Answer
The vector field $\overrightarrow{F}$ is not conservative.
Work Step by Step
The vector field $F(x,y)=Pi+Qj$ is a conservative field throughout the domain $D$, when
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
Here, $P$ and $Q$ are the first-order partial derivatives on the domain $D$.
The work integral field $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}$ will be independent of the path if and only if $\int_C \overrightarrow{F} \cdot \overrightarrow{dr}=0$ for every closed curve $C$.
Since, the work done represents the line integral of force, then the work done $\overrightarrow{F}$ along two different paths $C_1$ and $C_2$ connecting the same two points will be different.
Thus, the line integral of $\overrightarrow{F}$ is not path independent.
Hence, the vector field $\overrightarrow{F}$ is not conservative.
