For a uniform field, $\mathbf v\mathbin{\cdot}\nabla\mathbf p=0$, and the stated flow has $\boldsymbol\Omega=0$ and
$$
\mathbf D\mathbin{\cdot}\mathbf p=\gamma(p_x,-p_y/2,-p_z/2).
$$
The equation $D\mathbf p/Dt=-\Gamma\mathbf h$ becomes
$$
\partial_t\mathbf p=-\Gamma\left[\mathbf h-\frac{\xi\gamma}{\Gamma}(p_x,-p_y/2,-p_z/2)\right].
$$
Thus
$$
\boxed{\widetilde{\mathbf h}=\mathbf h+\alpha(p_x,-p_y/2,-p_z/2)},
\qquad
\boxed{\alpha=-\frac{\xi\gamma}{\Gamma}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 3 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-344/3/c/i/solution