Stokes' Theorem Example // Verifying both Sides // Vector Calculus

In this video we verify Stokes' Theorem by computing out both sides for an explicit example of a hemisphere together with a particular vector field. Stokes Theorem was introduced in the previous video in the vector calculus playlist (see below) and equated the line integral giving the circulation around the boundary curve with the surface integral of the proportion of the curl of the vector field in the normal direction. We finish with a really cool trick, where the old surface can be replaced with an entirely new one that is actually easier to computer.
0:00 Recalling Stoke's Theorem
2:40 Computing Circulation
5:02 Computing Surface Integral
10:40 Replacing the old surface with a new one
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[@andytsang4515]

I've been skipping classes since the beginning of the semester and this playlist saved my ass. Binged the entire playlist in just a few days. Tysm Dr. Bazett

[@DrTrefor]

Hope it helps!

[@apoorvbagal2936]

lol same here

[@dakotahrivers6640]

you ain't the only one lol

[@richardvalentinonainggolan328]

:v same here

[@user-dm1zj5zs3x]

You forgot the units

[@nebulouscalamity6338]

Holy guacamole. The bit at the end is the coolest wrap up I have seen in a while. It blew my socks off and fired me up with excitement. I love you and the immense excitement, empathy and curiosity that you radiate.

[@ycombinator765]

You know you love Math when these kind of phenomenona show up on your recommendations within an hour of posting! Thanks Sir!!!

[@DrTrefor]

haha you're telling the youtube algorithm good things!

[@MGTOW-nn9ls]

Thank you Dr Bazett. Explanation that curls are canceling each other helps a lot.After watching your video I can say now that I really understand the physical aspects of the Stocks Theorem

[@AbhishekKumar-jg7gq]

You are making me feel the science of mathematics by explaining the inside of every topic

[@paulgillespie4908]

Bloody brilliant video. I did this stuff forty years ago at university and had no clue what was happening. So glad that I now finally get it. Thanks so much.

[@glennnakamura4831]

Same thing happened to me when I took vector calculus and E&M theory 43 years ago. Picturing what was happening with the interactions of the E and B fields was tough. This gives me so much of a better understanding of what's happening.

[@robertoberidojr.435]

I really have hard times deciding when to use implicit, explicit and parametric. I found this very helpful since I now understand when to use those formulas

[@amyhughes2149]

Super helpful video, I've been learning through correspondence and all of the material seems so esoteric. Thank you for making something that is easy to understand!

[@briandwi2504]

This topic is amazing. I am working through the calculus and linear algebra as I want to learn more about Classical physics, post Newton. It's clear that vector calculus is a fundamental tool to understand Maxwell's equations. I am sure your work will have given me a good foundation to understand Physics properly. Brilliant work Dr Trefor. Thanks.

[@Kicsa]

Really great content, thank you for your enthusiasm and way of teaching this subject.

[@___xid9992]

What an amazing resource this channel is. I've watched so many of your vids, thanks for making them. :)

[@DrTrefor]

Thank you!!

[@coltonlemasters]

The visualizations you provide are amazing. I'm already doing very well in my vector calculus class, but I still like to visit your videos to gain an even more intuitive understanding. Please do more upper level math material. No need to actually do any problems, just conceptual videos would be amazing for added learning.

[@jacobwilliams8634]

thank you! Stokes Theorem would be such a difficult concept to learn without your videos

[@DrTrefor]

Happy to help!

[@user-dm1zj5zs3x]

This proof is BS since he does not include the units.

[@Kubikrup]

Great channel and explanations professor, thank you!! I have an exam and I hope I can express myself perfectly just like you while solving the questions:))

[@tanishkumar6682]

Thankyou so much, on clearing my doubt on the concept of replacing the old surface with the new one. Lots of love from INDIA

[@nikhilsingh5233]

the third example was genius . thankyou sir

[@mayathesheriff9100]

Thank you for the detailed graphics that help students visualize complex 3D concepts.

[@DrTrefor]

Glad they help!

[@MishaShvartsman]

Beautiful job!!

[@user-po5gy7le3s]

WONDERFUL VIDEO. thank you SO SO SO much.

[@ketaksingh5465]

This is awesome stuff ...thank you so much!!

[@shaheersherif7366]

Thank you , very helpful!

[@sergiolucas38]

Excellent video :)

[@cherrygarcia5931]

I have a cal3 final tmw and your videos are saving my life thank u so much :,)

[@DrTrefor]

Good luck!!

[@karlmudsam2834]

Emphasozed

[@hussamalanesi7432]

This is amazing sir. Could you make a video explaining more how to find n and d sigma for various cases?

