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!
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.
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.
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.
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.
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](/img/n.gif)
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!
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.
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!
I need that video explanation about Green Theorem and some application Too. Please.
I'm really stoked for this video
Brilliant explanation of one of the most abstract topics in calc..great upload!
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.
I haven't taken a course on PDEs, so it's not on my roadmap!
This is an awesome explanation man, thank you for this
Amazing lecture😍 Thank you😊
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.
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.
Amazing!
I am probably too stupid to understand this. Some people computing the curl, some people don‘t.