Let a uniform flow with velocity =$V_{\infty}$.

$\overrightarrow{V}=V_{\infty}\hat{i},V_{\infty}=u=\mathrm{constant}$

$$\nabla \cdot \overrightarrow{V}=\frac{\partial u}{\partial x}+\frac{\partial v }{\partial y}+\frac{\partial w}{\partial z}=0+0+0=0$$

Therefore the flow is incompressible.

$$\nabla \times\overrightarrow{V} =\begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k}\\ \frac{\partial }{\partial x} &\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ u& v& w \end{vmatrix}$$

$$\Rightarrow\overrightarrow{i}\left ( 0-0 \right )-\overrightarrow{j}\left ( 0-0 \right )+\overrightarrow{k}\left ( 0-0 \right ) =0$$

Therefore

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

The flow is irrotational.