Volume of a ball in n dimensions
HTML-код
- Опубликовано: 25 авг 2024
- In this video I explicitly calculate the volume of a ball of radius r in R^n. The method I’m presenting uses only multivariable calculus and the disk method from single-variable calculus, but we’ll also visit some other goodies like the Beta function and the Gamma function. Enjoy!
![Tee Grizzley - Detroit (Feat. 42 Dugg) [Official Video]](http://i.ytimg.com/vi/aeKfTtt0x14/mqdefault.jpg)
Link to sequel: Surface area of a sphere in n dimensions: ruclips.net/video/-uaVHzMgxw8/видео.html
Absolutely brilliant energy! Without this kind of energy it is would be hard to watch such a long derivation.
Thank you!!! :D
What really surprises me is that for n->inf the volume Vn(1) goes to 0.
Abstraction.
It is a surprise to me too
That's true, but it's of little real significance, because it's based on making the radius of every such ball the same (R=1). And while we tend to fixate on the radius as *the* way to characterize the size of a sphere or ball, you could just as well choose some other thing.
If you choose the diameter (so that each ball fits snugly inside the unit hypercube), the comparison (the ratio) from dimension to dimension changes; although, still, Vᵢ → 0 as i→∞.
You could choose some fraction of the radius, and get yet a different set of ratios, one to the next; but in all of those cases, Vᵢ → 0, because the factorial in the denominator will eventually overwhelm the n'th power of any constant in the numerator.
So this property, at least, is robust.
iam sorry sir. do you care to elaborate on why does this happen(geomtricly) and what it means for a shape to have 0 ''volume'' ?
+ dimos vallis: None of those shapes has 0 volume.
Their limiting volumes, as the number of dimensions goes to infinity, go to 0.
This is essentially no different from a (positive) geometric sequence whose ratio r is between 0 & 1: 0 < r < 1.
All the terms are > 0; no term = 0; but their limit is 0.
i think he has a part time job in a bakery. it's not chalk, it's powdered sugar and flour. it's tough to "differentiate" between them. :-)
Or... some other white powder, Mr. Walter White... :)
I think he wipes the board with his body
He wipes the board with chalk
100th like. And you know he is something of a board himself
Fascinating! From the N=2 and N=3 cases, one might have guessed they would all be of the form k pi R^N for some rational constant k, but it seems those cases are special in having only a single power of pi.
Interestingly, though, it seems that, despite appearances, you never actually get a fractional power of pi. Using gamma(x+1) = x gamma(x), we can unravel gamma(N/2 + 1) one step at a time until we get to either gamma(1) = 1 for even N or gamma(1/2) = sqrt(pi) for odd N. In the odd case, the sqrt(pi) cancels half a power of pi from the numerator, leaving an integer power of pi.
I worked out the next couple of cases and got:
V_4(R) = (1/2) pi^2 R^4
V_5(R) = (8/15) pi^2 R^5
Gaining a power of pi every *two* dimensions like this is really surprising, and makes me wonder whether there is some deep geometrical reason for it.
Apparently for every even number of dimensions (2n) the unit ball volume formula is ((π/4)^n)/n!
so if you add up volumes of unit balls in even number of dimensions the sum converges to exp(π/4)
I like the angle of these videos. Side is better than straight.
Thank you! I’ll keep that in mind when I’ll do my next batch of videos!
people don't know linear algebra but are fine with multi variable calculus, that s weird :D
That is the first thought I got too.
I too
I agree that Intro Linear Algebra is a prerequisite for Vector Calculus.
Dr. Peyam chalkboards are very 20th century. Digital whiteboard already please better even than bprp’s amazing pen handling skills! Maybe one of your grad students can hook it up.
At my school (Rutgers), Introductory Linear Algebra wasn't a requirement for Multivariable Calc, but it was a requirement for Elementary Diff Eq's. In the more advanced track, Introductory Linear Algebra wasn't required for either course, cuz the courses actually covered all the Intro Lin Alg material. Super weird...
that's the same as you choose matrix mechanics or wave mechanics
When I first watched this video (right when it came out) I was just in third semester of physics, and thought I would never understand such cool maths like the ones displayed in this video. Now I'm in ninth semester, ready to graduate and I came back to this video because I needed it to calculate the entropy of the ideal gas via statistical mechanics, and I understood everything. I just find it strange how you could think that you haven't grown or mature as life goes on, but when you look back you realize that in fact you have grown and matured a lot. Thanks for always making such cool videos Dr. Peyam, I've been following you since you started in youtube, keep it up!
