(a) For a velocity field $\mathbf{u}$, show that $\mathbf{u} \cdot \boldsymbol{\nabla} \mathbf{u}=\boldsymbol{\nabla}\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \times \boldsymbol{\omega}$, where $\boldsymbol{\omega}$ is the flow vorticity.

(b) For a scalar field $H(\mathbf{r})$, show that if $\mathbf{u} \cdot \nabla H=0$, then $H$ is constant along the flow streamlines.

(c) State the Euler equations satisfied by an inviscid fluid of constant density subject to conservative body forces.

(i) If the flow is irrotational, show that an exact first integral of the Euler equations may be obtained.

(ii) If the flow is not irrotational, show that although an exact first integral of the Euler equations may not be obtained, a similar quantity is constant along the flow streamlines provided the flow is steady.

(iii) If the flow is now in a frame rotating with steady angular velocity $\Omega \mathbf{e}_{z}$, establish that a similar quantity is constant along the flow streamlines with an extra term due to the centrifugal force when the flow is steady.