  We formulate explicitly the necessary and sufficient conditions for the local
invertibility of a field transformation involving derivative terms. Our
approach is to apply the method of characteristics of differential equations,
by treating such a transformation as differential equations that give new
variables in terms of original ones. The obtained results generalise the
well-known and widely used inverse function theorem. Taking into account that
field transformations are ubiquitous in modern physics and mathematics, our
criteria for invertibility will find many useful applications.
