Invertible field transformations with derivatives: necessary and sufficient conditions

Understand

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.