13. Consider a 2D vector field F, whose
domain consists of those points of $\mathbb{R}^2$
which are not the origin.
Let F = (P,Q).
Assume that $P_y = Q_x$ on the domain of
definition of F, and assume further that
F is not conservative on its domain.
Given are the (interiors of the) three re-
gions A, C, D sketched in the picture at
right. From this information, which of
the following statements are true? Mark
all the true statements. (3 points)
For every closed continuous curve $\gamma$ contained in A, the line integral $\int_{\gamma} \mathbf{F} \cdot d\mathbf{r}$ must be zero.
For every closed continuous curve $\gamma$ contained in C, the line integral $\int_{\gamma} \mathbf{F} \cdot d\mathbf{r}$ must be zero.
There exists some closed continuous curve $\gamma$ contained in C, such that the line integral $\int_{\gamma} \mathbf{F} \cdot d\mathbf{r}$ must
be non-zero.
For every closed continuous curve $\gamma$ contained in D, the line integral $\int_{\gamma} \mathbf{F} \cdot d\mathbf{r}$ must be zero.
There exists some closed continuous curve $\gamma$ contained in D, such that the line integral $\int_{\gamma} \mathbf{F} \cdot d\mathbf{r}$ must
be non-zero.
No such vector field exists.
