I think the signed aspect isn't crucial to understand the Jacobian determinant. Neither really is the formalism of the wedge product (yes it becomes handy in dimensions greater than two, but the fundamental intuition can be garnered by just considering examples in R^2) I think a simple explanation for calc 4 students just involves them understanding a bit of linear algebra. The central intuition is that we are transforming a vector of variables (x,y) to another vector of variables (r, theta). This is a non-linear transformation, but since it is differentiable, locally, within a small range (x,y) to (x+dx,y+dy), the transformation is linear. A linear transformation can be described as a matrix, that matrix is the Jacobian. This can also be seen by introducing to students the multivariable Taylor expansion, showing that the Jacobian is the first term, analogous to a simple derivative, but for more general transformations from R^n to R^m Now that we know the Jacobian is the transformation that takes tangent basis vectors x,y to r,theta, a natural question is what happens to the area under such a transformation. Here is where you introduce to students the idea of a determinant as a signed scale factor for areas under linear transformations. The signed aspect can be understood by giving students a couple simple examples (does the transformation flip the orientation of the basis vectors?) In many ways it is just a more general, multi-dimensional, form of "u-substitution" students are already intimately familiar with.
@Zxv9752 жыл бұрын
@@afroohar Yes, the Jacobian can be explained in a consistent way through what you just described. I just don't see how this is a *better or more intuitive* explanation than simply including orientation in 2D+ integration (something we know it's supposed to have anyway) and doing basic algebra to get the answer we expect. Everything falls out for free, and orientation doesn't really make things more difficult. Doing mental gymnastics and bookkeeping to avoid orientation is significantly more confusing. At least in my personal experience, even though I learned the Jacobian explanation you described in 2nd year uni, because I could never just derive it directly it never stuck with me. If I was expected to substitute x and y and I could come up with expressions for dx and dy, then why couldn't I just directly substitute dx and dy?! "Am I doing something wrong?" "Do differentials not obey algebra?" I had vague notions of 2D Infinitesimal area elements and sometimes remembered there was an absolute value in the formula for some reason, but the fact that none of it seemed like a proper, direct generalisation from the 1D case (where does the absolute value come from...?) meant it was hopeless for me to properly and intuitively grasp it. As a result, I also failed to grasp all subsequent ideas like Green's theorem and Divergence theorem because of the shaky fundamentals. They all seemed like disconnected ideas and I am heavily against rote learning. Maths is beautiful and connected, and if I couldn't see the connection between ideas then I must not be looking hard enough, or I must be misunderstanding something. I blamed myself for not being smart enough, but in reality that disconnect was just an artefact of the poor framework I had been taught. Under a more suitable framework they're all just various expressions of Stoke's theorem, but that's completely non-obvious without the wedge product. Now, there are obviously applications where having an orientation agnostic definition of 2D integration is useful, but it should be understood that those are a specific case and that's not the most general rule. In the same way that ∫f(x)|dx| is sometimes a useful quantity to calculate for certain applications (it's useful for measure theory, for example, where you are largely interested in non-negative quantities), it is *not* the most general 1D integral and shouldn't be treated as such. Many students carry an implicit and incorrect assumption that 2D+ integrals don't inherently include orientation, and it's not until much later (or never, in the case of many engineering and non pure math streams) in a course on differential geometry where we correct the misunderstanding. Why wait that long? Why not just show the proper, most general framework from the beginning? Especially if that framework is a direct generalisation of the 1D case, it allows the results to be obtained easily through basic algebra *and* it fundamentally links all of the other topics (Stokes, Divergence, Green's theorems) covered in multivariable calculus course in a unified framework? Honestly, the explanation I suspect is status quo and legacy. We teach multivariable calculus and differential geometry as two artificially disconnected subjects because we've always done it that way and existing textbooks are written assuming it will be taught that way. Imo that's a very poor reason.
@afroohar2 жыл бұрын
@@Zxv975 You gave a really good explanation. I guess it's kind of to each their own, I can see why to some who really enjoys abstract algebra, ideas like orientation and formalism like wedge products are much more natural. But for someone like me who is a huge fan of geometry, I find such ideas unfamiliar, and prefer to think about manifolds and tangent spaces :D
@vneger23772 жыл бұрын
@@afroohar your explaination is very clear, thank you.
@benhoppe3386 Жыл бұрын
Thank you!!! This makes me finally understand a concept I was confused about for quite a while now. The visual interpretations in this video are really helpful!
@garrettledford1147 Жыл бұрын
I knew I needed to just sit down and find somebody who can explain this stuff conceptually but it's still surprising when I'm doing it
@parsahamidi2 жыл бұрын
By far one of the best videos I've seen.
@JonathanLang-nu2lx9 ай бұрын
Would love seeing a vid about why using the jacobian determinant works. Now this just feels like magic
@yeast4529 Жыл бұрын
Best video there is on Jacobians
@connorgoosen24682 жыл бұрын
Such a great combination of using animations for visual understanding and actually showing how the Maths gets computed! Count me subscribed ;)
@alijoueizadeh2896 Жыл бұрын
Thank you. Visualization is the key.
@lordhelmet97097 ай бұрын
Cannot thank you enough for this explanation
@user-nu9xn4jh9b7 ай бұрын
I love you so much. This genuinely help me a ton.
Жыл бұрын
wonderful, i finally understood it, you are a very good teacher it is just crazy
@mathalysisworld6 ай бұрын
You got me in 2:49. I was so easily going with your flow. Haha
@MathManMcGreal2 жыл бұрын
bravo my guy, these visualizations are so good!
