Stokes' Theorem | MIT 18.02SC Multivariable Calculus, Fall 2010

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MIT OpenCourseWare

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@Juxtaroberto
@Juxtaroberto 11 жыл бұрын
It's the factor you have to introduce when switching to spherical coordinates. Technically it should be ro^2*sin phi dro*dphi*dtheta, but since ro=1 in this case, he was able to omit it. Whenever you switch from one kind of coordinate system, such as from rectangular to polar, or cylindrical, or spherical, or your own made up system, you need to introduce multipliers. Sometimes they're just constants, but sometimes they also involve variables or functions.
@stefanossalas3333
@stefanossalas3333 7 ай бұрын
That’s wrong
@stefanossalas3333
@stefanossalas3333 7 ай бұрын
Stop spreading misinformation
@stefanossalas3333
@stefanossalas3333 7 ай бұрын
It is “total nonsense”
@aldenrj
@aldenrj 7 ай бұрын
Odeeeee
@mmmeliih
@mmmeliih 13 жыл бұрын
I really love the scene when the guy went and then came back :D
@MrMathExpert
@MrMathExpert 10 жыл бұрын
apparently, it takes 10 sec or less for students at MIT to solve the problem at 2:00.
@Corvanor
@Corvanor 7 жыл бұрын
Lmao
@rodrigosimoes185
@rodrigosimoes185 7 жыл бұрын
its quite simple
@Falcon4610
@Falcon4610 12 жыл бұрын
I love how this guy does intergration and the cross product like its simple arithmetic
@SP-qi8ur
@SP-qi8ur 4 жыл бұрын
How is your life after 7 years
@RKatz121
@RKatz121 10 жыл бұрын
im pretty sure you are allowed to change the surface when you are using stokes theorem as long as you keep the same boundary. if you had changed it to a disc at the base of the surface, the normal becomes and the integral is just the area of the unit disc which is pi(1)^2 which is equals to pi
@tomctutor
@tomctutor 7 жыл бұрын
he's not working out a area, there's a vector field there? and pi/2 =/= pi
@arnurmakenov4350
@arnurmakenov4350 7 жыл бұрын
by Stokes' theorem you can take any other (as long as requirements for the theorem are met) surface which has the same boundary as the original one. When taking that "another" surface as a unit disk centered at (0,0,0), the surface integral happens to be the area of that disk. pi * radius^2 = pi
@marcusrosales3344
@marcusrosales3344 6 жыл бұрын
@@tomctutor Yes, but after taking the dot product, you get 1, so all that is left is the integral and dS. This is the area. You also miss read their response on the last part... He put pi; the 2 is an exponent.
@zino-kader
@zino-kader 5 жыл бұрын
This is correct and it's how I prefer to solve these since it involves less complicated parametrizations so the risk of screwing up isn't as high
@yaymynameispete
@yaymynameispete 13 жыл бұрын
@MrVandeju We have that z=0 and dz=0. So, in his calculation, he does 2z*dx + x*dy + y*dz. Thus we are left with x*dy.
@robertcummings3935
@robertcummings3935 9 жыл бұрын
Uh uh, he be skipping steps, I'm not advanced enough for those. Where PatrickJMT at lol
@sfdko3291
@sfdko3291 7 жыл бұрын
facts
@xoppa09
@xoppa09 7 жыл бұрын
What step did he skip? I would be happy to fill in the details, if you have a specific question.
@Boimeirelles
@Boimeirelles 7 жыл бұрын
when he does curl F he doesnt show how he did it haha and that is hard lol
@thechuggs8997
@thechuggs8997 4 жыл бұрын
spring1 could you explain the limits for fi? I don’t get why is it in between 0 and pi/2
@Danni4096
@Danni4096 4 жыл бұрын
Ian Tan we want to calculate upper half of sphere, so z>0. When he parametrized z=rcos(fi) you can ask your self: for which values cos is greater then 0? When fi goes from 0 to pi/2.
@gummybear92cjc
@gummybear92cjc 8 жыл бұрын
4:51 "You have to remember way back to 1801" Yep yep, it definitely feels like it's been 215 years since single variable calc. Haha
@LiamPattersonPlus
@LiamPattersonPlus 8 жыл бұрын
when he actually means MIT course 18.01 LOL
@Shyzah
@Shyzah 6 жыл бұрын
Liam Patterson pretty sure he was joking
@NedStarkZA
@NedStarkZA 3 жыл бұрын
5:51 "So that was the Line Integral, very straightforward thing!".... I laughed so hard!
@sinisasladojevic6419
@sinisasladojevic6419 8 жыл бұрын
briliant insructor
@MrGiratina999
@MrGiratina999 9 жыл бұрын
why parameterise at 11:40?? at that point, he has ∫∫z dS BUT that = ∫∫z dA/ n.k = ∫∫z dA/z because n.k = z, so ∫∫z dA and then go ∫∫ r dr dtheta and that gives the answer in 2 more lines.
