In fluid dynamics, the Euler equations govern inviscid fluid flow and provide quantitative statements on the conservation of mass, momentum, and energy. The continuity equation is given by

\[\frac{\partial ho}{\partial t}+abla \cdot(ho \mathbf{v})=0\]

where \(ho(x, y, z, t)\) is the mass density and \(\mathbf{v}(x, y, z, t)\) is the fluid velocity. The momentum equations are given by

\[\frac{\partial ho \mathbf{v}}{\partial t}+\mathbf{v} \cdot abla(ho \mathbf{v})=\mathbf{f}-abla p\]

Here, \(p(x, y, z, t)\) is the pressure and \(\mathbf{f}\) is the external force per volume.

a. Show that the continuity equation can be rewritten as

\[\frac{\partial ho}{\partial t}+ho abla \cdot(\mathbf{v})+\mathbf{v} \cdot abla ho=0\]

b. Prove the identity \(\frac{1}{2} abla v^{2}=\mathbf{v} \cdot abla \mathbf{v}\) for irrotational \(\mathbf{v}\).

c. Assume that

- the external forces are conservative \((\mathbf{f}=-ho abla \phi)\),

- the velocity field is irrotational \((abla \times \mathbf{v}=\mathbf{0})\),

- the fluid is incompressible ( \(ho=\) const), and

- the flow is steady, \(\frac{\partial \mathrm{v}}{\partial t}=0\).

Under these assumptions, prove Bernoulli's Principle:

\[\frac{1}{2} v^{2}+\phi+\frac{p}{ho}=\text { const. }\]