Show that Green's Theorem is satisfied for $\vec{G}: \mathbb{R}^2\rightarrow\mathbb{R}^2$ $\vec{G}=y\vec{i}-x\vec{j}$ and the path $C: x=\sin t, y=\cos t$ for $0\leq t \leq 2\pi$.
We want to show $$\oint_{\partial C} (Pdx+Qdy)=\iint_C \frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\,dx\,dy$$
LHS is \begin{align} \oint_{\partial C} (ydx-xdy)&=\int_{0}^{2\pi} \cos^2(t)+\sin^2(t) dt\\ & = 2\pi\\ \end{align}
RHS is \begin{align} \iint_C \ 2 \,dx\,dy&=2\pi\\ \end{align}
Is this correct working/logic?
Edit: where $P=y$ and $Q=-x$.