Stokes's Theorem

Stokes's Theorem is kind of like Green's Theorem, whereby we can evaluate some multiple integral rather than a tricky line integral. This works for some surface integrals too. Let's see how it works!
Script by Howard Whittle
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[@slimegolem]

Solution to the problem: r(x) x r(y) = < -2x,-2y,1> del x F = , (z = x^2 + y^2 --> -z = -(x^2 + y^2) = -x^2 - y^2) Integral: 0

[@meltheodelacruz8652]

Can you tell me what are the r(x) and r(y)? I cannot figure it out why r(x) × r(y) = Thank you so much!

[@slimegolem]

It has been a while since I've done this and fully understood this subject. But I think r(x) = and r(y) = . Because r is given by the plane x^2 + y^2 = z. So r = . The derivative with respect to both x and y will result in r(x) and r(y). I do not know if this is 100% correct though.

[@jack6914]

yeaaaaaaaaah if u see me i am clapping with my lips

[@quamos99]

shouldnt the limit for y be 0 to sqrt(4-x^2)?

[@talipgunes7022]

@@quamos99 Since y is already in our integral we can take upper limit as 2. No more additional expressions are required for this one

[@frentz7]

Hey Professor Dave. first time viewer, was just passing through. I used to teach at University. good video! impressed by two things #1 your overall presentation has a strong "tightness." no unnecessary audio or visual; the student can focus. #2 you gave the students an exercise! bravo [ claps ] .. so rare that people remember that they are talking to a human being, and that there may actually be a learning outcome LOL

[@fuge511]

Soooo who else has a calc 3/4 final???? lol

[@bonnet-bandit]

Me 😭🙋🏾‍♀️

[@danielpipersburgh1740]

calc 4 😢😢😢

[@graoliveira6688]

I come f from the future to say Me! 🙋🏽‍♀️

[@Beckenbauer9999]

Me

[@starrsk6690]

3 years late, but meee🔥

[@ph3043]

i’m cooked i fear

[@brylinsundae7464]

Same bro

[@AzAzMusic]

Lock in

[@ronbhattacharya3183]

Unbelievably so

[@zunzwak4482]

how was the exam bruv?

[@sooryaprakash6390]

Tomorrow I have a physics exam .thanks for uploading this dave. I am learning physics, biology, chemistry , maths from you. All your videos are short and provides knowledge with clarity. Thanks for everything man your videos mean a lot to me.

[@AGhostyProduction]

bringing clarity to those in the midst of uncertainty and confusion is so powerful. thank you sir don't stop your passion

[@jahjahjah213]

when someone finishes your video and thinks "it's just that?", it means you're a good teacher.

[@rsc8295]

exactly, Stokes and Divergence theorem was like the Tripitaka written in russian and now after watching these videos they're so easy to remember! 😂

[@yashovardhandubey5252]

Thanks a lot... I used to hate organic chemistry.... But because of you I'm starting to get better.... Professor dave, you the real MVP!!

[@shahinabeevis5853]

Thank you Dave.. I have no idea about my mathematics exam.But this give excellent study material.

[@farhanalam7642]

Professor Dave, you are probably the best youtuber at explaining concepts logically. Everytime I come to this channel to learn something, I learn it well. Thanks a lot

[@sergeiivanov5739]

What remains identical or similar in the Green, Gauss, and Stokes theorems is the logic which can be seen in the generalized Stokes theorem.

[@johnszwanke7175]

Very clear and easy to follow explanations. Thank you!

[@gavin0703]

Shout out to all the homies who are reviewing these the night before a Calc final. Goodluck to us all.

[@sanjaychacko7654]

Is it possible to calculate the surface area of a curve using Stoke's Theorem ? ie, Is it possible to calculate the surface area as a line integral?

[@alexlau6559]

If it is negative orientation,then how can I calculate it? Can I just simply reverse contour and gain a minus sign?

[@danika1425]

Excellent explanation, thank you

[@mrmister3507]

Extremely clear video, thanks a lot.

[@HarryJohnson69]

Holy shit, I just watched your discussion with Jesse Lee Peterson. That was hilarious. I had no idea you were also the same guy that helped me pass Calc 3. Thanks so much sir. Please keep continuing your work. You’ve been such great help

[@zinebfaiz2114]

thank you for your amazing explanations !! i m trying to understand the last example: why didn t we write y in function of x for the integral´s boundaries. what is different to the previous example where we put y=1-x

[@Meenameme]

I am eternally grateful for this. Just saved my whole major. 🙏

[@ivowehsely9131]

To be honest you are single handedly saying my calculus grade! Love the videos and the presentation! Keep going!

[@Partyontenz]

THANK YOU I NEEDED THIS

[@tommayor5038]

great video. Thank you heaps!

[@tianlouw8505]

Thanks for the vid on Stoksis Therim!