@karlyohe63792 жыл бұрын
Nicely done; I am proficient at using the Jacobian. But that explained WHY it works really well
@rafaelortega13766 ай бұрын
What I do not think is explained is why the determinant of this is precisely what you need to make the conversion. Maybe it is based on top of a prior video where this meaning is explained. Anyway, good video and well explained. I also like the comment about how this can be understood without Jacobians. I always have used that way of thinking, but thinking about the same topic in multiple ways is helpful
@aperson6389 Жыл бұрын
Worth noting that, because of how determinants work, you can swap the rows/cols and transpose the matrix as much as you want and the |J| won't change
@darcash1738 Жыл бұрын
True, the only way you could actually mess up the det is if you put the partials of x and y on the same row
@keyyyla2 жыл бұрын
This is a wonderful video! Very clearly explained.
@啟馼陳5 ай бұрын
This is god damn clear. Amazing!
@warisulimam34402 жыл бұрын
I can't pint-point a particular reason but I absolutely love your teaching method. By "teaching method", I'm not really referring to using animations, or the particular example you used. I'm referring to something more abstract and even I am not sure what it is that I am referring to. I just wanted to let you know that it was a brilliant video although I wouldn't really be able to give a solid reason to back my remark, if asked. I hope you continue creating videos and create playlists on different subjects in mathematics. I wish you all the best!
@jonashallgren44462 жыл бұрын
Absolute banger of a video.
@NXT_LVL_DVL Жыл бұрын
Best explanation ever ❤❤❤❤
@mastershooter64 Жыл бұрын
Great video! very nice explanations but I think you could've made the video even better by including a 5 minute segment on what the total derivative and the jacobian matrix are and what they represent and from that one can easily understand why the jacobian determinant tells one how much an infinitesimal area gets scaled when you do a coordinate transformation. From that you can easily talk about general curvilinear coordinate systems.
@clayton973302 жыл бұрын
Nice application of a shell balance to show the differential element in the theta direction is r*d(theta).
@ull893 Жыл бұрын
Sir, just keep going. Your channel is amazing 🙏❤️
@Teo-uw7mh10 ай бұрын
but you didnt explain why the determinant of the jacobian scales it correctly
@KYosco15 күн бұрын
Food for thought
@MS-cj8uw10 ай бұрын
الف رحمة على والديك ...😊
@Z-eng02 жыл бұрын
Love the animation, but just needed to say that it took me a while to understand what the transformation actually does in variable terms, that it takes r and theta from the r-theta domain and transforms/teleports the x-y domain *while* remaining r and theta
@yash11522 жыл бұрын
0:00 Thanks to Texas Uni for funding. Great!!
@boidaemon87332 жыл бұрын
you earned a subscription with this one
@michaelaristidou26057 ай бұрын
Excellent video on the topic
@bra1nsen7 ай бұрын
11:27 can’t you visualize a 3rd diagramm once Jacobi applied? Or in other words, visualizing Jacob matrix depending on r and theta 🤔
@ayooluwakolawole Жыл бұрын
This is so beautiful, thank you
@mukeshsharma74552 жыл бұрын
nice explanation with good graphics
@theillusionzone Жыл бұрын
This is a truly fantastic video, with great explanations and animations to match. The only part I don't understand is as follows: Why do you keep referring to this as a "volume double integral" at say 10:37, when you are actually simply computing the area of a circle, hence the double (not triple) integral?
@symonneliu243410 ай бұрын
THIS VIDEO IS DIVINE CONTENT
@hacerdemirel9833 Жыл бұрын
Oh thank you! Helped me so much.
@billmichae10 ай бұрын
Fantastic explanation!
@amirderakhshan644410 ай бұрын
Fantastic, Amazing ❤💥🙏🙌💯
@jacksonyu5322 Жыл бұрын
Amazing, just amazing!
@ggeremia Жыл бұрын
Brilliant lesson. Thanks.
@howitfeelslike53813 ай бұрын
A true masterpiece
@GimbertLane2 жыл бұрын
Beautifully done
@suhasisroy72402 жыл бұрын
I am surprised why you have so less views. You have extraordinary sense of visualisation of maths like 3b1b
@vigneshbalaji21 Жыл бұрын
Awesome explanation :)
@zukofire6424 Жыл бұрын
amazing explanation, thank you!
@bloom9458 ай бұрын
Amazing! Reminds me of Morphocular EDIT: I KNEW IT
@mechadense2 жыл бұрын
Why the determinant? IIRC because (in 3D) it corresponds to the triple product and thus the volume of the differential element relative to the volume of the untransformed differential element.
@justsomegirlwithoutamustac58372 жыл бұрын
UR AMAZING PLZ KEEP MAKING AWESOME VIDEOS LIKE THIS ONE
@Pauldyke Жыл бұрын
Brooo. Very Impressive video
@artinzareie48062 жыл бұрын
Why this page is so underrated
@cfebresmol2 жыл бұрын
Excellent video, thank you!
@mohammadsadeghashrafpoor6737 Жыл бұрын
Finally i undrestand jacobian 😂 Thank you ❤
@RAyLV179 ай бұрын
Does the Jacobian (scaling factor) always exist within the integral expression when we're changing the coordinate system? For example if I am "changing" the coordinates from x-y to y-x, the Jacobian should just be 1, since there isn't going to be any shearing or change in shape of the plane.
@steelguard2 жыл бұрын
Fantastic video! Thank you from your newest subscriber!
@aryansudan2239 Жыл бұрын
simply brilliant
@Yakubian12 күн бұрын
at 2:20 shouldnt it be 9+r^2? because x^2-y^2 cant be rewritten as r^2
@abdulelahaljeffery6234 Жыл бұрын
1:44 why are we interested in points that are closer to the origin?
@briandwi25042 жыл бұрын
That was just soooo good!
@Ноунеймбезгалочки-м7ч8 ай бұрын
isn't 0≤O≤2π(I can't find the fancy O) the same as 0≤O