斯托克斯定理

Green's theorem leaves the plane

A worked pairing

The boundary is the only thing that matters

Curl flux as circulation density

$\mathbf{F} = (-y, x, 0)$ and $S$ is the unit disk oriented upward with counterclockwise boundary. What does Stokes's theorem give?

A wavy bowl and a flat disk share the same boundary rim, in a smooth field. The disk's curl flux is $5$. What is the bowl's?

A surface is oriented with normal pointing generally east. Which boundary direction does Stokes pair with it?

A field has constant curl $(0, 0, 3)$. A disk of radius $2$ in the $xy$-plane is oriented upward. Give the curl flux in the form a*pi.

By Stokes, a surface with boundary rim has curl flux $-8$. Give the boundary circulation under the right-hand pairing, as a plain number.