Polya - Vector Field [better]

Let ( f(z) = u(x,y) + i,v(x,y) ) be a complex-valued function of a complex variable ( z = x + iy ), defined on some domain ( D \subseteq \mathbbC ).

[ u_x = v_y, \quad u_y = -v_x. ]

Hence:

Specifically, residue theorem:

[ \oint_C f(z) dz = 2\pi i \sum \textRes \quad \Rightarrow \quad \textCirculation + i,\textFlux = 2\pi i (\textsum of residues). ] polya vector field