27: Stokes' Theorem - Valuable Vector Calculus

[@MuPrimeMath]

Something I didn't mention in the video: Green's theorem is a special case of the curl theorem! To derive Green's theorem, consider a surface S which is only in the xy-plane. In other words, z=0 for the entire surface S. Then we take the normal vector to be [0, 0, 1]. Because the surface is just a region in the xy-plane, the "surface integral" is the same as an ordinary double integral!

[@abrahamsweetvoice7687]

Have you still considered making a seperate video for the green's theorem. I understand your explanation, but I think it would fit perfectly for your channel cuz thats the only type of theorem ( special integral equation ) you haven't made a video of.

[@MuPrimeMath]

I don't plan to make a video on Green's theorem any time soon. I'm going to start a new series in a week or two!

[@MrChai74]

I need that video explanation about Green Theorem and some application Too. Please.

[@Hobbit183]

I'm really stoked for this video

[@daniellabinjo6046]

Brilliant explanation of one of the most abstract topics in calc..great upload!

[@chanduyasodar6445]

Videos are really good, they helped me to pass my undergrad end sem exams

[@eswyatt]

Thanks for answering all my questions, some of which I realize have been ill-conceived. I made 18 pages of careful notes just from videos 24--27! Do you ever plan on doing a series on PDEs? I would love a different "take" on that stuff.

[@MuPrimeMath]

I haven't taken a course on PDEs, so it's not on my roadmap!

[@kingmunch7252]

This is an awesome explanation man, thank you for this

[@eswyatt]

Am I correct in thinking there is no requirement that the boundary of the surface lie on a 2-dimensional plane in order for Stokes' Theorem to work? It could be an object like an N95 facemask--- set on a table concave up (?), its edge wouldn't lie entirely on the table. If the boundary *does lie on a plane, Stokes' theorem becomes an instance of Green's Theorem, *even if it's not the xy-plane (i.e. even if we can't use k = 1 as the normal vector). Otherwise we could just do a change of basis and make whatever plane the boundary lies on the "xy-plane"? Just wondering if I'm thinking about this right.

[@MuPrimeMath]

You're correct that the boundary can be any curve in 3 dimensions! It is theoretically possible to perform some transformations on the coordinates such that a specific plane is mapped to the xy-plane while preserving shapes and function values.

[@assammukhtar2716]

Amazing lecture😍 Thank you😊

[@franktaktywolf970]

Amazing!

[@long8398]

I am probably too stupid to understand this. Some people computing the curl, some people don‘t.