(1) Show that \[ \nabla \times (\nabla^2 \vec{u}) = \nabla^2 (\nabla \times \vec{u}) \]

(2) If \(\vec{u}\) is irrotational, express the surface integral \[ \iint_S \vec{u} \times \nabla f \cdot \vec{n} \, dS \] as a line integral.

(3) Consider a fluid with density \(\rho(\vec{r}, t)\) flowing with velocity \(\vec{u}(\vec{r}, t)\). Let \(V\) be an arbitrary volume fixed in space, with surface \(S\) and outward normal \(\vec{n}\)

The total mass of the fluid contained in \(V\) is \(\quad\) The rate of mass enters \(V\) is
The law for conservation of mass of a fluid is
If the density of the fluid is independent of time and space, then the equation simplifies to