From the general mass-conservation equation, show that the velocity field $\mathbf{u}(\mathbf{x})$ of an incompressible fluid is solenoidal, i.e. that $\nabla \cdot \mathbf{u}=0$.

Verify that the two-dimensional flow

$$\mathbf{u}=\left(\frac{y}{x^{2}+y^{2}}, \frac{-x}{x^{2}+y^{2}}\right)$$

is solenoidal and find a streamfunction $\psi(x, y)$ such that $\mathbf{u}=(\partial \psi / \partial y,-\partial \psi / \partial x)$.