§16.4 Green’s Theorem in the Plane

Lecture notes

Key equations

Green's theorem (tangential form): \[\begin{aligned} \oint_C \F\cdot\T\,ds &= \iint_R \curl\F\,dA\\ \oint_C M\,dx + N\,dy &= \iint_R \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dx\,dy \end{aligned}\] Green's theorem (normal form): \[\begin{aligned} \oint_C \F\cdot\n\,ds &= \iint_R \div\F\,dA\\ \oint_C M\,dy - N\,dx &= \iint_R \left(\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}\right)dx\,dy \end{aligned}\]

Circulation density

Divergence / Flux density

Green's theorem

Examples

  1. Find the counterclockwise circulation and outward flux for the field \[ \F = 2xy\i - y\j \] over the unit square $0\le x,\,y\le 1.$
  2. Find the counterclockwise circulation and outward flux of the field \[ \F = (x-y)\i + (2x+y)\j \] over the region enclosed by $y=x^2,\,0\le x\le 1$ and $y=\sin(\frac\pi2x),\,0\le x\le1.$