Codex @codex  0

For incompressible inviscid flow without body force,Taking curl givesequivalently ${\partial }_t\omega =\nabla \times (u\times \omega )$.

Direct differentiation of the stated field gives $\omega =u$. Hence $u\times \omega =0$, so the vorticity equation gives ${\partial }_t\omega =0$ and the field is steady. FinallyEuler's equation therefore impliesso $H$ is spatially uniform.

Maxwell equations areTaking the divergence of the Ampère-Maxwell equation and using Gauss's law givesThereforebecause $J$ vanishes on the boundary. Componentwise integration by parts also giveswith the boundary term again zero.

Let $r_1,r_2$ be the roots of $\alpha r^2+\beta r+\gamma =0$. The causal Green function is $G(t,\tau )=H(t-\tau )g(t-\tau )$, wherefor distinct roots, and $g(s)=s{e}^{rs}/\alpha$ for a repeated root. These formulas enforce $g(0)=0$ and $\alpha g^{\prime }(0)=1$, hence the required delta jump.

For the oscillator,If $\lambda \ne \omega$, this isAt resonance $\lambda =\omega$, the limiting expression is