= Incompressible magnetic displacement energy identity
{title2=$\sigma^2\int\rho|\xi|^2=\int\xi_i^*\xi_j(\Pi_{,ij}+\rho\Phi_{,ij})+\mu_0^{-1}\int|(B\cdot\nabla)\xi|^2$}
For a static <ideal magnetohydrodynamics> equilibrium in fixed gravity, let $\nabla\cdot\xi=0$ and $\Pi=p+B^2/(2\mu_0)$. Frozen flux gives $b=(B\cdot\nabla)\xi-(\xi\cdot\nabla)B$ and continuity gives $\rho'=-\xi\cdot\nabla\rho$. Multiplying the linear momentum equation by the conjugate displacement and integrating by parts gives the displayed identity when surface terms vanish. The two magnetic cross terms cancel; the remaining magnetic equilibrium derivative combines with the <mass density>-gradient force to give the two <Hessian matrices>. Real symmetric <Hessian matrices> and the nonnegative <magnetic tension> term make $\sigma^2$ real.
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