[@sumitkumarsahoo7601]

well done I wish if I could give a presentation like you

[@yasirarafat1872]

best video ever

[@srinivasansrivilliputhur4704]

Dr. Bazett: Thank you for this wonderful series on Vector Calculus. Very helpful review for me. A question though. Will you please geometrically illustrate the transformation from dSigma to dA. You jumped too quickly around 8:35.

[@manishbhargao965]

Really awesome explanation

[@DrTrefor]

Thank you!

[@NavneetKumar-yb3gu]

Thank you sir

[@elgatito00]

OMG you are the best 😭😭😭

[@nightking8490]

Thanks man. I was confused about n ( unit normal) in methods 2 and 3. I was confused which normal to take, like the normal vector pointing in different directions on spherical surface or normal vector pointing from the plane area projection like XY plane has k as its normal unit vector. Thanks 2 you, my doubt now could rest in peace.

[@sahil-pu3cc]

Doubt cleared bro thnks

[@asterixx6878]

Quite often it's hard to actually identify the boundary of a surface. As an example example, lets say that we have a surface defined by z=1-x^2-y^2, x>0, z>0. This is a paraboloid, confined to the first and forth octants. What curve is a boundary of this thing? Is the straight line segment on the y-axis (-1≤y≤1) a part of a curve that's bounding the surface? As always - very nice videos. Thank you💛

[@eggxecution]

this helps

[@PhysicsBanglaTutorial]

Which apps or software you use to make this video? Can you tell me pls?

[@vaginalarthritis1753]

Have you considered doing a Real Analysis play list when this is done?

[@gerardsagliocca6292]

Your lectures should be numbered

[@DrTrefor]

They are all in order if you go the playlist for vector calculus:)

[@RSS18]

Respected sir , what if we change the orientation of curve and use normal as -k cap for the surface?

[@quantumaravinth]

Prof Trefor Bazett, do you know that there are organizations in India that sell videos at high price with same amount of animation (may be just a little more, we can say) that you made. Yours is much nobler and serious considering a membership to your channel.

[@DrTrefor]

Thank you for your kind words. For me any money from memberships etc has always been secondary, my first goal is to help provide a high quality mathematics education and if I’m able to help with that then I’m happy:)

[@margaretehrich7084]

Hey Trefor, thanks for all the help! Stem teacher to stem teacher, your handwriting could use some work.

[@DrTrefor]

ha well I can't disagree with you there:D

[@rowangoebel-bain3447]

bless up

[@ishraquearshad3243]

If our plane is in XYZ then is it possible to find the surface integral(circulation) of a vector field using the stokes theorem by projecting it on any plane XY, YZ, XZ?? Or there are some limitations: it should be a base plane projection from any particular direction!!

[@thefrenchiestfry8147]

How would one parameterise the circle if it wasn't a unit circle, e.g. (x + 1)^2 + (y + 1)^2 = 1? And also, at 7:38, why is the k unit vector included?

[@moinshaikh1963]

Great video professor! A question, around 10:00 why isn't the region to be considered the surface area of sphere instead of the circular area on which the shadow of sphere is falling?

[@fardeenrazif9377]

Total curl depended only from the boundary surface, (circle in xy plane) , so any surface with the same boundary must also give the same total curl. By boundary I mean the outside perimeter of the surface

[@frakkx3491]

@@fardeenrazif9377 it's pretty weird that he uses this in his example problem then explains why as if it's a totally different way of solving it

[@fardeenrazif9377]

Yeah I also thought about that

[@Rzsstrikerszr]

hi sorry if this off topic from this video but im stuck on a question: I have to find the transfer function and rearrange it to produce an expression for i(s) for V(t) = i(t) R + L di/dt + V0 if you or anyone can help me on this it be a big help thanks

[@emerald_eyes]

I don't quite understand, isn't the integral Fdr a line integral? Why do we get the area when integrating it?

[@gilbert4004]

When you are computing the circulation (at around 4:30), how did you parameterize the vector field F to get M = sin(theta) and N = -cos(theta)?

[@benmcginn8519]

Remember he is taking the dot product of F and r prime. He is taking the matching components, multiplying them, then adding them to the products of the other components. So in this case, the i component of F is sin theta in this parameterization. Taking the dot product with the i component in r prime, which is -sin theta, gives you negative sin squared theta. Calculate the other components in the same way, -x = -cos theta, j component of r prime is cos theta, giving -cos squared theta. Finally, z = 0, so the z term goes away.

[@johnkapauo2995]

Could you please try and verify the surface for the cylinder following that idea or concept?

