  We discuss a field transformation from fields $\psi_a$ to other fields
$\phi_i$ that involves derivatives, $\phi_i = \bar \phi_i(\psi_a,
\partial_\alpha \psi_a, \ldots ;x^\mu)$, and derive conditions for this
transformation to be invertible, primarily focusing on the simplest case that
the transformation maps between a pair of two fields and involves up to their
first derivatives. General field transformation of this type changes number of
degrees of freedom, hence for the transformation to be invertible, it must
satisfy certain degeneracy conditions so that additional degrees of freedom do
not appear. Our derivation of necessary and sufficient conditions for
invertible transformation is based on the method of characteristics, which is
used to count the number of independent solutions of a given differential
equation. As applications of the invertibility conditions, we show some
non-trivial examples of the invertible field transformations with derivatives,
and also give a rigorous proof that a simple extension of the disformal
transformation involving a second derivative of the scalar field is not
invertible.
