# Deriving Green's theorem

The reasoning leading to Green's theorem in my course makes a step I don't understand, with no justification.

We have a function $$P:R\to \mathbb{R}$$ that has a partial derivative with respect to $$y$$ over $$R$$. We're computing $$\displaystyle \iint \limits _R\dfrac{\partial P}{\partial y}(x,y)\,dA$$. Say $$R$$ is regular with respect to the $$x$$-axis. We can easily compute that it's equal to $$\displaystyle \int \limits _a^bP(x,f_2(x))\,dx-\int \limits _a^bP(x,f_1(x))\,dx$$ for $$a\leq x\leq b$$. These are line integrals along the curves $$y=f_1(x)$$ and $$y=f_2(x)$$. We'll call them respectively $$C_1$$ and $$C_2$$.
So our original double integral becomes $$\displaystyle \int \limits _{C_2}P(x,y)\,dx-\int \limits _{C_1}P(x,y)\,dx$$.

The next step is the one I don't understand: my course states that this is equal to $$\displaystyle -\oint \limits _{C^+}P(x,y)\,dx$$.

Why?

For Green to hold, the boundary of $$R$$ must be positively oriented, and if you draw a picture, that would mean the lower half $$C_1$$ is positively oriented, as it's from smaller $$x$$ to bigger ones, while the upper half $$C_2$$ is not. Therefore $$\oint_{C^+}=\int_{C_1} - \int_{C_2}$$.