What does it mean for a vector field $\mathbf{F}$ to be irrotational ?

The field $\mathbf{F}$ is irrotational and $\mathbf{x}_{0}$ is a given point. Write down a scalar potential $V(\mathbf{x})$ with $\mathbf{F}=-\nabla V$ and $V\left(\mathbf{x}_{0}\right)=0$. Show that this potential is well defined.

For what value of $m$ is the field $\frac{\cos \theta \cos \phi}{r} \mathbf{e}_{\theta}+\frac{m \sin \phi}{r} \mathbf{e}_{\phi}$ irrotational, where $(r, \theta, \phi)$ are spherical polar coordinates? What is the corresponding potential $V(\mathbf{x})$ when $\mathbf{x}_{0}$ is the point $r=1, \theta=0$ ?

$$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta} \mid \begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right] .$$