That is so sweet, congratulations! Yes, the thing is that at every point in life you only see dx but actually the things you’ve learned are integral f(x) dx :)
Hahaha you're right! Good analogy 😂 thank you so much for your videos, your evident love for teaching, I kid you not, has been one of the reasons I love math more than I thought I did before I found your channel.
"I guess some people don't know linear algebra yet, so lets do it using multivariable calculus". Where I live, Linear algebra comes way before multivariable calculus from pretty much every perspective possible lol :P
I totally agree, but here it’s done in a different order :(
now that i can understand everything in this video(because i watched it the last year and didnt understand )....THIS IS THE BEST VIDEO ON RUclipsEEEEEEEEEEEEEEEEEEEEEEEEEEE
What pre-requisites did you take to get all the understanding needed
you could also have mentioned that for all integer n>0 the output happens to always have an integer exponent of pi, since the gamma function at half-integers (science term for specifically integer + 1/2) has a pi^.5 term to cancel the same in the numerator
After troubling with my minor subjects, satisfied by another mathematics video, inspiring me to go ultimate professional with mathematics, after all. "When dr.Peyam lectures it's a show ❤"
Great video! Excellent explanation! I really like watching people that love what they're doing!😊
I love using beta and gamma functions in calculus exercises :D Amazing video!!!
In 11th grade I forgot my tennis shoes and had to sit in the hallway during gym class, I remember trying to generalize this. I don’t really remember how far I got
Always compelling videos, and a pleasure to watch. You, bprp and Michael Penn and my favs
Very clear, motivated explanation of high-dimensional ball, good job!
It helps me to study statistical mechanics.
Thank you doctor Peyam. You always makes mathematics very tasty!
Absolutely beautiful! ❤️
Congratulations, you just won the title of: the most confusing guy that I have ever seen 😂. Don’t take it personally, I don’t know anything about what you are trying to explain, yet I watched it because of you 😂😂
I have been wanting to see a video like this forever! I've seen the formula on Wikipedia, but none of the derivations made sense to me. Thank you!
You’re welcome :)
So using this formula we can actually find the volume of a ball in negative dimensions. If the dimension is a negative even number then the volume is 0. This leads me to guess that the functional form of the zeta function, whose "trivial zeros" are at negative even numbers, has a factor of 1/Gamma(s/2 + 1).
Amazing! All my support!
Merci pour votre travail.
OMG, Euler discovered the gamma and beta functions! We statisticians use them a lot!
I loved that proof!...and even more the fact you sincerely didn't know about the beta!
Maybe in my last post i didn't make it clear....i love these kind videos! More of them, thank you, pi-em!!!!
P.S. That zi-ti pasta's joke was lovely! I love the "ziti"!!!
P.S.2 I know that you're right and we can multiply the integrals..yet i love the old way of putting the dz in the end :)
P.S.3 I always had a feeling that Laplace took someway a "point" from the gamma function when he discovered the L.T.....it's just a feeling because i already know that many other mathematicians (contemporary to L.) "felt" that e^? might have been a great choice for a tranform. Whaddayathinkbout?
Thank you! :D And I’m actually not sure how Laplace came up with his transform! My only guess is that because the solution of y’ = ky is y = e^(kx), it follows that y e^(-kx) is constant and hence its integral? But it’s still pretty cool, you can use it to transform the wave equation into the heat equation!
beautiful result
This is really great, you seemed to put a lot of effort into this.
The idea of the ball in that graph at 8:13 is really cool and starts the whole recursion process.
Then, the beta and gamma function are super funny because there are cool properties to deal with.
Good job
Thank you!!! :D
I'm very like your video about maths. It make me easy understand and it funny :D.
That's such a lovely treatment of a simple problem. The quickest solution is π/6(d³) ... but I bet you were never taught that in school. You see, d represents the diameter of the ball, and d cubed represents the volume of the cube that contains it. As you probably know, the length of d runs from a speck longer than zero to infinity. When are mathematicians going to be taken behind the wood shed?
