Let $\displaystyle P(x,y),Q(x,y)\in C^1(R^2)$ and

$\displaystyle \oint\limits_{L(M_0,R)}^{}P(x,y)dx+Q(x,y)dy=0$, where

$\displaystyle L(M_0,R):y=y_0+\sqrt{R^2-(x-x_0)^2}$ (uper semicircle)

where $\displaystyle M_0(x_0,y_0)\in {\Bbb R}^2$ and $\displaystyle R\in {\Bbb R}^+$ are arbitrary.

Prove that

$\displaystyle P(x,y)\equiv 0$ and $\displaystyle \frac{\partial {Q(x,y)}}{\partial x}\equiv 0$

Hint: Prove first that $\displaystyle P(x,y)\equiv 0$ , by use parametric integrals. What must we do to apply Green's formula?