Green's Theorem special application

Question

If I want to evaluate the flux of a vector field over an unfinished square (basically over three sides of a square), instead of parametrezing the 3 sides and computing the flux 3 times, can I use Green's theorem over the full square and subtract from it the flux over the line integral of the side that was not available before?

Answer 1

After re-reading I think you seem to suggest \begin{align} \int\limits_\sqsubset A \cdot dr &= \int\limits_{\square = \partial S} A \cdot dr - \int\limits_{.\vert} A \cdot dr \\ &= \int\limits_S\text{curl } A \cdot dS - \int\limits_{.\vert} A \cdot dr \\ &= \int\limits_S (\partial_1 A_2 - \partial_2 A_1) \lVert dS \rVert- \int\limits_{.\vert} A \cdot dr \\ \end{align} where we extended to three dimensions in between by giving the square surface a unit normal in positive 3-direction: $S = \lVert S \rVert e_3$ and using Kelvin-Stokes instead of Green.