= Solution
Integrating the oscillatory term together with the constant components gives a <potential of a conservative vector field>
$$
\boxed{F(x,y)=x+2y+\frac{\sin(2\pi(x+y))}{2\pi}.}
$$
Direct <partial derivatives> recover both components of $\mathbf V$, proving it is a <conservative vector field>. Its <line integral> is therefore <path independent>. At an integer endpoint $(m,n)$, the sine term vanishes, so \b[for any of the specified paths]
$$
\boxed{\int_C\mathbf V\cdot d\mathbf x=m+2n.}
$$
Any other potential has the same <gradient>, so differs from $F$ by a constant on the connected plane. In particular every potential changes by $1$ under $x\mapsto x+1$ and by $2$ under $y\mapsto y+1$. Equivalently the integral from $(0,0)$ to $(1,0)$ is $1$, whereas a unit-periodic potential would make that difference zero. \b[There is no potential periodic with period one in both coordinate directions.] This is an example of how <periods obstruct a periodic potential>: a periodic <vector field> need not have a periodic potential.
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