선적분

(a) $P$ is the $x$-component of $\mathbf{F}$. As we travel vertically upward through $A$, that $x$-component decreases. Hence $P_y<0$ at $A$.

(b) $Q$ is the $y$-component of $\mathbf{F}$. As we travel horizontally to the right through $A$, that $y$-component increases. Hence $Q_x>0$ at $A$.

(c) ${ \boldsymbol{\nabla} }\times\mathbf{F} = (Q_x-P_y)\hat{\mathbf{k}}$ and $Q_x-P_y>0$ at $A$, so that the curl of $\mathbf{F}$ at $A$ is in the direction of $+\hat{\mathbf{k}}$.

(d) Along the curve $\mathcal{C}_1$ the magnitude of the angle between $\mathbf{F}$ and $\mathrm{d}\mathbf{r}$ is less than $90^\circ$, so that $\mathbf{F}\cdot\mathrm{d}\mathbf{r}>0$ and $\int_{\mathcal{C}_1}\mathbf{F}\cdot\mathrm{d}\mathbf{r}>0$.

(e) Along the curve $\mathcal{C}_2$ the magnitude of the angle between $\mathbf{F}$ and $\mathrm{d}\mathbf{r}$ is greater than $90^\circ$, so that $\mathbf{F}\cdot\mathrm{d}\mathbf{r}<0$ and $\int_{\mathcal{C}_2}\mathbf{F}\cdot\mathrm{d}\mathbf{r}<0$.

(f) If $\mathbf{F}$ were conservative, we would have $\int_{\mathcal{C}_1}\mathbf{F}\cdot\mathrm{d}\mathbf{r} =\int_{\mathcal{C}_2}\mathbf{F}\cdot\mathrm{d}\mathbf{r}$. As these two integrals have opposite signs $\mathbf{F}$ is not conservative. (Since $\mathbf{F}$ is not conservative, it is not the gradient of some function. At $A$, $P_x>0$ and $Q_y>0$. So $\mathbf{F}$ is not divergence free and is not the curl of a vector potential.)