(i) Define what is meant by a conservative vector field. Given a vector field $\mathbf{A}=\left(A_{1}(x, y), A_{2}(x, y)\right)$ and a function $\psi(x, y)$ defined in $\mathbb{R}^{2}$, show that, if $\psi \mathbf{A}$ is a conservative vector field, then

$$\psi\left(\frac{\partial A_{1}}{\partial y}-\frac{\partial A_{2}}{\partial x}\right)=A_{2} \frac{\partial \psi}{\partial x}-A_{1} \frac{\partial \psi}{\partial y}$$

(ii) Given two functions $P(x, y)$ and $Q(x, y)$ defined in $\mathbb{R}^{2}$, prove Green's theorem,

$$\oint_{C}(P d x+Q d y)=\iint_{R}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) d x d y$$

where $C$ is a simple closed curve bounding a region $R$ in $\mathbb{R}^{2}$.

Through an appropriate choice for $P$ and $Q$, find an expression for the area of the region $R$, and apply this to evaluate the area of the ellipse bounded by the curve

$$x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$$