3.7: Green's Theorem
( \newcommand{\kernel}{\mathrm{null}\,}\)
Ingredients: C a simple closed curve (i.e. no self-intersection), and R the interior of C.
C must be piecewise smooth (traversed so interior region R is on the left) and piecewise smooth (a few corners are okay).
If the vector field F=(M,N) is defined and differentiable on R then
∮CM dx+N dy=∫∫RNx−My dA.
In vector form this is written
∮CF⋅dr=∫∫RcurlF dA.
where the curl is defined as curlF=(Nx−My)