@Pomme843
@Pomme843 8 жыл бұрын
+MrGiratina999 What's n.k?
@MrGiratina999
@MrGiratina999 8 жыл бұрын
n.k = normal vector dotted with the unit vector in the k or z direction.
@sanjaykrish8719
@sanjaykrish8719 8 жыл бұрын
love his voice..
@willdoesmusic8402
@willdoesmusic8402 3 жыл бұрын
I've been confused with Stokes' Theorem for the past few days, and you just made it clear to me. Thank you!
@aangthejeweller
@aangthejeweller 8 жыл бұрын
Who's man is this? He is fantastic! Thank you sir, for blessing me with your knowledge.
@biplabghosh4219
@biplabghosh4219 7 жыл бұрын
if z=o then fig would be 2 dimensional and limit willl be 0-π/2 and 0-2π results whole circle.
@stumbling
@stumbling 6 жыл бұрын
13:26 "Let me just check I'm not doing anything silly..." *fades to black*
@068LAICEPS
@068LAICEPS 4 жыл бұрын
So if we have had a complete sphere there would not be a line integral to do because we would have need to be equal to zero? And it doesn't matter the height of the upper part of the surface because we the same contour we will need to obtain the same line integral and the same result?
@hourtbora3072
@hourtbora3072 9 жыл бұрын
what don't you use x=rcost and y=rsint? i like your style of showing this section. it's so cool--fast and clear.
@zuzusuperfly8363
@zuzusuperfly8363 9 жыл бұрын
Hourt Bora He did.
@Mumfin
@Mumfin 9 жыл бұрын
Hourt Bora r is 1.
@hourtbora3072
@hourtbora3072 9 жыл бұрын
ok i see. thanks guys!
@yonatanable
@yonatanable 13 жыл бұрын
Oh. God this theorem is so interesting, verifying stoke theorem gives some basic ideas that what Stokes Theorem is all about.
@ElTurbandito
@ElTurbandito 6 жыл бұрын
@4:57 why does F*dr = xdy, there is no explanation
@tsshamoo2376
@tsshamoo2376 6 жыл бұрын
Sorry if I'm too late but since the path is entirely contained within the xy plane, z=0. This also means dz=0 since the derivative of 0 is still 0. So F= and dr=. F•dr=0*dx+x*dy+y*0=xdy Hope that helps and again, sorry if im too late to help you for your final or something like that
@christineclement8497
@christineclement8497 6 жыл бұрын
Tssha MoO2 k
@lugia8888
@lugia8888 Жыл бұрын
There is explanation you just don’t pay attention.
@lugia8888
@lugia8888 Жыл бұрын
@@tsshamoo2376 dont be sorry tsshamoo its explained in the video. these people should know better - instead of paying attention they just want the answer spoon fed to them. tik tok brain
@riccardofasano1040
@riccardofasano1040 2 жыл бұрын
@8:18 i see how you get your normal vector by using special simmetries of the sphere, but I was looking for a more general way to find a normal vector for a surface. In my mind n is something orthogonal to the tangent space, so i should probably do the computations to find a vector which gives 0 when multiplied for a vector tangent to the surface. Nonetheless i find this process a bit difficult in higher dimensions, because i can't visualize a tangent vector fast enough. Anyway, very nice video :))))
@ashutoshpal3145
@ashutoshpal3145 7 жыл бұрын
at 13:20 , would there be limit of z from -pi/2 to pi/2
@juanholguin8783
@juanholguin8783 9 жыл бұрын
eurekkkkaaaaa!!!!!!!!! I have found a video with no racist comments conspiracy theories, blames jews for everything, atheist comments, et., calculus, you have gain a new friend for eternity.
@davidjohnson-my6sr
@davidjohnson-my6sr 7 жыл бұрын
Is he missing another surface? namely, the surface at the bottom of the hemisphere, which is the circle on the xy plane. we would have to compute the surface integral of this too right (of course, this will equal to zero)?
@tsshamoo2376
@tsshamoo2376 6 жыл бұрын
Sorry if Im too late with this reply, but no, the bottom would not equal 0. If you do it correctly, it should be -π since now the divergence theorem applies and the div(curl(F))=0. For the Kelvin-Stokes theorem, the surface doesnt have to be closed. It just has to be a piecewise-smooth surface that fills the region made by the curve
@musazephania9194
@musazephania9194 6 жыл бұрын
Very nice explanation about the verification of stoke theorem
@davidjohnson-my6sr
@davidjohnson-my6sr 7 жыл бұрын
does anyone know why dS was equal to that? surely we dont multiply by the jacobian since were not transforming the region of integration... simply just parametrizing..