Excellent video
very nice
Great lexture - that last drawing of the sphere with the slice reminded me of the Death Star LOL
I've just subscribed your channel, which help to add motivation for your team :D.
Thank you!
Using the property Gamma(x+1) = x*Gamma(x), you get the nice recursion:
V_n = 2*pi/n * V_(n-2),
with V_0 = 1 and V_1 = 2.
Great stuff. thank you.
Thanks a lot!!!! You're an angel!
😇
This was so satisfying! And soooo coooool!!!! I love these videos!!!
Omg this is amazing
Volume of a ball in n dimensions? More like "Very amazing derivation which can stoke our imaginations!" 👍
😃yay Dr Peyam! Can you please make more videos specific on multivar calc or linear algebra or something cool? Because it gets hard to find things that helps when in higher classes...
There are plenty of linear algebra videos on my channel, just keep browsing my videos :) And of course many more to come!
Oh thanks! Also, do you think learning linear algebra on my own is a good idea?
Definitely! It’s one of the subjects that’s super accessible; you don’t even need to know calculus to learn linear algebra
now generalise to non-euclidean metrics :P but for realsies if you did some L-p stuff i reckon that'd be super interesting.
This was mind blowing!
The chalk sounds from 9:25 to 9:39 are pretty musical
When N is even so N=2K ... the gamma functions become factorial and the formula becomes: V_(2K)(R) = ( pi^K * R^2K ) / (K+1)! so for example when N =8, K=4 and the volume = pi^4 * R^8 / 5! so the volume of a sphere in 8 dimensions is pi^4 * R^8 / 120
i was speechless seen this.
however, where is dr peyam set theory vaganza?
I actually don't know how this is on a larger scale but at my school in particular, calc 3 comes right before or around the same time as linear algebra. I wonder if there's a "quadratic algebra?" Very nice video :D
I imagine that would be analogous to 2-to-1 functions, so there can be two points that are mapped onto the same point after a ”quadratic transformation”?
Nobel prize should be started for maths and given to u. Agree???
Hahaha, it’s called a Fields medal :P
The change of variables u = zt, v = z - tz is basically polar coordinates with the taxicab (l^1) metric. Solving for z and t we get z = u + v and t = u/(u + v). z is the taxicab "radius" and t, which is similar to slope, is the "angle". My professor called these taxicab coordinates but that isn't standard.
Deepto Chatterjee That’s the comment I was talking about!
Dr. Peyam's Show thanks, I think I'm beginning to understand it. I understand how z is the taxicab radius but I'm trying to figure how numerically t relates to the angle; I sort of understand how it's like the x-coordinate divided by the taxicab distance which gives an indication to what the angle is but I can't figure out how to quantify it
Hi Deepto. I'll elaborate on the connection between t and the angle. First, it's important to keep in mind that this change of variable is only used for x,y >= 0. If either is negative then x + y is no longer the taxicab norm. Now for any value of z the graph x + y = z is a line segment from (x,y) = (0,z) to (z,0). The variable t = y/(x+y), which ranges from 0 to 1, is essentially the taxicab angle because 1) it identifies a point on this arc and 2) it is proportional to the arc length (which I only realized just now). Hope that helps!
nice video!
I love it ❤️
10:55 , how you can use the fact R=sqrt(1-x^2) in the general case Vn(R) ? R is obtained from 2 dimensional case !
Vn(r)=integral Vn-1(r), that I understand but
how you can put r=sqrt(1-x^2) in this ?
The way you obtained 'r' is assuming the ball as 3D !
Its because the two things that defined r are independent of the dimension. The radius is always 1, and x1 is the variable used in integration.
You'll notice that the case of a 3D ball is not unique, by seeing that you can use the same method for a 2D ball
I remember doing this ages ago. It was so satisfying, and then figuring out n->∞, the measure goes to 0 was even crazier.
EDIT: I wasn't as rigorous, but it hey, it was still very fun!
I really liked this. Can something similiar be done for SUM of n^k where n=1,2,3...N and k=1,2,3...K?
I don’t know if it’s similar, but you can evaluate that sum, it’s called Faulhaber’s formula
Why didn't you just to a trig substitution in computing the square root function? You would have been able to reduce it down via a recursion relation.
Do you have a reference for the disk method?
Amazing mathematix!
But what about the "gama(n/2+1)" , can you comput it? or make a more simplifide expretion?