[@j.o.5957]

Alright, cool computations. Struggled a bit with doing them myself, but did it after some effort. Question to self: we have the implicit n = nabla_g/|nabla_g|. What are the parametric and explicit? I remember parametric being r_u x r_v, but what was the explicit again? I don't quite remember, but wouldn't it just be k_hat? Perhaps in this example if we reduce the surface, but I'm not sure. Also found it weird that the k_hat went to the bottom, what happened there? d_sigma = 1/k_hat dA. So d_sigma is the normals on the surface. Dividing it by k_hat, you only get the normal to the xy-plane left, meaning you dA, which is dxdy. Aight, I think that's it.

[@golagha1959]

Please explain how curl F.n dsigma changes to curlF times some unit vector times dA. I can't see how dsgima changes to dA.

[@akhmalfaris4097]

i love you

[@baolongngo3486]

I want to ask why in the second way of the example , why we need unit normal vector , i thought this was be cancelled when dS becomes dA

[@Festus2022]

Trevor, thx for the great video. Since a line integral can represent WORK or CIRCULATION, does this mean that circulation IS work??. I not, what is the distinction?

[@DrTrefor]

It must means the same mathematical ideas apply in different physical contexts

[@Festus2022]

@@DrTrefor Thx so much for the response!

[@skcenior4227]

Sir do you have a calculus playlist purely for physics because I'm little confused with your playlist (IF I SAID ANYTHING WORNG ABOVE IM SORRY)

[@DrTrefor]

This playlist is actually intended for a course that physics students would take.

[@skcenior4227]

@@DrTrefor thanks 😊

[@kaveenmanusha9843]

cant understand what happened at 10:12 the s to r thing

[@skcenior4227]

Hi I'm from india your vedios really helpful for me THANKYOU but now it is midnight I'm going to sleep so I'm gonna watch this vedio tomorrow sorry!!

[@DrTrefor]

haha good night!

[@suryanarayanan757]

Is d sigma is equal to mod del g times dA and n ds is equal to del g times dA

[@devashishshah9021]

Stokes theorem was very hard to visualize until this happened....

[@ricardobautista-garcia8492]

Third. And nice video as always.

[@anushkatilekar5216]

I didn't understand what you did at @4:08 ..... The vector r had values in terms of theta ... The vector F had values in terms of x and y .... Then how did you write the vector F in terms of theta in the integral ?.......

[@frakkx3491]

Change to polar coordinates, y becomes sin(theta) and x becomes cos(theta)

[@ejisip2091]

No to brag, but I am able to solve the problem on myself. You're a great lecturer Dr. Trefor. More power!!

[@JohnVKaravitis]

The surface can't exceed the boundaries of an imaginary "cylinder" that extends upward from the line curve, yes? If you were to shine a flashligh from above, there would be no shadow, yes?

[@willardsavage2980]

why you didnt make the limits of integration you confused me, cuz the limits of integration in my opinion are radius 1, and angle 2pi, but since you did that thing of pi*rsquare, i didnt get what you did so i dont know if i did wrong or well, why im gettin 4 pi in the final answer i dont understand can you explain this please, i get that you use the formula of the area, but i wanna know how to do it with limits of integration, since limits of integration is what i will need to use in the exam, not a formula of an area, thank you

[@THKNTV]

i checked my teams

[@theshibu214]

🙂ngl i am having a mental break down ..howwwwwww am i supposed to do this

[@engineers_hub]

what is k-hat? where does it come from?

[@whyargon]

it is the normal of the boundary (since the boundary is a flat circle, the normal is always going to be just khat)

[@jrjr1313jrjr]

It appears you used a surface in the x,y plane in both calculations. You did not use a vector normal to the hemisphere.

[@lizzardnannerzz]

is he canadian? do canadians also pronounce 'z' like 'zedd' ?

[@DrTrefor]

Haha yup!

[@liannalim4521]

You skip through smaller parts way too much. Oftentimes I don't know what part correlates to what. It can get confusing. Like a 4:55 how is it suddenly all in sin and cos? F is in x and y.

[@rogersndegwa-ck3ud]

Do more examples..I hate theory

[@user-dm1zj5zs3x]

This proof is obtuse since the units of the integrals are omitted since a length is not equivalent to a surface area. Is there an ABC mouse for math teachers? Unless I am wrong but all the girls I know say I have cute dimples which is similar to a surface area and the lower length is enormous. Do you ever get that sinking feeling that what you do is entirely a waste of time. What if you did the text of your video in a seminar and I was there. What would you say? Would you cry? Mommy mommy why do you not write?

[@Alians0108]

Go back to physics

[@GOODTHINKER691]

Thanks you sir

[@DrTrefor]

You're most welcome!