@keithdow8327
@keithdow8327 4 жыл бұрын
He made the F dot dr integral too difficult. The integral from 0 to 2 pi is the same for cosine squared and sin squared. add in sin squared and then divide by two. cos squared plus sin squared = one. THe integral from 0 to 2 pi divided by two is pi.
@mathstoinfinityclassinTamizhla
@mathstoinfinityclassinTamizhla 5 жыл бұрын
I couldnot understand 13.26 how do you got the ds value ???can anyone help
@obsidiantechz1371
@obsidiantechz1371 4 жыл бұрын
how to verify for not unit sphere
@albertmendoza8330
@albertmendoza8330 5 жыл бұрын
This guy is awesome!
@snadbad
@snadbad 12 жыл бұрын
Equals Sign: 10/10 would watch again
@FarazJananKhattak
@FarazJananKhattak 12 жыл бұрын
Most of us on the youtube do are not from the class, if you could go a bit in the background that would be great; instead of testing it right away
@alexwilson8034
@alexwilson8034 3 жыл бұрын
Why is MIT uploading 360p videos...
@abeimnida
@abeimnida Жыл бұрын
How are we left with x dy? I dont get it whatt?
@lugia8888
@lugia8888 Жыл бұрын
He explains it. You can replay the video multiple times too it’s not hard to miss. Pay attention.
@MegumiTheGreat
@MegumiTheGreat Жыл бұрын
I'm sitting here watching cat videos all of a sudden KZbin's like you know what here calculus.
@XweienX
@XweienX 8 жыл бұрын
Just a heads up, this is not for calculus beginners. You'd be lost when he advances without writing down steps.
@MazwiKhoza
@MazwiKhoza 11 жыл бұрын
why is dS equals to that? anyone knows?
@mithunmahato309
@mithunmahato309 7 жыл бұрын
Mfanafuthi Khoza this is the same confusion I face...
@davidjohnson-my6sr
@davidjohnson-my6sr 7 жыл бұрын
Mfanafuthi Khoza does anyone know the answer to this? i thought d (boldface S) = n dS = n * |N| * dphi dtheta ?!?!
@bilalakram3521
@bilalakram3521 4 жыл бұрын
Where did come ds from?
@iamthesak
@iamthesak 12 жыл бұрын
God damn I LOVE THIS GUY I WANT TO TAKE ALL HIS CLASSES.
@dustinengel2315
@dustinengel2315 9 жыл бұрын
when he went to spherical coordinates, he made the jacobian sin(phi) only. is this because even though the jacobian should be rho^2 sin(phi), rho is 1 and 1^2 is 1 so he just left it off?
@aleespazx
@aleespazx 8 жыл бұрын
yep
@codyfan4070
@codyfan4070 7 жыл бұрын
It's a unit sphere so ro is just 1
@BambiChub
@BambiChub 9 жыл бұрын
Can someone be kind enough to explain to me why, upon parametrization, x=cos θsin φ etc.?
@AvenueFifth
@AvenueFifth 9 жыл бұрын
Assata Amaechi Osborne Those are the basic spherical coordinates. Check wikipedia about it
@MaxG628
@MaxG628 9 жыл бұрын
+Assata Amaechi Osborne For a given φ, you can think of sin φ = r, where r is the radius of the horizontal circular cross section of the sphere, at height z = cos φ. Now the parameterizations of x and y are exactly the same as for the line integral.
@sshannon1948
@sshannon1948 6 жыл бұрын
is dS interchangeable with dA?
@massexploder
@massexploder 12 жыл бұрын
I love this guy!
@KumarHemjeet
@KumarHemjeet 7 жыл бұрын
why is z zero ?
@Stayawayfrommyname
@Stayawayfrommyname 7 жыл бұрын
the circle is on the z=0 plane
@xoppa09
@xoppa09 7 жыл бұрын
Excellent teacher. Please make complex analysis videos.
@orhanaziz5477
@orhanaziz5477 8 жыл бұрын
You deserve subscription.
@FarazJananKhattak
@FarazJananKhattak 12 жыл бұрын
Love you guys!! Thanks
@adrianamishell5309
@adrianamishell5309 3 жыл бұрын
the best teacher!
@emir2591
@emir2591 4 жыл бұрын
yo isn’t ds rho^2 sin(phi) not sin(phi)?!!!???!!?!!??!!?!??!!?!?!?
