The hardest question on the hardest calc 3 test
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- เผยแพร่เมื่อ 29 ม.ค. 2020
- Are you ready for the hardest question on the hardest multivariable calculus exam? This is the exam question that I gave both in my vector calculus course in Fall 2018 and my Partial Differential Equations course in Fall 2019, and among the 170 students who took that exam, only 1 person solved it correctly! But the question is: Can you??? Watch this video and get ready for the challenge!
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Divergence theorem and Stokes theorem are the heart of multivariable calculus.
it's all Stokes to me
@@Czeckie
My courses in Engineering Physics always seem to use the Divergence Theorem - I have seen it in Partial Differential Equations, Continuum Mechanics and Electromagnetic Field Theory.
@@Peter_1986 what I wanted to say is that Divergence theorem (and Green's theorem too) are just special cases of the general Stokes' theorem on integration of differential forms
@@Czeckie
I actually didn't know that, that's cool.
I did study this stuff a few years ago, but I had forgotten about that relationship.
Gauss, Green and Stokes all are important
It is not a that bad question. When you see div, just triple integrate it over some closed solid. When you see curl, just do surface integral. All they need is some experience with problem-solving technique. Maybe it would be more appropriate to put it in an analysis class
Can we get videos on tensor calculus.
Yes I would love that as well
@@thedoublehelix5661 Trying to understand the lie derivative.
No, I won't even try ... I can see it's a no from what knowledge of Maths I posses. I will be happy for you to explain it to me, as always, Dr Peyam !!! (Great name, by the way !!!)
You don't even need it to be a ball. The same proof works for any closed surface. The value of the field inside a given volume is always some kind of average of its value at the boundary.
Somebody said 0,1 and infinity are at the heart of functions
Damn, I do not want to prove that.
@@Alians0108 the proof in the video works for arbitrary connected regions with the same conditions at the boundary. Of course lots of details are missing, because you have to justify using the divergence theorem. For example the rehion might have a fractal-like boundary, in which case you can still require that f and grad f be 0 on the boundary, but the divergence theorem can't br used because you can't integrate on the surface.
I lost it at vector field
Hahaha
Students with background in Laplace equations or electrostatics should not find much difficulty in this question. You can invite your students to study electrostatics ha ha
Well, as an electrical engineering student I confirm this
It makes intuitive sense when we think about this in terms of physics. Specificaly the heat equation. The Laplace operator of the temperature (in this Peyam's case the scalar function f) is a measure of how much the temperature at a given point differs from its neigbouring points. This is then proportional to the rate at which the temperature changes in time (in Peyam's case zero on the right-hand side of the equation). Zero rate of change implies constant temperature and if we require continuity f has to be zero everywhere.
Similarly with Gauss' law in electrostatics... Electric field can be written as gradient of the electric potential which is proportional to the charge density (zero). But zero charge density requires constant potential which turns out to be zero when we account for the boundary.
I mean, Gauss law is quite literally the divergence theorem.
It only looks hard when you dont know how.
@Sad Face - I guess everything is, really.
Thank you so much for this hard work! I've been following You, sir and blackpenredpen since the beginning and you showed to me how beautiful math is and now I'm the 3rd year in math uni. Going to be math bacc soon this year, hopefully. 😁 I just wanted to thank you for all the effort which is motivating so many people to study math. Have a great year, sir!
Thanks so much!!!
Please, keep making more and more videos on pds because it is the most usefull tools for physics. Best redards
thank you dr. peyam🥰
"……but maybe you can solve it."
Oh?
"……so first of all, if you have a vector field F …"
I can't solve it.
Me: “Maybe I can manage this question, I did pretty well in multivariable calc last semester”
Also me: doesn’t remember what the divergence operator does
I remember doing this question in my calc 3 class. It was super fun hahah
It's a tricky one, butI'm really happy I got it! But as a physist I can give another explanation without the use of the first part of the question. I would be happy if you tell me how true is this approach.
We start by choosing an arbitrary closed volume of space inside the ball; let's call it V. Then we integrate the laplacian of f on V; let's call the result of this integral I. Because lapl(f) =div(grad(f)), and by virtue of the divergence theorem, we have that the surface integral over the boundry of V of grad(f) is equal to I. But I=0, because the integrand is the laplacian of f, which is 0 everywhere in the ball and also V is in the ball. Then the surface integral over the boundry of V of the gradient of f is 0 too. But because V is arbitrary, then the boundry of V can be any closed surface in the ball.
(and here is the part I want to have verified)
If the surface of integration is arbitrary, and the value of the integral is 0, then the integrand is identically equal to 0.
(here the part, which I want to have verified, ends)
From this follows, that grad(f)=0 everywhere inside the ball => f=const.(inside the ball).
So what do you think? I'll be happy to read your response! Thank you in advance!
And, as always, continue to do this amazing videos! I've always liked them and I always learn something from them!
We did this in my PDE class. If I were to be in my Multivariable class, I would have no clue too lol.
Calculus 3 is awesome.
There is something hugely satisfying about extending things from ℝ² to ℝⁿ and drawing all those pretty 3D graphs.
