For a source flow ;

$$\overrightarrow{V}=V_{r}\overrightarrow{e}_{r}=\frac{\Lambda }{2\pi r}\overrightarrow{e_{r}}$$

In polar coordinates :

$$\nabla \cdot \overrightarrow{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( rV_{r} \right )+\frac{1}{r}\frac{\partial V_{\theta}}{\partial \theta} \\\Rightarrow\nabla \cdot \overrightarrow{V}=\frac{1}{r}\frac{\partial }{\partial r}\left [ r\;\frac{\Lambda }{2\pi r} \right ]+\frac{1}{r}\frac{\partial(0)}{\partial \theta} \\\Rightarrow\nabla \cdot \overrightarrow{V}=0+0=0$$

This shows that the flow is incompressible.For the flow to be irrotational

$$\nabla\times \overrightarrow{V}=0$$

$$\nabla \times \overrightarrow{V}=\frac{1}{r}\begin{vmatrix} \overrightarrow{e}_{r} &r\overrightarrow{e}_{\theta} & \overrightarrow{e}_{z}\\ \frac{\partial }{\partial r}& \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z}\\ V_{r}& rV_{\theta} & V_{z} \end{vmatrix}=\frac{1}{r}\begin{vmatrix} \overrightarrow{e}_{r} &r\overrightarrow{e}_{\theta} & \overrightarrow{e}_{z}\\ \frac{\partial }{\partial r}& \frac{\partial }{\partial \theta} & \frac{\partial }{\partial z}\\ \frac{\Lambda }{2\pi r}&0 & 0 \end{vmatrix}$$

$$\Rightarrow\nabla\times \overrightarrow{V}=-r\overrightarrow{e_{\theta}}\left ( \frac{\partial (0)}{\partial r} -\frac{\partial }{\partial z}\left ( \frac{ \Lambda }{2\pi r} \right )\right )+\overrightarrow{e}_{z}\left ( \frac{\partial (0)}{\partial r} -\frac{\partial }{\partial \theta}\left ( \frac{ \Lambda }{2\pi r} \right ) \right )$$

$$\Rightarrow\nabla\times \overrightarrow{V}=-r\overrightarrow{e_{\theta}}\left (0-0\right )+\overrightarrow{e}_{z}\left ( 0-0 \right )=0$$

This shows that it is irrotational everywhere.