Stokes' theorem intuition | Multivariable Calculus | Khan Academy

Conceptual understanding of why the curl of a vector field along a surface would relate to the line integral around the surface's boundary

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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.

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[Ali]

In class this made me experience something which I would define as "brain death."
You're my resuscitation, Sal.

[devikabsree]

This is the most outstanding explanation of Stoke's theorem. So clearly explained. Thank you so much.

[panikostiritas]

Crazy how 3 hours of lectures amounted to me retaining 0 knowledge. Then 10 minutes of this and I understand it like crazy

[bawol-official]

My final exam is in a week, I just got assigned homework for the sections on Vector Field, Green’s, Stokes, and Divergence theorem . Pray for me.

[sudarshanseshadri]

Hours of reading the book and listening to prof lecture and this 10 mins was more effective than all that. And it’s not like Sal had the online video advantage - my prof also recorded and posted his lecture. Sal really has a gift.

[coco_jae]

wow... finally, i understand the stoke's theorem.

[kal]

30% of my exam for university 2 weeks ago was on stokes and greens theorem. Thank you so much for these videos :)

[amandaferguson]

Wow I came here for Stoke's Theorem and I get an actual explanation of curl also

[sbullock]

Wow. The simplicity of this explain blew my mind! Great video.

[EldonSchoop]

When I took multivariable calculus, I never got an intuitive understanding of Stokes' Theorem. Now I do. Thanks, Sal. :)

[carmelpule]

I am now an old man and over 65 years ago I saw it all in this manner,
The Curl is the amount of circulation behaviour around the smallest element dxdy. So if we total all the circulations on the elemental area, we find the circulation around the outer contour.
This is no different from finding the total mass of a rod, If we know the mass per unit length then we integrate along the length to find the total mass.

I believe that the following " activities have similar/related building blocks/ logic to produce the tacit differences. .

1. Cauchy Riemann relations
2. The Grad operator.
3. The curl operator.
4. the Divergence operator.
5 . Green's Curl theorems of circulation
6. Green's Divergent theorem of flux
7. Stoke's Curl theorem involving circulation
8. Divergence theorem involving divergences through volumes/surfaces,
I always thought that students should see the close links there are in how these derivatives are combined to produce their " engineered" activity.

dU/dx dU/dy dU/dz
dV/dx dV/dy dV/dz
dZ/dx dZ/dy dZ/dz and reduced to two dimensions
dU/dx dU/dy
dV/dx dV/dy
.

[robromijnders]

Sal, you're a genius. Thank you!

[battleangelgally]

Oh My Goodness! All those equations have suddenly started to make so much sense...
Thanks a lot Sal!!!

[robertmatuschek]

reasoning for dotting with N. …We do this because curl is a vector whose direction is orthogonal to the Counter clockwise rotation and the dot product calculates the amount of curl in S

[bijoythewimp]

My college didn't reach here. I am way ahead. I lost my faith in teachers long time ago. Sal and other youtube teachers are my only hope and I am contended to have them as my virtual teachers.

[MeshalWinehouse]

I’m on second year as a physicist 😮‍💨 can’t wait to graduate 😭😭 thanks for ur help.

[elizabetheckenrod]

And now I can pass my final... Bless you, Khan Academy!

[Virtualexist]

I get so happy with him when the vector fields and path are in the same direction !!!

[mmzzcc]

I wish I could have seen this video when I was at the university! Thanks Sal!

[ParthPaTeL-wmkt]

This is best video about stokes theorem in whole YouTube, Great, thanks dude