[@renswieffer2645]

great, short video. Thanks mate

[@luxeternity]

You know it would be awesome if we can see the detail calculation to the problem. Not all of us can suddenly pick it up at lightning speed. Some of us really need to see how it was done

[@winterl8241]

Thank you!!! It's really helpful

[@MikeyMyra]

I have my calcII exam tomorrow, thanks for the last minute help!

[@pradojonardpalamos5985]

May I ask sir just a new to this how did you get the derivative of xy with respect to y? but equals to x? how is that happen if the x is constant? Hope you can answer me sir

[@fatimafadel7810]

thank you professor its really helpful!

[@bendavis2234]

Thanks Dave! I got kind of stuck on that problem at the end, but realized I made a stupid arithmetical mistake that was driving me crazy. Still stoked that I got the process right, even if the I didn't get the exact answer. Hopefully my mental calculator gets better so that I can check both boxes!

[@anotheryoutuberperson38]

Stoked.. haha get it?

[@vatsalparikh633]

Could you make a video on how do you make such interesting videos ? I am interested in knowing how to make the animation parts and such wonderful slides.

[@alishashoukat1682]

I have a little confusion, from where we get the value of F in ∆xF? Thank you!

[@AlbinoJedi]

Thank you. Actually kinda surprised I got the comprehension answer on the first try.

[@lovesonmukombiwa3093]

Thank you proff , thank you once again

[@blazepower7625]

Watching science in another language is fun so much stuff changes

[@arik9842]

so from the comments I've gathered that the boundaries are taken as is in the question (0

[@stephtephrite]

not a cone. z^2 = x^2 + y^2 is a cone. z = x^2 + y^2 is an ellipsoid

[@lavanyagoski464]

It's helpful sir....

[@shahinabeevis5853]

Sir, Can you apload video about abstract algebra?

[@imaginary8168]

I thought that line integrals were easier, why didn't we calculate it instead? Was the long route with that surface integral really necessary?

[@ipodtouch470]

Line integrals can become a pain if it isn't conservative vector field.

[@debparnosarkar2193]

Oh I love your intro music :)

[@atifsultan_mech]

When curlF . (Rx x Ry) is calacualted we cannot simply put the limits of x and y equal 0 to 2. this is because the projection in x-y plane is is circle and we need to use the polar coordinates to sove it. I guess the answer should -16pi

[@WilliamSiffer]

I keep getting 0 for the comprehension problem. Is there somewhere I can see a solution

[@slimegolem]

r(x) x r(y) = < -2x,-2y,1> del x F = , (z = x^2 + y^2 --> -z = -(x^2 + y^2) = -x^2 - y^2) Integral: 0

[@saurabhsingh-ow7ue]

thank you sir....

[@mahmutakdemir9964]

hello.. ... I know little English.there are a few questions.can you help me

[@ShivamVerma-gq2sm]

Hey Dev, just wanted to tell you that your video lessons had been helping me from class 7th or 8th I guess. You explain topics very easily and I get more insights into the topic . I am in my 2nd year of engineering presently and watching the relevant materials here. This video was so good. This entire series is good and its explanations had not been good enough without the supporting videos in the this series. I was looking for the exact lessons. Please continue uploading lessons on various topics as you always do . Thanks alot man! :) Love from India.

[@anabolicchicken4115]

You mean to tell me you're in 8th grade?

[@ShivamVerma-gq2sm]

@@anabolicchicken4115 no , I am in 2nd year of my engineering

[@edidiongmoses2889]

@@ShivamVerma-gq2sm nigga you scared the shit outta me😂😂....i was like what demonic school is doing stokes theorem in 8th grade😂😂. I'm doing stokes right now and i'm in my 3rd year of engineering

[@wongkachun3097]

how do you figure out r= at 3:43??

[@FaranAiki]

basically r = for 2nd dimension and r = for third dimension, since we know what z is we can substitute

[@Monafide3305]

Absolute life saver 🙏

[@lordyabo9910]

Guys where does the x and y in the r = (x,y,1-x-y) came from?

[@malikwasim4291]

How r is calculated? Kindly ans me!

[@chrispirilloiseboola]

doesnt the normal vector need to be a unit normal vector?

[@imgood8519]

What's different have n and no n?

[@sikespiegel9583]

There's actually an easier method to calculate the normal vector to the phi--> x+y+z -1 plane. Using grad (phi) or (del operator)*(phi), where del = (∂/ ∂x)i + ( ∂/ ∂y)j + ( ∂/ ∂z)k , so doing the (del operator)*(phi) = 1i +1j +1k. This method can be used to find out the normal of all planes provided we have their equations.

[@tf8896]

You need to know that the plane is x+y+z=1 in the first place. The most formal method of doing this needs 2 vectors and 1 point, where you get the cross product of two vectors that are on the plane (I chose point (1,0,0) and k-i and j-i as the plane vectors) to get their normal vector N, which indeed is i+j+k. The coefficients of this vector are the components of the equation for the plane ax+by+cz=d, and to get d, plug in (1,0,0), or get the dot product of N and the position vector of the point you chose. d=1, therefore the equation of the plane is x+y+z=1.

