Let R be a closed region in the xy plane bounded by a simple closed curve C, and if ϕ(x, y) and ψ(x, y) are continuous functions having continuous partial derivatives in R, then \(\displaystyle\int_{C}\)(ψ dx + ϕ dy) equals :
1
\(\displaystyle\iint_{R}\left(\frac{\partial ϕ}{\partial x} + \frac{\partial \psi}{\partial y}\right)\) dx dy
2
\(\displaystyle\iint_{R}\left(\frac{\partial ϕ}{\partial x} − \frac{\partial \psi}{\partial y}\right)\) dx dy
3
\(\displaystyle\iint_{R}\left(\frac{\partial ϕ}{\partial x} ⋅ \frac{\partial \psi}{\partial y}\right)\) dx dy
4
\(\displaystyle\int_{C}\left(\frac{\partial ϕ}{\partial x} − \frac{\partial \psi}{\partial y}\right)\) dx