Totally agree!
I got it!
- Apply the already proven fact to show the divergence of f times the gradient of f is just the abolute value of the gradient of f, squared (because of Laplace's equation within B)
- Use Gauss' divergence theorem to integrate the expression and notice the surface integral is zero, due to f being zero at the boundary dB.
- Now we got a volume integral with an at least zero integrand, that is zero. Thus the integrand is zero and so is the gradient of f.
- f doesn't change so it's a constant, but we know f=0 at the boundary, thus the only possibility is that f is identically equal to 0.
QED :)
Nice!
Geez that second part is super super tricky, there are a lot of steps that are not exactly obvious😅
Greetings Dr. Can we get videos on the Variational methods as well? That would be great.
Already done, search for calculus of variations
6:10 I immediately know that the trick is to use Green’s Formula. Why? Because I got stumped by *THIS EXACT QUESTION* two years ago in a calculus exam, and I thought about it for a while before figuring it out. After that, I never forgot the way to do it, and I never will. The only difference is that my professor wasn’t kind enough to give us a hint with part a; he went straight to part b!
I thought about attending UC Berkeley just to take your math course
I will recheck this video after i have taken calc 3 exam😂✋🏃🏃🏃🏃🏃
I couldn’t even tell when he stopped explaining the question and started explaining the answer.
How do you write so good I’m left handed as well
If the laplacian of f is zero on a ball there will be a complex function of a complex variabel with real part f and the function is analytical and an analytical function max will be reached at its boundry and its zero so the function is zero so use this theorem on expo(i*G) (G is the complex function with real part f)
Dude, I missed it. Like, should I rewatch?
With Dr Peyam even the hardest question becomes easy.
If you use the maximum principle for harmonic functions the question is trivial. A harmonic function (that is C^2 in the region and C^0 on the boundary) attains its max and min on the boundary, but the max and min for f are both 0 so we get 0
But it's harmonic... which means it should satisfy the maximum modulus principle right?
first, love your channel
What’s your PhD in and how long did it take you to get it?? Just curious cus you seem really young.
Love your vids by the way
Peyam's 100th video special! th-cam.com/video/HvLScjQlCV0/w-d-xo.html
Wow, thank you 🥰 Took me 6 years!
Dr Peyam awesome! Really appreciate this as a chemical engineering student
ok now find the this-ball'th derivative of x
The random inflections of your voice terrifies me but your math question intrigues me.
haji ghablesh ye bar tamrin koni bad nista! :)))I know you're super smart
Very nice!
It's more difficult but not impossible to prove without using vector analysis concepts. For a start, you could separate the variables x, y and z, (or more usefully rho, theta and phi) and solve the 3 boundary conditions. I remember doing that in Cartesian coordinates on a rectangular domain and ending up with sums of sin(ax)*sinh(by) which forces b to be always 0. That was a more basic PDE approach.
Anyway, great video as always!
Love this video! I probably wouldn't have got it back when I took Calc 3 either lol. But now, I know that solutions to the Laplace equation have their extremal points on the boundary of their domain, so if f=0 on S, f has to be zero in the ball, and you can use the divergence theorem to show this.
I wish I could be able to take multivariable calculus
maximum principle?
Dr Peyam can you find the antiderivative of
(e^x)(ln x) because i can't find it anywhere.
It's (e^x)(ln(x)) - Ei(x) + C
Where Ei(x) is the integral with respect to t of (e^t)/t from 0 to x.
Try it out using integration by parts.
@@joelsagflaatholmberg3922 e^t/t is seperable you take the LN of it t - ln(t) the intergral ln(t) being ln(t)t - 1 exponentializate it again.
@@sadface7457 That doesn't work. You can't drag the ln inside the integral, or else the integral of x would be e^(x ln x - x)
@@joelsagflaatholmberg3922 thanks for the reply!
Multivariable calculus is great!!
By the way, 02.02.2020 can be read back and forth. How many dates can be written this way?
Any dates of the form ab.cd.dcba with the appropriate restrictions on cd (can only go from 01 to 12) and ab (can only go from 01 to 31 depending on cd)
In other words:
cd = 01 gives you ab.01.10ba
Where ab goes from 01 to 31
cd = 02 gives you ab.02.20ba
Where ab goes from 01 to 28
(29.02.2029 is not a leap year)
Etc, and finally up to cd = 12
So I think there are 365 possibilities
Which makes sense, Hahahaha, take any date of the year, like 31.03 and reverse it to get 31.03.3013
@@drpeyam Nice answer. Why don´t you make a video with dates? Like these: jasonmccreary.me/articles/frequency-of-date-equations/ or www.smithsonianmag.com/science-nature/last-sequential-date-century-121314-fun-math-lovers-180953413/
Why not convert the equation to spherical coordinates and then separate variables? Although in 3 dimensions it might be a bit tedious
It is much nicer to keep everything in vector form rather than splitting it into coordinates
@@henryginn7490 well sure my post was more of a question regarding the validity of the proposed solution (because I'm bored to try it)
I got it right! 😃
Hey Dr. Peyam, I like your solution a lot, but I thought you might appreciate this alternative solution in sentences :) So assume there is a nonzero solution f. Then over the entire interior of the region, there is either a point where f is maximum or minimum. WLOG consider a point r=(x0,y0,z0) where f is minimum. It is in particular a local minimum. Consider a ball centered at r with radius epsilon, arbitrarily small. By definition of local minimum, grad(f) points outward everywhere on this ball. If we take the limit as this ball's radius goes to zero, we see that the div(grad(f)) must be strictly positive at r, a contradiction.