@lugia8888
@lugia8888 Жыл бұрын
rho is 1 stupid
@yaohuadong1271
@yaohuadong1271 5 жыл бұрын
Thank you very much :)
@57kikky
@57kikky 13 жыл бұрын
thank you
@aamirgeelani1814
@aamirgeelani1814 4 жыл бұрын
I watched this at 2X playback speed. :-D
@banana144
@banana144 7 жыл бұрын
curl(F) is wrong, it should be not ; although you get the same result anyway
@Stayawayfrommyname
@Stayawayfrommyname 7 жыл бұрын
nope, it's correct
@banana144
@banana144 7 жыл бұрын
wow, i was wrong! Thank you very much (:
@annanyatyagip
@annanyatyagip 5 жыл бұрын
thanks!!
@sandun001
@sandun001 12 жыл бұрын
Thank you sir, you made my life easier !
@rahulgouni9686
@rahulgouni9686 8 жыл бұрын
very nice!
@dracodiemnuntiat
@dracodiemnuntiat 10 жыл бұрын
He forgot to mention the radius when parametrizing, if the radius was 2 this process would be slightly wrong.
@Tschaegger79
@Tschaegger79 10 жыл бұрын
No, because it's the unit sphere, so radius is 1. So you can ignore it in this case.
@dracodiemnuntiat
@dracodiemnuntiat 10 жыл бұрын
Tschaegger79 In this case, but it would be better to have an all encompassing explanation than to just ignore it.
@bugsbunny278
@bugsbunny278 6 жыл бұрын
That is what I was wondering
@6thHorseMan
@6thHorseMan 11 жыл бұрын
My three year old son informed it is actually a train track.
@alokchaudhari28
@alokchaudhari28 13 жыл бұрын
very helpful keep the good work up
@aslamkmns91
@aslamkmns91 13 жыл бұрын
4 hours lecture = 17 minutes video. =='
@paulg444
@paulg444 6 жыл бұрын
great video but he rushed through the dot product way too fast.
@lugia8888
@lugia8888 Жыл бұрын
Its dot product… you want him to help you how to add too? Dumb criticism
@vd853
@vd853 13 жыл бұрын
14:00 nice equal sign lol
@deadcoww
@deadcoww 12 жыл бұрын
You're suppose to know that already..
@PikelsMH3
@PikelsMH3 9 жыл бұрын
Thanks man.
@mathematicsbyahsanmohsan8849
@mathematicsbyahsanmohsan8849 8 жыл бұрын
sir how i can download this lecture
@jlsmatejuanluisramirez
@jlsmatejuanluisramirez 6 жыл бұрын
I love your explanations!!!
@Algebrainiac
@Algebrainiac 2 жыл бұрын
This whole time I thought it was “strokes’ theorem” and not “stokes’ theorem” 🤦🏻‍♂️🤦🏻‍♂️🤦🏻‍♂️ Also… my brain 💀💀💀💀
@vblackkpr
@vblackkpr 12 жыл бұрын
That is the most Awkward Silence of ALL TIME 2:00 XDDDDDDD Nice Explanation !! Loved it
@Cody_Pulliam
@Cody_Pulliam 8 жыл бұрын
quit watching once he left out the rho's in parametrization
@sasquatchjim3198
@sasquatchjim3198 8 жыл бұрын
+Cody Pulliam Hey genius, he was integrating over the unit sphere where rho would be one. Also when using Stoke's theorem, at least in Calc 3, you can only use up to two substitution variables.
@HT-rq5pi
@HT-rq5pi 8 жыл бұрын
+Cody Pulliam fail.
@jackbryant6087
@jackbryant6087 8 жыл бұрын
Well maybe if you actually gave it 2 seconds of thought you'd understand why he "left out the rho's".... Good luck in your finals.....
@carleto-y8q
@carleto-y8q 10 ай бұрын
M-I-T-K-E-Y M-O-U-S-E
@sarahromero8090
@sarahromero8090 7 жыл бұрын
mind blown
@UchihaKat
@UchihaKat 12 жыл бұрын
Isn't it? ;-P
@poperman18
@poperman18 8 жыл бұрын
he computed the curl wrong, its , not
@kimcheelegacy
@kimcheelegacy 10 жыл бұрын
oh my, thank you so much. This is well explained and helped me study better for my upcoming exam.
@Rayquesto
@Rayquesto 8 жыл бұрын
7:26 Just write out the determinant man!!!
@drewpierpont3361
@drewpierpont3361 6 жыл бұрын
Well, thanks for helping, and also thanks for reminding me I am UT-Dallas material and not MIT.
@maheshtom9003
@maheshtom9003 7 жыл бұрын
sounds like the joker
@carleto-y8q
@carleto-y8q 10 ай бұрын
Horse shit, a length is not equal to a surface area. Why do you omit the units of the integrals?
@mr.anirbangoswami
@mr.anirbangoswami 7 жыл бұрын
it's not pheeeeeeeee, it's phi -_-
@thecornydocc
@thecornydocc 7 жыл бұрын
Thank you
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