What is a conservative vector field on $\mathbb{R}^{n}$ ?

State Green's theorem in the plane $\mathbb{R}^{2}$.

(a) Consider a smooth vector field $\mathbf{V}=(P(x, y), Q(x, y))$ defined on all of $\mathbb{R}^{2}$ which satisfies

$$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$$

By considering

$$F(x, y)=\int_{0}^{x} P\left(x^{\prime}, 0\right) d x^{\prime}+\int_{0}^{y} Q\left(x, y^{\prime}\right) d y^{\prime}$$

or otherwise, show that $\mathbf{V}$ is conservative.

(b) Now let $\mathbf{V}=(1+\cos (2 \pi x+2 \pi y), 2+\cos (2 \pi x+2 \pi y))$. Show that there exists a smooth function $F(x, y)$ such that $\mathbf{V}=\nabla F$.

Calculate $\int_{C} \mathbf{V} \cdot d \mathbf{x}$, where $C$ is a smooth curve running from $(0,0)$ to $(m, n) \in \mathbb{Z}^{2}$. Deduce that there does not exist a smooth function $F(x, y)$ which satisfies $\mathbf{V}=\nabla F$ and which is, in addition, periodic with period 1 in each coordinate direction, i.e. $F(x, y)=F(x+1, y)=F(x, y+1)$.