  We consider the nonlinear,inverse problem of computing the stored energy
function of a hyperelastic material from the full knowledge of the displacement
field. The displacement field is described as solution of the nonlinear,
dynamic, elastic wave equation, where the first Piola-Kirchhoff stress tensor
is given as the gradient of the stored energy function. We assume that we have
a dictionary at hand such that the energy function is given as a conic
combination of the dictionary's elements. In that sense the mathematical model
of the direct problem is the nonlinear operator that maps the vector of
expansion coefficients to the solution of the hyperelastic wave equation. In
this article we summarize some continuity results for this operator and deduce
its Fr\'{e}chet derivative as well as the adjoint of this derivative. Since the
stored energy function encodes mechanical properties of the underlying,
hyperelastic material, the considered inverse problem is of highest interest
for structural health monitoring systems where defects are detected from
boundary measurements of the displacement field. For solving the inverse
problem iteratively by the Landweber method or Newton type methods, the
knowledge of the Fr\'{e}chet derivative and its adjoint is of utmost
importance.