Yeah i was thinking (n/2)!
B is for beautiful! now I know why math is ez! I think it will be easier for me to learn category theory!
Gamma Function!
Very interesting!
Why we didn't use the trigonometry in the integral
So which different dimensional spheres have the same volume? If you include fraktional dimensions.
Very nice...so, extending this result, we could guess that the volume of an ellipsoid in R^n should be equal to K * a1*a2*...*an, where K is the coefficient that you have found with pi and Gamma, while a1, a2...an are the measures of semiaxes of the (hyper)ellipsoid. Can it be right?
So you can calculate the volume of a ball in real dimensions?
Or even substitute complex values here, if you're prepared to raise complex R to complex N
holy shit, imaginary-dimensional ball?
what could it even mean?
We are taking the absolute value of the volume function anyways (negative area is just positive area) so maybe it would all resolve to the volume of a ball in the absolute real dimension of the magnitude of the complex dimension?
Imaginary ball = mind-blown! :D
Dimensions in complex numbers...
Instead of using lines to "define" a dimension, you would use a plan?
intersections of (hyper)planes is already how dimensions are defined I think
Thank you doktor, video was great, but it becomes weird for the linear algebra section,
If we dont like Gamma and Beta functions we can derive reduction formula for this integral by parts
Int(sqrt(1-x^2)^n,x)=xsqrt(1-x^2)^n-Int(n sqrt(1-x^2)^{n-1}(-x)/sqrt(1-x^2) x,x)
Int(sqrt(1-x^2)^n,x)=xsqrt(1-x^2)^n-n Int((1-x^2-1)/sqrt(1-x^2)sqrt(1-x^2)^{n-1},x)
Int(sqrt(1-x^2)^n,x)=xsqrt(1-x^2)^n-n(Int(sqrt(1-x^2)^n,x)-Int(sqrt(1-x^2)^{n-2}),x))
Int(sqrt(1-x^2)^n,x)=xsqrt(1-x^2)^n-n Int(sqrt(1-x^2)^n,x)+n Int(sqrt(1-x^2)^{n-2}),x)
(1+n)Int(sqrt(1-x^2)^n,x)=xsqrt(1-x^2)^n+n Int(sqrt(1-x^2)^{n-2}),x)
Int(sqrt(1-x^2)^n,x)=1/(n+1)xsqrt(1-x^2)^n+n/(n+1)Int(sqrt(1-x^2)^{n-2}),x)
I_{n}=1/(n+1)xsqrt(1-x^2)^n+n/(n+1)I_{n-2}
Dear Dr. Peyam, could you do a video on where the gamma function came from? I know the part with factorial but dont know where the integral definition came from. And also this video was great :)
Just look the video he made about half-derivative, there he explains what the Gamma-Function is and where it comes from: ruclips.net/video/gaAhCTDc6oA/видео.htmlm34s
thx m8
Hello Dr.Peyam ,
Could you please teach us a *one* method of solving deferential equations that can solve most of deferential equations. No matter whether it is simple or complex.
Thank you in advance.
tareq ttatt I wish there were such a way, unfortunately differential equations are complicated, and there isn’t one size that fits them all! There are 2 videos on variation of parameters on my channel that are really useful, though, you can watch them if you want :)
I'd love to see the lin Algebra argument in the beginning! Is there some resource where I can look it up?
Check out The Jacobian
I love that ( Ta Da) 😂
Cant we substitute x1=sin(theta) and use induction to generalize the formula
V1(1)= pi^1/2 . 1^1 / gama(3/2) = pi^1/2 /((3/2). pi^1/2) = 2/3, but V1(1) should be 2, isn't it?
can someone explain to me why I can't instead of summing all the x values of the sphere just take all the radiuses from 0 to 1 and multiply by 2, I mean, I'm still going through all the values of R. but when I calculated the integral it came out wrong.
I literally understood nothing of the proof of why R can be separated... Just gonna accept it.
This is the moment I noticed the video is more than half an hour long
This. If you assumed nothing, the variables dx and dy are differ only by R, by virtue could could swap them around; why is the integral dy = V(1) and integral dx=V(R)?
This might help a little bit? ruclips.net/video/MIxTvKXG1jY/видео.html (in that video, let a = b = c = R).