Btw, you should try asking a classroom of physics grad students. It's a common problem in E&M ;)
Hey Peyam. Probably you know this already since you're a math guy but deriving identities related to div grad and curl becomes much easier if you use Levi Civitas and kronekar deltas, incorporated with Einstein sum. The writing becomes more succinct that way.
Ewwww, I can’t stand those
Figures. Well, that was expected of a mathematician so disappointed but not surprised :3
I’d be able to do that. This is just the uniqueness theorem for the boundary value problem for the Poisson’s equation.
Wow Ohannes you must be really smart.
Could the solution be given by invoking the mean value property of harmonic functions ?
Why can’t it be discontinuously non-zero at some point and still solve both equations?
Well this proof is actually harder than most of vector calculus (if I remember correctly almost all proofs there are pretty straightforward)
Hm , I just made the decision to go back to school at 27 for biomedical engineering. Seems like I going to be stretched beyond belief, but I assume it will be worth it .
Hey peyam ur pretty smart
there are other ways to proof that?
Maximum principle!
but this class was for engineers? or for math and physics students?
A mix of engineers, math, and physics students
1:32 The most difficult part of this problem is to state it.
This looks like Problem A4 in 2019 Putnam exam
Why should the length of the gradient squared be zero outside of the ball?
I never said that! All I claimed is that u = 0 inside the ball
@@drpeyam So, you don't care and have not been talking about f outside the ball?
Exactly
OK I get it being hard for cal 3, but I'd really exepct pde class to be able to apply such a standard energy argument
Where do you teach?
UC Irvine
@@drpeyam my dad went there
Woooow, what a coincidence!!!
In Calc 3, no way. Having taking Electricity and Magnetism 1 and 2 now, easy! I used this exact argument involving surface current density for a proof I was assigned.
That was trivial 😅
What will be the intregation of cosx/x?
Its non-elementary, however the cosine integral (which is definied as the negative of the integral from x to infinity of cos(t)/t) does the job. Theres some pretty interesting stuff with that special function, I'd reccomend looking it up if you're interested. Hope that helps!
Me looking at this while I’m in middle school like 👁👄👁
tfw u thought it was clickbait but it's actually a hard question
arxiv.org/abs/1806.02231 the paper for the complex fib series that has been requested.
Why is capital F the gradient of little f ?
Convention, f stands for functions and F for vector fields
OUCH!
When you don't know M of University mathematics but you're still watching it
My mind: “Maximum principle f=0
So f=0. Yes. This is one of those. with no calculations required! “
Dr. Peyam: ...with multivariable calculus...
Me: oh shit here we go again
I transformed div(f*div(f)) into Δ(f^2)/2 and got dead lost
Actually, if U are a physics student it's pretty obvious. Love
I live in usa student learn fractions at University
How many points for that?
15 out of 100 :)
@@drpeyam Did you write WOAH when marking the exam of the student that solved it?
I did 😂😂😂
@@drpeyam awesome ... I hope he or she passed. Would be a real bummer if he/she solved this problem as the only one, and still failed 😂
f must be equal to zero...it must be
Not 100% sure about this, but you should really write down the domains of your functions. Because if i recall correctly, this equation does have nontrivial solutions, if f is only defined on the ball. But you probably mean, that f is defined on the entire R^3 right?
Nevermind, guess i was wrong. (well you should still mention, that by "everywhere" you mean the ball.^^) I thought these Eigenvalue problems for the Laplacian always had a (nontrivial) solution on bounded regions...
This was supposed to be hard?
I don't think this would be terribly hard. Like, I'm in high school and I understand a good portion of where everything comes from I just don't know what any of it means.
Smart people in the comments, if you wanna help me be a pro at calc 3 before I'm even in college hit me up lol
A is trivial and you dont need it for B, claim: just like picard theorem laplace equation answer is one and only (very easy to prove) you can acatually geuss 0 and be right. Other wise gauss law for electrisity should be finishing this. As well As another easy lema that is the div of the inside equals the flux on the surface. If the flux on an erea is 0 the field is const than cause of the 0 of the ball it is sero every where. This is a shame on you that only one of your pde class didnt solve it, this is first weak question and the easiest of them in my pde class.
For someone claiming to teach a pde class, you have horrific spelling and grammar.
What even was the question? Did I miss it because I'm that stupid?
The question was to prove the divergence fF formula, and then to prove the statement that u = 0
You lost me at (a)...
what I'm learning from this video: don't take a test from Peyam ! lol jk love the videos