Question: Is there any formula that bounds the line and double integrals other than the Green one? My guess: No! We know: $$ \int_V \operatorname{div} \vec{F}\, dx\,dy\,dz = \int_{\partial V} \vec{F} \cdot \vec{n} \cdot dS$$ Here: $n$ denotes the unit normal vector of $dS;$ div stands for divergence and defined by the formula through limit, as known. This formula is not the same as the Stokes one, in which one may discern curl. My guess is supported by defining the vector function $$ \vec{F} = (\varphi(x, y) ; \psi(x, y)). $$ So, $$ \operatorname{div} \vec{F} = \vec{\nabla} \cdot \vec{F} = \frac{\partial \varphi}{\partial x} + \frac{\partial \psi}{\partial y}$$
Taking an attempt to get an analogous, we have
$$\int_A \operatorname{div} \vec{F} \,dx\,dy = \int_{\partial A} \vec{F} \,\vec{dl} $$
Taking into consideration that $\vec{dl} = \sum x_k \vec{e_k} $, we get $$\int_A \left(\frac{\partial \varphi}{\partial x} + \frac{\partial \psi}{\partial y}\right) \, dx\,dy = \int_{\partial A} (\varphi \,dx + \psi \,dy) $$ That does not correspond to the Green formula, which implies that this should not work.