Will You Swapping them around actually gives the same result in the end, try it out :)
What i understood is that to calculate the volume of n dimensional ball you need n dimensional board
My brain exploded.
yay! though i am still searching for a geometric proof of the surface "volume" of a 4d sphere/ball.
i remember the proofs for the circumference of a circle and the surface area of a sphere, so what would the 4d equivalent be?
The derivative of the volume of a ball in R^5 :)
Also I’ll definitely do the surface area proof one day, it’s super interesting!
no... that's not "geometric".
the geometric proof for the surface of a 3d sphere: ruclips.net/video/6EzQEdBX_30/видео.html
and there exists a general proof that the volume of any N dimensional sphere is r/N times the surface area, since the volume of any pyramid in N dimensions is base*height/N, and you can divide any sphere's surface into (dx) sized parts, and create pyramids of height r, so the sum of all their volumes is surface*r/N
OLD CHALKYAM WAS DOPEEE
Can you explain how you determined u and v in terms of z and t
There’s a comment below that explains it really well, I’ll tag you in it!
Love it!.. but wish I could follow it all :(
I tried to figure this out with spherical coordinates, and this is super complicated!
Wouldn't you be calculating N-olume and not volume? (I've made up that word)
Yeah, basically :)
It looks like you just had a little bit of a notational flub between the volume of the ball and the ball itself, but that's ok.
I just finished watching. Very nice video.
I'm watching this to derive the pdf of the chi squared derivation for k degrees of freedom haha
that was absolutely amazing, but I have a question:
you assumed that there is a ball in 9999th dimension and what I wonder is: is that possible/is it proven? can you have a ball in N dimensions?
answer: yes or no
(if you want to)
and give either a link of the proof or name of it.
Yes, a ball is defined as the set of points x in R^n such that ||x|| = r for a given number r that is such that sqrt(x1^2 + ... + xn^2) = r, where x = (x1,...,xn), which allows us to define balls in n dimensions!
Actually, that's the definition of an (n-1)-sphere, which is the boundary of an n-ball.
Also, it should technically be defined in Eⁿ (Euclidean n-space), which is Rⁿ with the usual (Euclidean) metric attached to it.
The n-ball would be the set of points for which ||x|| = √(∑xᵢ²) ≤ r
.
Dr. Peyam's Show oh wow, that was simple! thanks!
Is it even called volume in higher dimension?
Hypervolume, I guess :)
Or "capacity"? Or "n-capacity"?
isn't gamma(n/2+1)=(n/2)! ??
so if n is even... just do the factorial
It is interesting that this goes to zero, for large N.
True!
I need volume of sphere model video
There already is one!
the B A A L L L
Yo no lo entiendos
Yo tampoco lo entiendos
Volveré prontos, cuando lo entiendas
Explica si lo entiendes
Oquei, vuelva prontos
What is the average age of your viewers?
Great question! I’d say around 24-ish? It ranges from 10 to 70 years roughly!
Nice. Keep develop the math.
This is very easy problem by the way, why it is consider as a hard one? Is it because of the higher math involved or?
Sorry for my forward writing.
Too long. I can calculate the volume of the N-sphere using only the Gamma function and the volume integral. In just 1 minute.
Math is messy work.
Hey, there you use ball and not sphere 😆 (You used "circle" for "disc" and replied you always use the former, for consistency, you should say sphere 🙃)
Nah, ball and disk are shorter to pronounce
@@drpeyam That makes no sense 🤣
Length of an interval cannot be a volume, because lines don't have volume - they're infinitely thin threads.
Same goes with discs in 2D: they're flat, so they don't have volume - they're infinitely thin sheets.
What we _can_ do, though, is to multiply the length of the interval (which is a number) by a unit of volume (e.g. a unit cube 1×1×1) and _then_ we have a volume. But this has to be explicitly said, otherwise it's not very rigorous and it can be misleading. It's the most common mistake being made by calculus people, because they are so used to "summing up infinitely thin rectangles to get the area under the curve" (another absurdity).
Another way out is, instead of calling it a "volume", call it an "interior" of a shape (i.e. something contained inside some boundary), as a generalization in which volumes, areas and lengths are special cases.
Here by volume I mean the Lebesgue measure or generalized volume, which in 1d is length, in 2d is area, and 3d is volume. It’s very common in measure theory just to call it volume :)