  Let $\Omega_\epsilon$ be a metallic plate whose top inaccessible surface has
been damaged by some chemical or mechanical agent. We heat the opposite side
and collect a sequence of temperature maps $u^\epsilon$. Here, we construct a
formal explicit approximation of the damage $\epsilon\theta$ by solving a
nonlinear inverse problem for the heat equation in three steps: (i) smoothing
of temperature maps, (ii) domain derivative of the temperature, (iii) thin
plate approximation of the model and perturbation theory.
  Our inversion formula is tested with realistic synthetic data and used in a
real laboratory experiment.
