= Boundary integral for kinetic energy of planar potential flow
{title2=$T=-\frac\rho2\oint_{\partial B}\phi\,d\psi$}
For a smooth constant-density exterior <potential flow> satisfying the <finite-energy condition for planar exterior potential flow>, use $\mathbf u=\nabla\phi$ and the planar <streamfunction> convention $u=\psi_y$, $v=-\psi_x$. Since $\nabla^2\phi=0$, $\nabla\cdot(\phi\nabla\phi)=|\mathbf u|^2$. Apply the <divergence theorem> to a truncated exterior region. The outer term vanishes; on a counterclockwise body boundary the body's outward normal satisfies $\partial_n\phi\,ds=d\psi$, while the fluid's outward normal points the opposite way. This gives the displayed <kinetic energy> per unit span. The formula yields the <added mass of a circular cylinder> directly from its surface values of $\phi$ and $\psi$.
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