= Thermostat X-ray transform
{title2=$I_F(h,\theta)(\gamma)=\int_\gamma(h(\gamma(t))+\theta_{\gamma(t)}(\dot\gamma(t)))\,dt$}
The transform integrates a base function and a velocity-evaluated one-form along the closed trajectories of a <surface thermostat>. Every <exact differential form> of degree one integrates to zero, because $F(f\circ\pi)=df(v)$. The relevant kernel theorem concerns these functions of velocity degree zero and one, rather than arbitrary functions on the <unit tangent bundle>.
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