a) Time rate of change of volume of a fluid element per unit volume is given as \[\frac{1}{\delta v}\frac{D\left ( \delta v \right )}{Dt}=\nabla \cdot \vec{V}\]In polar coordinates \[\nabla \cdot \vec{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( rV_{r} \right )+\frac{1}{r}\frac{\partial V_{\theta} }{\partial \theta}\]

On transforming by using \[x=rcos\theta
\\y=rsin\theta
\\V_{r}=ucos\theta+vsin\theta
\\V_{\theta}=-usin\theta+vcos\theta\]

Let for a source flow \[u=\frac{cx}{x^{2}+y^{2}}=\frac{crcos\theta}{r^{2}}=\frac{ccos\theta}{r}
\\v=\frac{cy}{x^{2}+y^{2}}=\frac{crsin\theta}{r^{2}}=\frac{csin\theta}{r}\]

Therefore \[V_{r}=\frac{c}{r}cos^{2}\theta+\frac{c}{r}sin^{2}\theta=\frac{c}{r}
\\V_{\theta}=\frac{-c}{r}cos\theta sin\theta+\frac{c}{r}cos\theta sin\theta=0\]

\[\nabla \cdot \vec{V}=\frac{1}{r}\frac{\partial }{\partial r}\left ( c \right )+\frac{1}{r}\frac{\partial (0)}{\partial \theta}=0\]

b) The vorticity

Vorticity is given by \[\nabla \times V=e_{z}\left [ \frac{\partial V_{\theta}}{\partial r} +\frac{V_{\theta}}{r}-\frac{1}{r}\frac{\partial V_{r}}{\partial \theta}\right ]\]

Therefore

\[\nabla \times V=e_{z}\left [ 0+0-0 \right ]
\\=0\]

The flow field is irrotational.