About this video
Sal begins a worked example of Stokes' theorem. The path C is where the plane y + z = 2 slices the infinite 'pole' x^2 + y^2 = 1, and there is a vector field F. Instead of computing the line integral around C directly, Stokes' theorem lets us use a surface integral of curl F over a surface whose boundary is C.
You'll be able to
- describe the path C as the intersection of the plane y + z = 2 and the cylinder x^2 + y^2 = 1.
- explain why x^2 + y^2 = 1 in 3D is an infinite pole, not a closed can.
- choose a surface whose boundary is C so Stokes' theorem can be applied.
Key words
- intersection
- The set of points that lie on both surfaces at once.
- cylinder x^2 + y^2 = 1
- In 3D, every point whose x and y lie on the unit circle, at any height z: an endless pole.
- Stokes' theorem
- The line integral of F dot dr around a boundary equals the surface integral of curl F dot dS over the surface it bounds.
Transcript
Let's now attempt to apply Stokes' theorem And so over here we have this little diagram, and we have this path that we're calling C, and it's the intersection of the plain Y+Z=2, so that's the plain that kind of slants down like that, its the intersection of that plain and the cylinder, you know I shouldn't even call it a cylinder because if you just have x^2 plus y^2 is equal to one, it would essentially be like a pole, an infinite pole that keeps going up forever and keeps going down forever so it would never have a top or a bottom but we slice that pole, with y plus z is equal to 2, to get where they intersect we get our path C. We also have this vector field defined in this way
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