I am trying to solve an exercise in where it is asked to show the divergence theorem, or also known as Gauss theorem, in $\mathbb R^2$ using Green's theorem.
I suppose that the divergence theorem in $\mathbb R^2$ is that for a $C^1$ vector field $F:\mathbb R^2 \to \mathbb R^2$ defined on a $3$ type region $D$ such that $\partial D$ is a closed curve, we have $$\iint_D div(F)dA=\int_{\partial D} Fds$$
I've tried to prove this theorem applying Green's theorem but I coudn't, I would appreciate if someone could provide a solution using Green's theorem (or at least the steps I should follow to prove the equality).
Best Answer
Recall Green's theorem $$\int_{\partial D} F ds=\int_{\partial D}F_1 dx+F_2 dy=\iint_D (\partial F_2/\partial x-\partial F_1/\partial y)dxdy$$ From this, setting $G_1=-F_2$ and $G_2=F_1$, we get $$\iint_D (\partial F_1/\partial x+\partial F_2/\partial y)dxdy=\iint_D (\partial G_2/\partial x-\partial G_1/\partial y)dxdy=$$$$\int_{\partial D} G_1 dx+G_2dy=\int_{\partial D} -F_2 dx+F_1 dy$$ So $$\iint_D\text{div} Fdxdy=\int_{\partial D} F\cdot\mathbf{n} ds$$ where $\mathbf{n}$ is an outward unit normal vector to $\partial D$.
Notice that this is not the theorem you stated!