[@pravingaikwad1337]

We have to divide phi by its magnitude to make it unit vector right?

[@pravingaikwad1337]

I mean unit vector would be ((i+j+k)/√3)

[@eggxecution]

Solved it!

[@curiousbit9228]

Bravo!!

[@florencetembo6905]

I love you from Zambia me passing calculus three because of you

[@atulmaniac8083]

Sir can you plz provide some videos to get to know about some Terms as log, e, uses of it and integration differentiation i.e. the basic calculus to use in physics cause sir I am not understanding where the things are coming from and what sudden change it is providing in the result ....... Plz help me while explaining these terms basically in physics to derive anything the question wants. And also allowing me to understand the concept from the starting and making the use of it... Thank you Sir Plz help me so that I can regain my interest in physics and mathematics (Due to the coming of integration deriving formula differentiating things which causes a trouble as after a lot of search nobody is explaining it and even without knowing their use properties I am skipping that part .) Plz help me sir I am in class 11 th n That sudden change is causing a great trouble ....

[@ProfessorDaveExplains]

buddy just go through my mathematics playlist, it's all in there for you!

[@atulmaniac8083]

@@ProfessorDaveExplains Sir Yes Sir.

[@wrench24]

I miss the old hair professor dave

[@woodychelton5590]

in the checking comprehension problem I am getting that the integral is equal to 0 ?

[@woodychelton5590]

nvm

[@frankfemino5376]

good video

[@pedritotuga2984]

damn, i still get confused what the line integral means!! haha i'll have to look it up! great video :)

[@ProfessorDaveExplains]

I cover that earlier in the series!

[@darcash1738]

Summary: - We have F, some vector space , where a, b, and c can each contain x, y, and z. - We have C, which is the boundary of our surface, some equation involving x, y, and z. - We have r, , where {x, y} is z in terms of x and or y [isolate z from our curve equation] 1) Find the curl: Del x F 2) Find the normal vector of our plane surface: d/dx (r) x d/dy (r) 3) Dot product 1) and 2) 4) Take the double integral of 3. Look at the surface x and y span, may need to sketch it out. If they are not constant (like the box at the end's sample problem), you will need to find x in terms of y or y in terms of x depending on which you want to integrate first. The part I'm least sure about is 4. In the main example, we have x + y + z = 1, and use (1, 0) and (0, 1). The reason we ignore z seems to be because we already have the integral in terms of x and y. Let's say we have ax + by + cz = d. More simply, we use ax + by = d. It seems like the points we use for the sketch is where the other variable = 0, or ax = d, or by = d. Idk if that is even right tho, is it? and what about other cases, too--besides something where its just constants for x and y like the sample prob at the end [And assuming the graphs of them are not just given to you to make it easy to see] OH and how do we calc whether or not it is positively oriented

[@nikhilverma9969]

Just put the limit 0 to 2 in both x and y

[@lizzardnannerzz]

okay but why do i keep getting -32/3 ???? D':

[@miketorlinski3462]

me too

[@vtrandal]

Stokes' Theorem

[@GS-ke8ex]

This seems 3 dimensional, yet only a double integral is used. Why?

[@harshitayadav950]

Love from India 🥰❣️

[@doctor99267]

The normal vector isn't unit though

[@erfanmohagheghian707]

The 2x2 square cannot be the boundary on the paraboloid. The question is wrong. You can take a quarter of a circle with radius 2 in the first quadrant as a legit example of a boundary curve.

[@avaritia1522]

Did the intro song change? And what happened to your hair, brooooo... 😭😭😭

[@ItsPouring]

Solution to the last problem may not be crystal clear but the answer sure is...😵

[@SamsungA04e-dp7kj]

book : ADVANCE CALCULUS

[@jack6914]

Solution to the problem: r(x) x r(y) = < -2x,-2y,1> del x F = , (z = x^2 + y^2 --> -z = -(x^2 + y^2) = -x^2 - y^2) Integral: 0

[@experimentbysaifanali7580]

Who else is learning it for physics?

[@shadrickmphande7074]

unasheko umungulu boi

[@jasonbroadway8027]

I disagree with your answer at the end of the video. Nonetheless, I enjoyed the lesson.

[@coreymonsta7505]

Stokes’ “tharem” -.-

[@alfredomancini3710]

the video is very consufing

[@sketchfam8037]

All his vids are

[@theallmightypeanut2939]

I think the answer is wrong but I'm not sure

[@jasonbroadway8027]

I believe that the upper limit must have the sqrt(4-x^2). I agree with you.

[@babarazamsucks]

First!

[@rahleigh5829]

Who else has the answer -256/3 😂😂

[@joaorosa9772]

u are wrong

[@ProfessorDaveExplains]

Um, no.

[@joaorosa9772]

@@ProfessorDaveExplains I had a stoke watching the video